{"id":41787,"date":"2023-07-06T16:01:56","date_gmt":"2023-07-06T10:31:56","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=41787"},"modified":"2025-11-19T19:04:33","modified_gmt":"2025-11-19T13:34:33","slug":"determinants-and-matrices-unit-i","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=41787","title":{"rendered":"DETERMINANTS AND MATRICES (UNIT &#8211; I)"},"content":{"rendered":"\n<p>SYLLABUS<\/p>\n\n\n\n<p>Definition and expansion of second and third order determinants \u2013 Solution of simultaneous equations using Cramer\u2019s rule for 2 and 3 unknowns \u2013 Types of matrices &#8211; Algebra of matrices \u2013 Equality, addition, subtraction, scalar multiplication and multiplication of matrices\u2013 Cofactor matrix \u2013 Adjoint matrix \u2013 Singular and non-singular matrices \u2013 Inverse of a matrix \u2013 Simple problems \u2013 Engineering applications of Determinants and Matrices.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} { DETERMINANTS}\\ \\hspace{20cm}\\] <script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {Definition\\ of\\ Determinant}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ system\\ of\\ Linear\\ equations\\ like\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a_1x + b_1y = c_1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a_2x + b_2y = c_2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p>Determinant is a square arrangement of numbers within two vertical lines.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {Eg:}\\ \\Delta =\\begin{vmatrix}\na_1 &amp; b_1 \\\\\na_2 &amp; b_2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {Order\\ and\\ Value\\ of\\ the\\ Determinant}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {Determinant\\ of\\ Second\\ order:}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A =\\begin{vmatrix}\na_1 &amp; b_1 \\\\\na_2 &amp; b_2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<p>                      consisting of two rows and two columns is called&nbsp; a determinant of second order.<\/p>\n\n\n\n<p>  Value of the Determinant&nbsp; is<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =\\begin{vmatrix}\na_1 &amp; b_1 \\\\\na_2 &amp; b_2 \\\\\n\\end{vmatrix}\\ = a_1b_2\\ -\\ a_2b_1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 1:}\\ \\color {red}{Find\\ the\\ determinant\\ of}\\ A =\\begin{vmatrix}\n2 &amp; -5 \\\\\n1 &amp; 3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ A = \\begin{vmatrix}\n2 &amp; -5 \\\\\n1 &amp; 3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ = 2(3) &#8211; (-5)(1) \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= 6 + 5 \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta= 11 \\hspace{12cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/g8cu8pE_9bU?list=PLQIom4Rz29vxcUyYeKo2Sw4cmEJ0RbBA2\" title=\"Determinants and Matrices - Part - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 2:}\\ \\color {red}{Find\\ the\\ value\\ of\\ x\\ if}\\ \\begin{vmatrix}\n2 &amp; 3 \\\\\n4 &amp; x \\\\\n\\end{vmatrix}\\ = 0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\begin{vmatrix}\n2 &amp; 3 \\\\\n4 &amp; x \\\\\n\\end{vmatrix}\\ = 0 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2(x)\\ -\\ 4 (3) = 0\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2x\\ -\\ 12\\ =\\ 0\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2x\\  =\\  12\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x\\  =\\ 6\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 3:}\\ \\color {red}{Find\\ the\\ value\\ of\\ x\\ if}\\ \\begin{vmatrix}\n3 &amp; 3 \\\\\n4 &amp; x \\\\\n\\end{vmatrix}\\ = 0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  April\\ 2025\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\begin{vmatrix}\n3 &amp; 3 \\\\\n4 &amp; x \\\\\n\\end{vmatrix}\\ = 0 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3(x)\\ -\\ 3 (3) = 0\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3x\\ -\\ 9\\ =\\ 0\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3x\\  =\\  9\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x\\  =\\ 3\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 4:}\\ \\color {red}{Find\\ the\\ value\\ of\\ x\\ if}\\ \\begin{vmatrix}\nx &amp; 1 \\\\\n4 &amp; x \\\\\n\\end{vmatrix}\\ = 0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\begin{vmatrix}\nx &amp; 1 \\\\\n4 &amp; x \\\\\n\\end{vmatrix}\\ = 0 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x(x) &#8211; 4 (1) = 0\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2 &#8211; 4 = 0\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x^2  = 4\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x  = \\pm\\sqrt{4}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x  = \\pm 2\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{\\boxed{x\\ =\\ 2\\ or\\ x\\ =\\ -\\ 2}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {Determinant\\ of\\ Third\\ order:}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ expression\\ \\begin{vmatrix}\na_1 &amp; a_2 &amp; a_3 \\\\\nb_1 &amp; b_2 &amp; b_3 \\\\\nc_1 &amp; c_2 &amp; c_3 \\\\\n\\end{vmatrix}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<p>                       consisting of three rows and three columns is called&nbsp; a determinant of third order.<\/p>\n\n\n\n<p>Value of the Determinant&nbsp; is :<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =a_1\\begin{vmatrix}\nb_2 &amp; b_3 \\\\\nc_2 &amp; c_3 \\\\\n\\end{vmatrix}\\ -\\ a_2\\begin{vmatrix}\nb_1 &amp; b_3 \\\\\nc_1 &amp; c_3 \\\\\n\\end{vmatrix}\\ +\\ a_3\\begin{vmatrix}\nb_1 &amp; b_2 \\\\\nc_1 &amp; c_2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =a_1(b_2c_3\\ -\\ b_3c_2)\\ &#8211; a_2 (b_1c_3\\ -\\ b_3c_1) + a_3(b_1c_2\\ -\\ b_2c_1)\\ \n\\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 5:}\\ \\color {red}{Find\\ the\\ determinant\\ of}\\ A =\\begin{bmatrix}\n3 &amp; 1 &amp; -1 \\\\\n2 &amp; -1 &amp; 2 \\\\\n2 &amp; 1 &amp; -2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ A =\\begin{bmatrix}\n3 &amp; 1 &amp; -1 \\\\\n2 &amp; -1 &amp; 2 \\\\\n2 &amp; 1 &amp; -2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =3\\begin{vmatrix}\n-1 &amp; 2 \\\\\n1 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 2 \\\\\n2 &amp; -2 \\\\\n\\end{vmatrix}\\ +\\ -1\\begin{vmatrix}\n2 &amp; -1 \\\\\n2 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{4cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =3(2\\ -\\ 2)\\ &#8211; 1 (-4\\ -\\ 4) &#8211; 1(2\\ +\\ 2)\\ \n\\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =3(0)\\ &#8211; 1 (-8) &#8211; 1(4)\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =0\\ +8 &#8211; 4\\ \n\\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =4\\ \n\\hspace{16cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/OHusIAlwnBo?list=PLQIom4Rz29vxcUyYeKo2Sw4cmEJ0RbBA2\" title=\"Determinants and Matrices - Part - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 6:}\\ \\color {red}{Find\\ the\\ value\\ of\\ m\\ if}\\ \\begin{vmatrix}\n3 &amp; 4 &amp; -2 \\\\\n-3 &amp; 6 &amp; 2 \\\\\n4 &amp; 1 &amp; m \\\\\n\\end{vmatrix}\\ =\\ 0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\begin{vmatrix}\n3 &amp; 4 &amp; -2 \\\\\n-3 &amp; 6 &amp; 2 \\\\\n4 &amp; 1 &amp; m \\\\\n\\end{vmatrix}\\ =\\ 0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\begin{vmatrix}\n6 &amp; 2 \\\\\n1 &amp; m \\\\\n\\end{vmatrix}\\ -\\ 4\\begin{vmatrix}\n-3 &amp; 2 \\\\\n4 &amp; m \\\\\n\\end{vmatrix}\\  -2\\begin{vmatrix}\n-3 &amp; 6 \\\\\n4 &amp; 1 \\\\\n\\end{vmatrix}\\ =\\ 0\\ \\hspace{4cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3(6m\\ -\\ 2)\\ -\\ 4 (-3m\\ -\\ 8)\\ -\\ 2(-3\\ -\\ 24)\\ =\\ 0\\\n\\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ 18m\\ -\\ 6\\ +\\ 12m\\ +\\ 32\\  &#8211; 2(-27)\\ =\\ 0\\\n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ 30m\\ +\\ 26\\  +\\  54\\ =\\ 0\\\n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ 30m\\ +\\ 80\\ =\\ 0\\\n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ 30m\\  =\\ -\\ 80\\\n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ \\boxed{m\\  =\\ \\frac{-8}{3}}\\\n\\hspace{14cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/dedPf-CdfQg?list=PLQIom4Rz29vxcUyYeKo2Sw4cmEJ0RbBA2\" title=\"Determinants and Matrices - Part - 5\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {SOLUTION\\ OF\\ SIMULTANEOUS\\ EQUATIONS\\ USING\\ CRAMERS\\ RULE}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a_1x\\ + b_1y\\ +\\ c_1z\\ = d_1\\ &#8212;&#8211; (1)\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a_2x\\ + b_2y\\ +\\ c_2z\\ = d_2\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a_3x\\ + b_3y\\ +\\ c_3z\\ = d_3\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ to\\ find\\ x,\\ y,\\ z\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Step\\ 1:\\ \\Delta = \\begin{vmatrix}\na_1 &amp; b_1 &amp; c_1 \\\\\na_2 &amp; b_2 &amp; c_2 \\\\\na_3 &amp; b_3 &amp; c_3 \\\\\n\\end{vmatrix}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Step\\ 2:\\ \\Delta_x = \\begin{vmatrix}\nd_1 &amp; b_1 &amp; c_1 \\\\\nd_2 &amp; b_2 &amp; c_2 \\\\\nd_3 &amp; b_3 &amp; c_3 \\\\\n\\end{vmatrix}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Step\\ 3:\\ \\Delta_y= \\begin{vmatrix}\na_1 &amp; d_1 &amp; c_1 \\\\\na_2 &amp;  d_2 &amp;  c_2 \\\\\na_3 &amp; d_3 &amp; c_3 \\\\\n\\end{vmatrix}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Step\\ 4:\\ \\Delta_z = \\begin{vmatrix}\na_1 &amp; b_1 &amp; d_1 \\\\\na_2 &amp; b_2 &amp; d_2 \\\\\na_3 &amp; b_3 &amp; d_3 \\\\\n\\end{vmatrix}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Solution\\ is\\ x=\\ \\frac{\\Delta_x}{\\Delta}.\\ y=\\ \\frac{\\Delta_y}{\\Delta},\\ z=\\ \\frac{\\Delta_z}{\\Delta}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{Example\\ 7:}\\ \\color{red}{Solve}\\ the\\ following\\ equations\\ x + y\\ +\\  z\\ =\\ 3,\\ 2x\\ -\\ y\\ +\\ z\\ =\\ 2\\ and\\ \\hspace{15cm}\\\\ 3\\ x\\ +\\ 2\\ y\\ -\\ 2\\ z\\ =\\ 3\\ \\color{red}{using\\ Cramers\\ Rule}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{22cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x + y\\ +\\  z\\ =\\ 3\\ &#8212;&#8212;&#8212;&#8212;&#8212; (1) \\hspace{7cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2x\\ -\\ y\\ +\\ z\\ =\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\ x\\ +\\ 2\\ y\\ -\\ 2\\ z\\ =\\ 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta = \\begin{vmatrix}\n1 &amp; 1 &amp; 1 \\\\\n2 &amp; &#8211; 1 &amp; 1 \\\\\n3 &amp; 2 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =1\\begin{vmatrix}\n-1 &amp; 1 \\\\\n2 &amp; &#8211; 2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 1 \\\\\n3 &amp; -2 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; -1\\\\\n3 &amp;  2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =\\ 1(2\\ -\\ 2)\\ &#8211; 1 (-4\\ -\\ 3)\\ +\\ 1(4\\ +\\ 3)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =1(0)\\ &#8211; 1\\  (-\\ 7)\\ +\\ 1(7)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =\\ 0\\ +\\ 7\\  +\\ 7\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta =\\ 14}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x = \\begin{vmatrix}\n3 &amp; 1 &amp; 1 \\\\\n2 &amp; -1 &amp; 1 \\\\\n3 &amp; 2 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =\\ 3\\begin{vmatrix}\n-1 &amp; 1 \\\\\n2 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 1 \\\\\n3 &amp;  -2 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; -1\\\\\n3 &amp; 2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 3(2\\ -\\ 2)\\ -\\ 1 (-4\\ -\\ 3)\\ + \\ 1(4\\ +\\ 3)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =\\ -3(0)\\ &#8211; 1 (-7)\\  +\\ 1 (7)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 0\\ +\\ 7\\ +\\ 7\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_x\\ =\\ 14}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y = \\begin{vmatrix}\n1 &amp; 3 &amp; 1 \\\\\n2 &amp; 2 &amp; 1 \\\\\n3 &amp; 3 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y =1\\begin{vmatrix}\n2 &amp; 1 \\\\\n3 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 3\\begin{vmatrix}\n2 &amp; 1 \\\\\n3 &amp; -2 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; 2\\\\\n3 &amp;  3 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y =1(-\\ 4\\ -\\ 3)\\ -\\ 3\\ (-\\ 4\\ -\\ 3)\\ +\\ 1(6\\ -\\ 6)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y =1(-7)\\ -\\ 3\\ (-\\ 7)\\  +\\ 1(0)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ -\\ 7\\ +\\ 21\\ +\\ 0\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_y\\ =\\ 14}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z = \\begin{vmatrix}\n1 &amp; 1 &amp; 3 \\\\\n2 &amp; -1 &amp; 2 \\\\\n3 &amp; 2 &amp; 3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z =1\\begin{vmatrix}\n-1 &amp; 2 \\\\\n2 &amp; 3 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 2 \\\\\n3 &amp; 3 \\\\\n\\end{vmatrix}\\ +\\ 3\\begin{vmatrix}\n2 &amp; -1\\\\\n3 &amp;  2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 1(-\\ 3\\ -\\ 4)\\ -\\ 1(6\\ -\\ 6)\\  +\\ 3(4\\ +\\ 3)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 1(-\\ 7)\\ -\\ 1 (0)\\ +\\ 3(7)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z =\\ -\\ 7\\ -\\ 0\\ +\\ 21\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_z\\ =\\ 14}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ Solution\\ is\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x=\\ \\frac{\\Delta_x}{\\Delta} =\\ \\frac{14}{14} =\\ 1\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y=\\ \\frac{\\Delta_y}{\\Delta} =\\ \\frac{14}{14} =\\ 1\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[z=\\ \\frac{\\Delta_z}{\\Delta} =\\ \\frac{14}{14}\\ =\\ 1\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[For\\ cross\\ verification\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Put\\ x\\ =\\ 1\\ y\\ =\\ 1\\ and\\ z\\ =\\ 1\\ in\\ equation (1)\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[LHS\\ =\\ 1 +\\ 1\\ +\\ 1\\]\\[ =\\ 3\\]\\[ = RHS\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {violet}{Example\\ 8:}\\ \\color {red} {Solve\\ the\\ following\\ equations\\ using\\ Cramers\\ Rule}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4\\ x\\ +\\ y\\ +\\  z\\ =\\ 6,\\ 2\\ x\\ -\\ y\\ -\\  2\\ z\\ =\\ -\\ 6\\ and\\  x\\ +\\ y\\ +\\ z\\ =\\  3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  October\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{22cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4\\ x\\ +\\ y\\ +\\  z\\ =\\ 6\\ &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;-(1)\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ x\\ -\\ y\\ -\\  2\\ z\\ =\\ -\\ 6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x\\ +\\ y\\ +\\ z\\ =\\  3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta = \\begin{vmatrix}\n4 &amp; 1 &amp; 1 \\\\\n2 &amp; -1 &amp; -2\\\\\n1 &amp; 1 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =4\\begin{vmatrix}\n-1 &amp; -2 \\\\\n1 &amp; 1 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; -2 \\\\\n1 &amp; 1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; -1\\\\\n1 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =4(-1\\ +\\ 2)\\ &#8211; 1 (2\\ +\\  2)\\ +\\ 1(2\\ +\\ 1)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta\\ =\\ 4(1)\\ &#8211; 1 (4)\\ +\\ 1(7)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =4\\ -\\ 4\\ +\\  3\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta\\ =\\ 3}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x = \\begin{vmatrix}\n6 &amp; 1 &amp; 1 \\\\\n-6 &amp; -1 &amp; -2 \\\\\n3 &amp; 1 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =6\\begin{vmatrix}\n-1 &amp; -2 \\\\\n1 &amp; 1 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n-6 &amp; -2 \\\\\n3 &amp; 1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n-6 &amp; -1\\\\\n3 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 6(-1\\ +\\ 2) &#8211; 1 (\\ -6\\ +\\ 6)\\ +\\ 1(-6\\ +\\ 3)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 6(1)\\ &#8211; 1 (0)\\ +\\ 1(-3)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x = 6\\ +\\ 0\\ -3\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_x\\ =\\ 3}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y = \\begin{vmatrix}\n4 &amp; 6 &amp; 1 \\\\\n2 &amp; -6 &amp; -2 \\\\\n1 &amp; 3 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ 4\\begin{vmatrix}\n-6 &amp;  -2 \\\\\n3 &amp; 1\\\\\n\\end{vmatrix}\\ -\\ 6\\begin{vmatrix}\n2 &amp; -2 \\\\\n1 &amp; 1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; -6\\\\\n1 &amp;  3 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ 4(-\\ 6\\ +\\ 6)\\ -\\ 6 (2\\ +\\ 2)\\ +\\ 1(6\\ +\\ 6)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ 4(0)\\ -\\ 6 (4)\\ +\\ 1(12)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ -\\ 0\\ -\\ 24\\ +\\ 12\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_y\\ =\\ -12}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z = \\begin{vmatrix}\n4 &amp; 1 &amp; 6 \\\\\n2 &amp; -1 &amp; -6 \\\\\n1 &amp; 1 &amp; 3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 4\\begin{vmatrix}\n-1 &amp; -6 \\\\\n1 &amp; 3 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; -6 \\\\\n1 &amp; 3 \\\\\n\\end{vmatrix}\\ +\\ 6\\begin{vmatrix}\n2 &amp; &#8211; 1\\\\\n1 &amp;  1 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 4(-\\ 3\\ +\\ 6)\\ -\\ 1 (6\\ +\\ 6)\\ +\\ 6(2\\ +\\ 1)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 4(3)\\ -\\ 1 (12)\\ +\\ 6(3)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 12\\  -\\  12\\ +\\  18\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_z\\ =\\ 18}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ Solution\\ is\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x=\\ \\frac{\\Delta_x}{\\Delta} =\\ \\frac{3}{3} =\\ 1\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y=\\ \\frac{\\Delta_y}{\\Delta} =\\ \\frac{-12}{3} =\\ -4\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[z=\\ \\frac{\\Delta_z}{\\Delta} =\\ \\frac{18}{3} =\\ 6\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[For\\ cross\\ verification\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Put\\ x\\ =\\ 1,\\ y\\ =\\ -4\\ and\\ z = 6\\ in\\ equation (1)\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[LHS\\ =\\ 4(1) &#8211; 4 + 6\\]\\[ = 4 &#8211; 4 + 6 = 6\\]\\[ = RHS\\] <\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/I8vKziN5qco?list=PLQIom4Rz29vxcUyYeKo2Sw4cmEJ0RbBA2\" title=\"Determinants and Matrices - Part - 7\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {violet}{Example\\ 9:}\\ \\color {red} {Solve\\ the\\ following\\ equations\\ using\\ Cramers\\ Rule}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\ x\\ +\\ y\\ -\\  z\\ =\\ 2,\\ 2\\ x\\ -\\ y\\ +\\  2\\ z\\ =\\  6\\ and\\  2\\ x\\ +\\ y\\ -\\ 2\\ z\\ =\\  -\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  October\\ 2023\\  April\\ 2025\\ June\\ 2025(Supp)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{22cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\ x\\ +\\ y\\ -\\  z\\ =\\ 2\\ &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;-(1)\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ x\\ -\\ y\\ +\\  2\\ z\\ =\\  6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ x\\ +\\ y\\ -\\ 2\\ z\\ =\\  -\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta = \\begin{vmatrix}\n3 &amp; 1 &amp; -1 \\\\\n2 &amp; -1 &amp; 2\\\\\n2 &amp; 1 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =\\ 3\\begin{vmatrix}\n-1 &amp; 2 \\\\\n1 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 2 \\\\\n2 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; -1\\\\\n2 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =\\ 3(2\\ -\\ 2)\\ &#8211; 1 (-\\ 4\\ -\\  4)\\ -\\ 1(2\\ +\\ 2)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta\\ =\\ 3(0)\\ &#8211; 1 (-8)\\ -\\ 1(4)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =\\ 0\\ +\\ 8\\ -\\  4\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta\\ =\\ 4}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x = \\begin{vmatrix}\n2 &amp; 1 &amp; -1 \\\\\n6 &amp; -1 &amp; 2 \\\\\n-2 &amp; 1 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 2\\begin{vmatrix}\n-1 &amp; 2 \\\\\n1 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n6 &amp; 2 \\\\\n-2 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n6 &amp; -1\\\\\n-2 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 2(2\\ -\\ 2) &#8211; 1 (-12\\ +\\ 4)\\ -\\ 1(6\\ -\\ 2)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 2(0)\\ &#8211; 1 (-8)\\ -\\ 1(4)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =\\ 0\\ +\\ 8\\ -\\ 4\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_x\\ =\\ 4}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y = \\begin{vmatrix}\n3 &amp; 2 &amp; -1 \\\\\n2 &amp; 6 &amp; 2 \\\\\n2 &amp; -2 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ 3\\begin{vmatrix}\n6 &amp;  2 \\\\\n-2 &amp; -2\\\\\n\\end{vmatrix}\\ -\\ 2\\begin{vmatrix}\n2 &amp; 2 \\\\  2 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 6\\\\\n2 &amp;  -2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ 3(-\\ 12\\ +\\ 4)\\ -\\ 2 (-4\\ -\\ 4)\\ -\\ 1(-4\\ -\\ 12)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ 3(-8)\\ -\\ 2 (-8)\\ -\\ 1(-16)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ -\\ 24\\ +\\ 16\\ +\\ 16\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_y\\ =\\ 8}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z = \\begin{vmatrix}\n3 &amp; 1 &amp; 2 \\\\\n2 &amp; -1 &amp; 6 \\\\\n2 &amp; 1 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 3\\begin{vmatrix}\n-1 &amp; 6 \\\\\n1 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 6 \\\\\n2 &amp; -2 \\\\\n\\end{vmatrix}\\ +\\ 2\\begin{vmatrix}\n2 &amp; &#8211; 1\\\\\n2 &amp;  1 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 3(2\\ -\\ 6)\\ -\\ 1 (-\\ 4\\ -\\ 12)\\ +\\ 2(2\\ +\\ 2)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 3(-4)\\ -\\ 1 (-16)\\ +\\ 2(4)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ -\\ 12\\  +\\  16\\ +\\  8\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_z\\ =\\ 12}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ Solution\\ is\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x=\\ \\frac{\\Delta_x}{\\Delta} =\\ \\frac{4}{4} =\\ 1\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y=\\ \\frac{\\Delta_y}{\\Delta} =\\ \\frac{8}{4} =\\  2\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[z=\\ \\frac{\\Delta_z}{\\Delta} =\\ \\frac{12}{4} =\\ 3\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[For\\ cross\\ verification\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Put\\ x\\ =\\ 1,\\ y\\ =\\ 2\\ and\\ z = 3\\ in\\ equation (1)\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[LHS\\ =\\ 3(1)\\ +\\ 2 -\\ 3\\]\\[ =\\ 3\\ +\\ 2\\ -\\ 3=\\ 2\\]\\[ = RHS\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Definition\\ of\\ a\\ Matrix}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<p>A matrix is a rectangular array of numbers arranged in rows and columns enclosed by brackets.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Ex:}\\ 1)\\ A =\\begin{bmatrix}\n3 &amp; 6 \\\\\n1 &amp; 2\\\\\n\\end{bmatrix}\\ \\hspace{2cm}\\ 2)\\ B =\\begin{bmatrix}\n1 &amp; -1 &amp; 2\\\\\n2 &amp; -2 &amp; 4\\\\\n3 &amp; -3 &amp; 6\\\\\n\\end{bmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Order\\ of\\ a\\ Matrix}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<p>If there are&nbsp; m rows and n columns in a matrix, then the order of&nbsp; the matrix is&nbsp; m \u00d7 n.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Ex:}\\ A =\\begin{bmatrix}\n1 &amp; 2 &amp; -1 &amp; 3\\\\\n2 &amp; 4 &amp; -4 &amp; 7\\\\\n-1 &amp; -2 &amp; -2 &amp; -2\\\\\n\\end{bmatrix}\\ is\\ a\\ matrix\\ of\\ order\\ 3\u00d74 \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {Square\\ Matrix}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<p>A matrix which has equal number of rows and columns is called a square matrix.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {Eg:}\\ 1)\\ A =\\begin{bmatrix}\n3 &amp; 6 \\\\\n1 &amp; 4 \\\\\n\\end{bmatrix}\\ is\\ a\\ square\\ matrix\\ of\\ order\\ 2\u00d72 \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {Eg:}\\ 2)\\ B =\\begin{bmatrix}\n1 &amp; -1 &amp; 2 \\\\\n2 &amp; -2 &amp; 4 \\\\\n3 &amp; -3 &amp; 6 \\\\\n\\end{bmatrix}\\ is\\ a\\ square\\ matrix\\ of\\ order\\ 3\u00d73 \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {Transpose\\ of\\ Matrix}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<p>Let A be a square matrix. The transpose of A is obtained by changing rows into columns and vise-versa and is denoted by&nbsp; A<sup>T<\/sup><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {Eg:}\\ 1)\\ A =\\begin{bmatrix}\n3 &amp;  6 \\\\\n1 &amp; 4 \\\\\n\\end{bmatrix}\\  \\hspace{10cm}\\]\\[A^T =\\begin{bmatrix}\n3 &amp;  1 \\\\\n6 &amp; 4 \\\\\n\\end{bmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {Eg:}\\ 2)\\ A =\\begin{bmatrix}\n3 &amp; 4 &amp; 1 \\\\\n0 &amp; -1 &amp; 2 \\\\\n5 &amp; -2 &amp;  6 \\\\\n\\end{bmatrix}\\  \\hspace{10cm}\\]\\[A^T =\\begin{bmatrix}\n3 &amp; 0 &amp; 5 \\\\\n4 &amp; -1 &amp; -2 \\\\\n1 &amp; 2 &amp;  6 \\\\\n\\end{bmatrix}\\  \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {6.\\ Unit\\ Matrix}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<p>Unit matrix is a square matrix in which the diagonal elements are all ones and all the other elements are zeros.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {Eg:}\\ 1)\\ I =\\begin{bmatrix}\n1 &amp; 0 \\\\\n0 &amp; 1 \\\\\n\\end{bmatrix}\\ is\\ a\\ unit\\ matrix\\ of\\ order\\ 2\u00d72 \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2)\\ I =\\begin{bmatrix}\n1 &amp; 0 &amp; 0 \\\\\n0 &amp; 1  &amp; 0 \\\\\n0 &amp; 0 &amp; 1 \\\\\n\\end{bmatrix}\\ is\\ a\\ unit\\ matrix\\ of\\ order\\ 3\u00d73 \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Operation\\ on\\ Matrices:}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<p>i) Addition and subtraction of matrices<br>ii) Multiplication of matrix by a scalar<br>iii) Multiplication of matrices<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {i)\\ Addition\\ and\\ Subtraction\\ of\\ Matrices:}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<p>Two Matrices can be added (or) subtracted if they have the same order. We can add (or) subtract two matrices by the corresponding element by element.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 10:}\\ \\color{red}{If}\\ A =\\begin{bmatrix}\n1 &amp; 2 &amp; 7 \\\\\n0 &amp; 4 &amp; 5 \\\\\n3 &amp; 1 &amp; 6 \\\\\n\\end{bmatrix}\\ ,\\ B =\\begin{bmatrix}\n1 &amp; 3 &amp; 1 \\\\\n2 &amp; 4 &amp; 0 \\\\\n1 &amp; 7 &amp; 5 \\\\\n\\end{bmatrix}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red}{Find\\ A + B}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ A + B=\\begin{bmatrix}\n1 + 1 &amp; 2 + 3 &amp; 7 + 1 \\\\\n0 + 2 &amp; 4 + 4 &amp; 5 + 0 \\\\\n3 + 1 &amp; 1 + 7 &amp; 6 + 5 \\\\\n\\end{bmatrix}\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A + B=\\begin{bmatrix}\n2 &amp; 5 &amp; 8 \\\\\n2 &amp; 8 &amp; 5 \\\\\n4 &amp; 8 &amp; 11 \\\\\n\\end{bmatrix}\\ \\hspace{7cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/CommcbgmLZg\" title=\"Addition of Matrices - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 11:}\\ \\color {red}{If}\\ A =\\begin{bmatrix}\n1 &amp; 3 &amp; 5 \\\\\n2 &amp; 0 &amp; 7 \\\\\n1 &amp; 5 &amp; 2 \\\\\n\\end{bmatrix}\\ ,\\ B =\\begin{bmatrix}\n7 &amp; 3 &amp; 4 \\\\\n1 &amp; -1 &amp; 5 \\\\\n0 &amp; 2 &amp; 4 \\\\\n\\end{bmatrix}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red}{Find\\ A &#8211; B}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ A &#8211; B=\\begin{bmatrix}\n1 &#8211; 7 &amp; 3 &#8211; 3 &amp; 5 &#8211; 4 \\\\\n2 &#8211; 1 &amp; 0 + 1 &amp; 7 &#8211; 5 \\\\\n1 &#8211; 0 &amp; 5 &#8211; 2 &amp; 2 &#8211; 4 \\\\\n\\end{bmatrix}\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A &#8211; B=\\begin{bmatrix}\n-6 &amp; 0 &amp; 1 \\\\\n1 &amp; 1&amp; 2 \\\\\n1 &amp; 3 &amp; -2 \\\\\n\\end{bmatrix}\\ \\hspace{7cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/Gp53R56K1uU\" title=\"Determinants and Matrices - Part - 9\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 12:}\\ \\color {red}{If}\\ A =\\ \\begin{pmatrix}\n4 &amp; 6 &amp; 2 \\\\\n0 &amp; 1 &amp; 5 \\\\\n0 &amp; 3 &amp; 2 \\\\\n\\end{pmatrix}\\ ,\\ B =\\begin{pmatrix}\n0 &amp; 1 &amp; &#8211; 1 \\\\\n3 &amp; &#8211; 1 &amp; 4 \\\\\n&#8211; 1 &amp; 2 &amp; 1 \\\\\n\\end{pmatrix}\\ \\hspace{7cm}\\]\\[\\color {red}{Prove\\ that\\ (A\\ +\\ B)^T\\ =\\ A^T\\ +\\ B^T}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  October\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Given\\  A =\\ \\begin{pmatrix}\n4 &amp; 6 &amp; 2 \\\\\n0 &amp; 1 &amp; 5 \\\\\n0 &amp; 3 &amp; 2 \\\\\n\\end{pmatrix}\\ ,\\ B =\\begin{pmatrix}\n0 &amp; 1 &amp; &#8211; 1 \\\\\n3 &amp; &#8211; 1 &amp; 4 \\\\\n&#8211; 1 &amp; 2 &amp; 1 \\\\\n\\end{pmatrix}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A + B=\\begin{pmatrix}\n4 + 0 &amp; 6 + 1 &amp; 2 &#8211; 1 \\\\\n0 + 3 &amp; 1 &#8211; 1 &amp; 5 + 4 \\\\\n0 &#8211; 1 &amp; 3 + 2 &amp; 2 + 1 \\\\\n\\end{pmatrix}\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A + B=\\ \\begin{pmatrix}\n4 &amp; 7 &amp; 1 \\\\\n3 &amp; 0 &amp; 9\\\\\n-1 &amp; 5  &amp; 3\\\\\n\\end{pmatrix}\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(A + B)^T\\ =\\ \\begin{pmatrix}\n4 &amp; 3 &amp; &#8211; 1 \\\\\n7 &amp; 0 &amp; 5 \\\\\n1 &amp; 9  &amp; 3\\\\\n\\end{pmatrix}\\ \\hspace{2cm}\\ &#8212;&#8212;&#8211; (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^T =\\begin{pmatrix}\n4 &amp; 0 &amp; 0 \\\\\n6 &amp; 1 &amp; 3 \\\\\n2 &amp; 5 &amp; 2 \\\\\n\\end{pmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B^T =\\begin{pmatrix}\n0 &amp; 3 &amp; &#8211; 1 \\\\\n1 &amp; &#8211; 1 &amp; 2 \\\\\n&#8211; 1 &amp; 4 &amp; 1 \\\\\n\\end{pmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^T\\ +\\ B^T\\ =\\begin{pmatrix}\n4 + 0 &amp; 0\\ +\\ 3 &amp;  0\\ -\\ 1\\\\\n6\\ +\\ 1 &amp; 1 &#8211; 1 &amp; 3 + 2 \\\\\n2 &#8211; 1 &amp; 5 + 4 &amp; 2 + 1 \\\\\n\\end{pmatrix}\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^T\\ +\\ B^T\\ =\\begin{pmatrix}\n4 &amp; 3 &amp;  &#8211; 1\\\\\n7 &amp; 0 &amp; 5 \\\\\n1 &amp; 9 &amp; 3 \\\\\n\\end{pmatrix}\\ \\hspace{2cm}\\ &#8212;&#8212;&#8212;- (2)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[From\\ (1)\\ and\\ (2), It\\ is\\ concluded\\ that  (A\\ +\\ B)^T\\ =\\ A^T\\ +\\ B^T\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {ii)\\ Matrix\\ Multiplication\\ by\\ a\\ scalar}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<p>We can multiply the matrix by any non-zero scalar [(value) number] obtain we get the matrix whose all the elements are multiplied by that same scalar.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 13:}\\ \\color {red}{If}\\ A =\\begin{pmatrix}\n5 &amp; &#8211; 6 \\\\\n3 &amp; 7\\\\\n\\end{pmatrix},  \\color {red} {Find\\ 3A}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\ A =\\begin{pmatrix}\n5 &amp; &#8211; 6 \\\\\n3 &amp; 7\\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ 3A = 3\\begin{pmatrix}\n5 &amp; &#8211; 6 \\\\\n3 &amp; 7\\\\\n\\end{pmatrix}\\  \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3A = \\begin{pmatrix}\n15 &amp; &#8211; 18 \\\\\n9 &amp; 21\\\\\n\\end{pmatrix}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 14:}\\ \\color {red}{If}\\ A =\\begin{bmatrix}\n9 &amp; 10 \\\\\n13 &amp; 20\\\\\n\\end{bmatrix}\\  and\\ B =\\begin{bmatrix}\n3 &amp; 5 \\\\\n8 &amp; 9\\\\\n\\end{bmatrix},\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red}{\\ find}\\ 2A\\ +\\ B\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  October\\ 2023,\\ June\\ 2025(Supp)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ 2A = 2\\begin{bmatrix}\n9 &amp; 10 \\\\\n13 &amp; 20\\\\\n\\end{bmatrix}\\  \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2A = \\begin{bmatrix}\n18 &amp; 20 \\\\\n26 &amp; 40\\\\\n\\end{bmatrix}\\ &#8212;&#8212;&#8212;- (1)\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B = \\begin{bmatrix}\n3 &amp; 5 \\\\\n8 &amp; 9\\\\\n\\end{bmatrix}\\ &#8212;&#8212;&#8212;- (2)\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2A\\ +\\ B = \\begin{bmatrix}\n18 &amp; 20 \\\\\n26 &amp; 40\\\\\n\\end{bmatrix}\\ +\\ \\begin{bmatrix}\n3 &amp; 5 \\\\\n8 &amp; 9\\\\\n\\end{bmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\begin{bmatrix}\n18\\ +\\ 3 &amp; 20\\ +\\ 5 \\\\\n26\\ +\\ 8 &amp; 40\\  +\\ 9\\\\\n\\end{bmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2A\\ +\\ B = \\begin{bmatrix}\n21 &amp; 25 \\\\\n34 &amp; 49\\\\\n\\end{bmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 15:}\\ \\color {red}{If}\\ A =\\begin{pmatrix}\n1 &amp; 2 \\\\\n3 &amp; 5\\\\\n\\end{pmatrix}\\  and\\ B =\\begin{pmatrix}\n&#8211; 5 &amp; 7 \\\\\n0 &amp; 4\\\\\n\\end{pmatrix},\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red}{\\ find}\\ 2A\\ +\\ B\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  October\\ 2024\\ October\\ 25\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ 2A = 2\\begin{pmatrix}\n1 &amp; 2 \\\\\n3 &amp; 5\\\\\n\\end{pmatrix}\\  \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2A = \\begin{pmatrix}\n2 &amp; 4 \\\\\n6 &amp; 10\\\\\n\\end{pmatrix}\\ &#8212;&#8212;&#8212;- (1)\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B = \\begin{pmatrix}\n-5 &amp; 7 \\\\\n0 &amp; 4\\\\\n\\end{pmatrix}\\ &#8212;&#8212;&#8212;- (2)\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2A\\ +\\ B = \\begin{pmatrix}\n2 &amp; 4 \\\\\n6 &amp; 10\\\\\n\\end{pmatrix}\\ +\\ \\begin{pmatrix}\n-5 &amp; 7 \\\\\n0 &amp; 4\\\\\n\\end{pmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\begin{pmatrix}\n2\\ -\\ 5 &amp; 4\\ +\\ 7 \\\\\n6\\ +\\ 0 &amp; 10\\  +\\ 4\\\\\n\\end{pmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2A\\ +\\ B = \\begin{pmatrix}\n-3 &amp; 11 \\\\\n6 &amp; 14\\\\\n\\end{pmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 16:}\\ \\color {red}{If}\\ A =\\begin{pmatrix}\n5 &amp; 2 \\\\\n16 &amp; 15\\\\\n\\end{pmatrix}\\  and\\ B =\\begin{pmatrix}\n5 &amp; 2 \\\\\n4 &amp; 6\\\\\n\\end{pmatrix},\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red}{\\ find}\\ 3A\\ +\\ 2B\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  April\\ 2025\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ 3A = 3\\begin{pmatrix}\n5 &amp; 2 \\\\\n16 &amp; 15\\\\\n\\end{pmatrix}\\  \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3A = \\begin{pmatrix}\n15 &amp; 6 \\\\\n48 &amp; 45\\\\\n\\end{pmatrix}\\ &#8212;&#8212;&#8212;- (1)\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B = \\begin{pmatrix}\n5 &amp; 2 \\\\\n4 &amp; 6\\\\\n\\end{pmatrix}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2B = \\begin{pmatrix}\n10 &amp; 4 \\\\\n8 &amp; 12\\\\\n\\end{pmatrix}\\ &#8212;&#8212;&#8212;- (2)\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3A\\ +\\ 2B = \\begin{pmatrix}\n15 &amp; 6 \\\\\n48 &amp; 45\\\\\n\\end{pmatrix}\\ +\\ \\begin{pmatrix}\n10 &amp; 4\\\\\n4 &amp; 8\\\\\n\\end{pmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\begin{pmatrix}\n15\\ +\\ 10 &amp; 6\\ +\\ 4 \\\\\n48\\ +\\  8&amp; 45\\  +\\ 12\\\\\n\\end{pmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3A\\ +\\ 2B = \\begin{pmatrix}\n25 &amp; 10 \\\\\n56 &amp; 57\\\\\n\\end{pmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 17:}\\ \\color {red}{If}\\ A =\\begin{bmatrix}\n1 &amp; 2 \\\\\n3 &amp; 5\\\\\n\\end{bmatrix}\\  and\\ B =\\begin{bmatrix}\n-5 &amp; 7 \\\\\n0 &amp; 4\\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red}{Then\\ Find}\\ 4A\\ -\\ 2B\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ 4A = 4\\begin{bmatrix}\n1 &amp; 2 \\\\\n3 &amp; 5\\\\\n\\end{bmatrix}\\  \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4A = \\begin{bmatrix}\n4 &amp; 8 \\\\\n12 &amp; 20\\\\\n\\end{bmatrix}\\ &#8212;&#8212;&#8212;- (1)\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ 2B = 2\\begin{bmatrix}\n-5 &amp; 7 \\\\\n0 &amp; 4\\\\\n\\end{bmatrix}\\  \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2B = \\begin{bmatrix}\n-10 &amp; 14 \\\\\n0 &amp; 8\\\\\n\\end{bmatrix}\\ &#8212;&#8212;&#8212;- (2)\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4A\\ &#8211; 2B = \\begin{bmatrix}\n4 &amp; 8 \\\\\n12 &amp; 20\\\\\n\\end{bmatrix}\\ -\\ \\begin{bmatrix}\n-10 &amp; 14 \\\\\n0 &amp; 8\\\\\n\\end{bmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\begin{bmatrix}\n4 + 10 &amp; 8 -14 \\\\\n12 &#8211; 0 &amp; 20 -8\\\\\n\\end{bmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4A\\ &#8211; 2B = \\begin{bmatrix}\n14 &amp; -6 \\\\\n12 &amp; 12\\\\\n\\end{bmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/N_1tOAb6Ito\" title=\"Matrix multiplication by a Scalar - 2\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 18:}\\ If\\ f(x)\\ =\\ 3\\ x\\ +\\ 2\\ and\\ A =\\begin{pmatrix}\n1 &amp; 0 \\\\\n2 &amp; -1\\\\\n\\end{pmatrix},\\  \\color {red}{\\ find\\ f(A)}.\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ Given\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ A =\\begin{pmatrix}\n1 &amp; 0 \\\\\n2 &amp; -1\\\\\n\\end{pmatrix}\\ and\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{3cm}\\ f(x)\\ =\\ 3\\ x\\ +\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[f(A) = \\ 3A\\ +\\ 2\\ I, Where\\ I\\ is\\ a\\ Identity\\ matrix\\ of\\ order\\ 2\\ &#8212;&#8212;&#8212;- (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3A\\ =\\ 3\\begin{pmatrix}\n1 &amp; 0 \\\\\n2 &amp; -1\\\\\n\\end{pmatrix}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3A\\ =\\ \\begin{pmatrix}\n3 &amp; 0 \\\\\n6 &amp; -3\\\\\n\\end{pmatrix}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[I\\ =\\ \\begin{pmatrix}\n1 &amp; 0 \\\\\n0 &amp; 1\\\\\n\\end{pmatrix}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2I\\ =\\ \\begin{pmatrix}\n2 &amp; 0 \\\\\n0 &amp; 2\\\\\n\\end{pmatrix}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3A\\ +\\ 2I\\ = \\begin{pmatrix}\n3 &amp; 0 \\\\\n6 &amp; -3\\\\\n\\end{pmatrix}\\ +\\ \\begin{pmatrix}\n2 &amp; 0 \\\\\n0 &amp; 2\\\\\n\\end{pmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\begin{pmatrix}\n3\\ +\\ 2 &amp; 0\\ +\\ 0 \\\\\n6\\ +\\ 0\\ &amp; -3\\  +\\  2\\\\\n\\end{pmatrix}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3A\\ +\\ 2I = \\begin{pmatrix}\n5 &amp; 0 \\\\\n6 &amp; -1\\\\\n\\end{pmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\color{green}{\\therefore\\ f(A)\\ = \\begin{pmatrix}\n5 &amp; 0 \\\\\n6 &amp; -1\\\\\n\\end{pmatrix}}}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {iii)\\ Multiplication\\ of\\ Matrices}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<p>The condition of multiplication of two matrices A and B is the number of columns in A is equal to the number of rows in B.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 19:}\\ \\color {red}{If}\\ A =\\begin{bmatrix}\n1 &amp; 2 \\\\\n3 &amp; 4 \\\\\n\\end{bmatrix}\\ ,\\ B =\\begin{bmatrix}\n5 &amp; 6 \\\\\n7 &amp; 8 \\\\\n\\end{bmatrix}\\ ,\\ \\color {red} {Find\\ AB}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\begin{bmatrix}\n1 &amp; 2 \\\\\n3 &amp; 4 \\\\\n\\end{bmatrix}\\  \\begin{bmatrix}\n5 &amp; 6 \\\\\n7 &amp; 8 \\\\\n\\end{bmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n1 \u00d7 5 + 2 \u00d7 7 &amp; 1 \u00d7 6 + 2 \u00d7 8 \\\\\n3 \u00d7 5 + 4 \u00d7 7 &amp; 3 \u00d7 6 + 4 \u00d7 8\\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n5 + 14 &amp; 6 + 16 \\\\\n15 + 28 &amp; 18 + 32\\\\\n\\end{bmatrix}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{AB = \\begin{bmatrix}\n19 &amp;  22 \\\\\n43 &amp; 50\\\\\n\\end{bmatrix}}\\  \\hspace{12cm}\\]<\/div>\n\n\n<p><iframe width=\"771\" height=\"434\" src=\"https:\/\/www.youtube.com\/embed\/ypa0ebBMCpc?list=PLQIom4Rz29vxcUyYeKo2Sw4cmEJ0RbBA2\" title=\"If two Matrices A and B are of Order 2, Find AB - Example - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 20:}\\ \\color {Red}{If}\\ A =\\begin{bmatrix}\n1 &amp; 1 \\\\\n-1 &amp; 1 \\\\\n\\end{bmatrix}\\ and\\ B =\\begin{bmatrix}\n1 &amp; 7 &amp;; 0\\\\\n4 &amp; 3 &amp; &#8211; 2; \\\\\n\\end{bmatrix}\\ ,\\ \\color {red} {Find\\ AB}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\begin{bmatrix}\n1 &amp; 1 \\\\\n&#8211; 1 &amp; 1 \\\\\n\\end{bmatrix}\\  \\begin{bmatrix}\n1 &amp; 7 &amp;; 0\\\\\n4 &amp; 3 &amp; &#8211; 2; \\\\\n\\end{bmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n1 \u00d7 1 + 1 \u00d7 4 &amp;amp 1 \u00d7 7 +  1\u00d7 3 &amp; 1 \u00d7 0 +  1\u00d7 -2 \\\\\n-1 \u00d7 4 + 1 \u00d7 4 &amp;amp &#8211; 1 \u00d7 7 +  1\u00d7 3 &amp; -1 \u00d7 0 +  1\u00d7 &#8211; 2 \\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n1 + 4 &amp; 7 +  3 &amp;; 0 &#8211;  2 \\\\\n&#8211; 4 + 4 &amp; &#8211; 7 +  3 &amp;;  0 &#8211;  2 \\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{AB\\ =\\ \\begin{bmatrix}\n5 &amp;  10 &amp; &#8211; 2\\\\\n0&amp; &#8211; 4 &amp; &#8211; 2\\\\\n\\end{bmatrix}}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 21:}\\ \\color {red}{Verify\\ (AB)^T\\ =\\ B^T\\ A^T}\\ If\\ A =\\begin{bmatrix}\n1 &amp; 0 &amp; 3 \\\\\n2 &amp; 1 &amp; &#8211; 1 \\\\\n1 &amp; -1 &amp; 1 \\\\\n\\end{bmatrix}\\ and\\ \\ B =\\begin{bmatrix}\n1 &amp; 0 &amp; 2 \\\\\n0 &amp; 1 &amp; 2 \\\\\n1 &amp; 2 &amp; 0 \\\\\n\\end{bmatrix}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  October\\ 2023,\\  June\\ 2025\\ (Supp)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ A =\\begin{bmatrix}\n1 &amp; 0 &amp; 3 \\\\\n2 &amp; 1 &amp; &#8211; 1 \\\\\n1 &amp; -1 &amp; 1 \\\\\n\\end{bmatrix}\\ and\\ \\ B =\\begin{bmatrix}\n1 &amp; 0 &amp; 2 \\\\\n0 &amp; 1 &amp; 2 \\\\\n1 &amp; 2 &amp; 0 \\\\\n\\end{bmatrix}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\begin{bmatrix}\n1 &amp; 0 &amp; 3 \\\\\n2 &amp; 1 &amp; &#8211; 1 \\\\\n1 &amp; -1 &amp; 1 \\\\\n\\end{bmatrix}\\  \\begin{bmatrix}\n1 &amp; 0 &amp; 2 \\\\\n0 &amp; 1 &amp; 2 \\\\\n1 &amp; 2 &amp; 0 \\\\\n\\end{bmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n1\\ \u00d7\\ 1\\ +\\ 0 \u00d7\\ 0\\ +\\ 3\\ \u00d7\\ 1&amp; 1\\ \u00d7\\ 0\\ +\\ 0 \u00d7\\ 1\\ +\\ 3\\ \u00d7\\ 2 &amp; 1\\ \u00d7\\ 2\\ +\\ 0 \u00d7\\ 2\\ +\\ 3\\ \u00d7\\ 0\\\\\n2\\ \u00d7\\ 1\\ +\\ 1 \u00d7\\ 0\\ +\\ -1\\ \u00d7\\ 1\\ &amp; 2\\ \u00d7\\ 0\\ +\\ 1 \u00d7\\ 1\\ +\\ -1\\ \u00d7\\ 2\\ &amp; 2\\ \u00d7\\ 2\\ +\\ 1 \u00d7\\ 2\\ +\\ -1\\ \u00d7\\ 0\\\\\n1\\ \u00d7\\ 1\\ +\\ -1 \u00d7\\ 0\\ +\\ 1\\ \u00d7\\ 1\\ &amp; 1\\ \u00d7\\ 0\\ +\\ -1 \u00d7\\ 1\\ +\\ 1\\ \u00d7\\ 2\\ &amp; 1\\ \u00d7\\ 2\\ +\\ -1 \u00d7\\ 2\\ +\\ 1\\ \u00d7\\ 0\\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n1\\ +\\ 0\\ +\\ 3\\ &amp; 0\\ +\\ 0\\ +\\ 6 &amp; 2\\ +\\ 0  +\\ 0\\\\\n2\\ +\\ 0\\ +\\ -1\\ &amp; 0\\ +\\ 1\\ -\\ 2\\ &amp; 4\\ +\\ 2\\ +\\ 0\\\\\n1\\ +\\ 0\\ +\\ 1\\ &amp; 0\\ -\\   1\\  +\\ 2\\ &amp; 2\\ -\\ 2\\ +\\ 0\\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ AB = \\begin{bmatrix}\n4 &amp; 6 &amp; 2 \\\\\n1 &amp; -1 &amp; 6 \\\\\n2 &amp; 1 &amp; 0 \\\\\n\\end{bmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(AB)^T\\ = \\begin{bmatrix}\n4 &amp; 1 &amp; 2 \\\\\n6 &amp; -1 &amp; 1 \\\\\n2 &amp; 6 &amp; 0 \\\\\n\\end{bmatrix}\\  &#8212;&#8212;&#8212;&#8212;&#8211; (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ B^T =\\begin{bmatrix}\n1 &amp; 0 &amp; 1 \\\\\n0 &amp; 1 &amp; 2 \\\\\n2 &amp; 2 &amp; 0 \\\\\n\\end{bmatrix}\\ and\\ \\ A^T\\ =\\begin{bmatrix}\n1 &amp; 2 &amp; 1 \\\\\n0 &amp; 1 &amp; &#8211; 1 \\\\\n3 &amp; -1 &amp; 1 \\\\\n\\end{bmatrix}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B^T\\ A^T\\ =\\begin{bmatrix}\n1 &amp; 0 &amp; 1 \\\\\n0 &amp; 1 &amp; 2 \\\\\n2 &amp; 2 &amp; 0 \\\\\n\\end{bmatrix}\\ \\begin{bmatrix}\n1 &amp; 2 &amp; 1 \\\\\n0 &amp; 1 &amp; &#8211; 1 \\\\\n3 &amp; -1 &amp; 1 \\\\\n\\end{bmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n1\\ \u00d7\\ 1\\ +\\ 0 \u00d7\\ 0\\ +\\ 1\\ \u00d7\\ 3&amp; 1\\ \u00d7\\ 2\\ +\\ 0 \u00d7\\ 1\\ +\\ 1\\ \u00d7\\ -1 &amp; 1\\ \u00d7\\ 1\\ +\\ 0 \u00d7\\ -1\\ +\\ 1\\ \u00d7\\ 1\\\\\n0\\ \u00d7\\ 1\\ +\\ 1 \u00d7\\ 0\\ +\\ 2\\ \u00d7\\ 3\\ &amp; 0\\ \u00d7\\ 2\\ +\\ 1 \u00d7\\ 1\\ +\\ 2\\ \u00d7\\ -1\\ &amp; 0\\ \u00d7\\ 1\\ +\\ 1 \u00d7\\ -1\\ +\\ 2\\ \u00d7\\ 1\\\\\n2\\ \u00d7\\ 1\\ +\\ 2 \u00d7\\ 0\\ +\\ 0\\ \u00d7\\ 3\\ &amp; 2\\ \u00d7\\ 2\\ +\\ 2 \u00d7\\ 1\\ +\\ 0\\ \u00d7\\ -1\\ &amp; 2\\ \u00d7\\ 1\\ +\\ 2 \u00d7\\ -1\\ +\\ 0\\ \u00d7\\ 1\\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n1\\ +\\ 0\\ +\\ 3\\ &amp; 2\\ +\\ 0\\ -\\ 1 &amp; 1\\ +\\ 0  +\\ 1\\\\\n0\\ +\\ 0\\ +\\ 6\\ &amp; 0\\ +\\ 1\\ -\\ 2\\ &amp; 0\\ -\\ 1\\ +\\ 2\\\\\n2\\ +\\ 0\\ +\\ 0\\ &amp; 4\\ +\\   2\\ +\\ 0\\ &amp; 2\\ -\\ 2\\ +\\ 0\\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B^T\\ A^T\\ = \\begin{bmatrix}\n4 &amp; 1 &amp; 2 \\\\\n6 &amp; -1 &amp; 1 \\\\\n2 &amp; 6 &amp; 0 \\\\\n\\end{bmatrix}\\  &#8212;&#8212;&#8212;&#8212;&#8211; (2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[From\\ (1)\\ and\\ (2)\\ (AB)^T\\ =\\ B^T\\ A^T\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 22:}\\ \\color {red}{Verify\\ (AB)^T\\ =\\ B^T\\ A^T}\\ If\\ A =\\begin{pmatrix}\n1 &amp; 2 &amp; 2 \\\\\n2 &amp; 1 &amp; 2 \\\\\n2 &amp; 2 &amp; 1 \\\\\n\\end{pmatrix}\\ and\\ \\ B =\\begin{pmatrix}\n1 &amp; 0 &amp; 2 \\\\\n0 &amp; 1 &amp; 2 \\\\\n1 &amp; 2 &amp; 0 \\\\\n\\end{pmatrix}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  April\\ 2025\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ A =\\begin{pmatrix}\n1 &amp; 2 &amp; 2 \\\\\n2 &amp; 1 &amp;  2 \\\\\n2 &amp; 2 &amp; 1 \\\\\n\\end{pmatrix}\\ and\\ \\ B =\\begin{pmatrix}\n1 &amp; 0 &amp; 2 \\\\\n0 &amp; 1 &amp; 2 \\\\\n1 &amp; 2 &amp; 0 \\\\\n\\end{pmatrix}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[AB =\\begin{pmatrix}\n1 &amp; 2 &amp; 2 \\\\\n2 &amp; 1 &amp; 2 \\\\\n2 &amp; 2 &amp; 1 \\\\\n\\end{pmatrix}\\  \\begin{pmatrix}\n1 &amp; 0 &amp; 2 \\\\\n0 &amp; 1 &amp; 2 \\\\\n1 &amp; 2 &amp; 0 \\\\\n\\end{pmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{pmatrix}\n1\\ \u00d7\\ 1\\ +\\ 2 \u00d7\\ 0\\ +\\ 2\\ \u00d7\\ 1&amp; 1\\ \u00d7\\ 0\\ +\\ 2 \u00d7\\ 1\\ +\\ 2\\ \u00d7\\ 2 &amp; 1\\ \u00d7\\ 2\\ +\\ 2 \u00d7\\ 2\\ +\\ 2\\ \u00d7\\ 0\\\\\n2\\ \u00d7\\ 1\\ +\\ 1 \u00d7\\ 0\\ +\\ 2\\ \u00d7\\ 1\\ &amp; 2\\ \u00d7\\ 0\\ +\\ 1 \u00d7\\ 1\\ +\\ 2\\ \u00d7\\ 2\\ &amp; 2\\ \u00d7\\ 2\\ +\\ 1 \u00d7\\ 2\\ +\\ 2\\ \u00d7\\ 0\\\\\n2\\ \u00d7\\ 1\\ +\\ 2\u00d7\\ 0\\ +\\ 1\\ \u00d7\\ 1\\ &amp; 1\\ \u00d7\\ 0\\ +\\ 2 \u00d7\\ 1\\ +\\ 1\\ \u00d7\\ 2\\ &amp; 2\\ \u00d7\\ 2\\ +\\ 2 \u00d7\\ 2\\ +\\ 1\\ \u00d7\\ 0\\\\\n\\end{pmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{pmatrix}\n1\\ +\\ 0\\ +\\ 2\\ &amp; 0\\ +\\ 2\\ +\\ 4 &amp; 2\\ +\\ 4  +\\ 0\\\\\n2\\ +\\ 0\\ +\\ 2\\ &amp; 0\\ +\\ 1\\ +\\ 4\\ &amp; 4\\ +\\ 2\\ +\\ 0\\\\ 2\\ +\\ 0\\ +\\ 1\\ &amp; 0\\ +\\   2\\ +\\ 2\\ &amp; 4\\ +\\ 4\\ +\\ 0\\\\\n\\end{pmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ AB = \\begin{pmatrix}\n3 &amp; 6 &amp; 6 \\\\\n4 &amp; 5 &amp; 6 \\\\\n3 &amp; 4 &amp; 8 \\\\\n\\end{pmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(AB)^T\\ = \\begin{pmatrix}\n3 &amp; 4 &amp; 3 \\\\\n6 &amp; 5 &amp; 4 \\\\\n6 &amp; 6 &amp;  8\\\\\n\\end{pmatrix}\\  &#8212;&#8212;&#8212;&#8212;&#8211; (1)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ B^T =\\begin{pmatrix}\n1 &amp; 0 &amp; 1 \\\\\n0 &amp; 1 &amp; 2 \\\\\n2 &amp; 2 &amp; 0 \\\\\n\\end{pmatrix}\\ and\\ \\ A^T\\ =\\begin{pmatrix}\n1 &amp; 2 &amp; 2 \\\\\n2 &amp; 1 &amp; 2 \\\\\n2 &amp; 2 &amp; 1 \\\\\n\\end{pmatrix}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B^T\\ A^T\\ =\\begin{pmatrix}\n1 &amp; 0 &amp; 1 \\\\\n0 &amp; 1 &amp; 2 \\\\\n2 &amp; 2 &amp; 0 \\\\\n\\end{pmatrix}\\ \\begin{pmatrix}\n1 &amp; 2 &amp; 2 \\\\\n2 &amp; 1 &amp; 2 \\\\\n2 &amp; 2 &amp; 1 \\\\\n\\end{pmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{pmatrix}\n1\\ \u00d7\\ 1\\ +\\ 0 \u00d7\\ 2\\ +\\ 1\\ \u00d7\\ 2&amp; 1\\ \u00d7\\ 2\\ +\\ 0 \u00d7\\ 1\\ +\\ 1\\ \u00d7\\ 2 &amp; 1\\ \u00d7\\ 2\\ +\\ 0 \u00d7\\ 2\\ +\\ 1\\ \u00d7\\ 1\\\\\n0\\ \u00d7\\ 1\\ +\\ 1 \u00d7\\ 2\\ +\\ 2\\ \u00d7\\ 2\\ &amp; 0\\ \u00d7\\ 2\\ +\\ 1 \u00d7\\ 1\\ +\\ 2\\ \u00d7\\ 2\\ &amp; 0\\ \u00d7\\ 2\\ +\\ 1 \u00d7\\ 2\\ +\\ 2\\ \u00d7\\ 1\\\\\n2\\ \u00d7\\ 1\\ +\\ 2 \u00d7\\ 2\\ +\\ 0\\ \u00d7\\ 2\\ &amp; 2\\ \u00d7\\ 2\\ +\\ 2 \u00d7\\ 1\\ +\\ 0\\ \u00d7\\ 2\\ &amp; 2\\ \u00d7\\ 2\\ +\\ 2 \u00d7\\ 2\\ +\\ 0\\ \u00d7\\ 1\\\\\n\\end{pmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{pmatrix}\n1\\ +\\ 0\\ +\\ 2\\ &amp; 2\\ +\\ 0\\ +\\ 2 &amp; 2\\ +\\ 0  +\\ 1\\\\\n0\\ +\\ 2\\ +\\ 4\\ &amp; 0\\ +\\ 1\\ +\\ 4\\ &amp; 0\\ +\\ 2\\ +\\ 2\\\\\n2\\ +\\ 4\\ +\\ 0\\ &amp; 4\\ +\\   2\\ +\\ 0\\ &amp; 4\\ +\\ 4\\ +\\ 0\\\\\n\\end{pmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B^T\\ A^T\\ = \\begin{pmatrix}\n3 &amp; 4 &amp; 3 \\\\\n6 &amp; 5 &amp; 4 \\\\\n6 &amp; 6 &amp; 8 \\\\\n\\end{pmatrix}\\  &#8212;&#8212;&#8212;&#8212;&#8211; (2)\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[From\\ (1)\\ and\\ (2)\\ (AB)^T\\ =\\ B^T\\ A^T\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 23:}\\  If\\ A =\\begin{bmatrix}\n1 &amp; 2 &amp; 2 \\\\\n2 &amp; 1 &amp;  2 \\\\\n2 &amp; 2 &amp; 1 \\\\\n\\end{bmatrix}\\ then\\ \\color{red}{show\\ that\\ A^2\\ -\\ 4\\ A\\ -\\ 5I\\ =0}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ A =\\begin{bmatrix}\n1 &amp; 2 &amp; 2 \\\\\n2 &amp; 1 &amp;  2 \\\\\n2 &amp; 2 &amp; 1 \\\\\n\\end{bmatrix}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^2 =\\begin{bmatrix}\n1 &amp; 2 &amp; 2 \\\\\n2 &amp; 1 &amp;  2 \\\\\n2 &amp; 2 &amp; 1 \\\\\n\\end{bmatrix}\\  \\begin{bmatrix}\n1 &amp; 2 &amp; 2 \\\\\n2 &amp; 1 &amp;  2 \\\\\n2 &amp; 2 &amp; 1 \\\\\n\\end{bmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n1\\ \u00d7\\ 1\\ +\\ 2 \u00d7\\ 2\\ +\\ 2\\ \u00d7\\ 2 &amp; 1\\ \u00d7\\ 2\\ +\\ 2 \u00d7\\ 1\\ +\\ 2\\ \u00d7\\ 2 &amp; 1\\ \u00d7\\ 2\\ +\\ 2\\ \u00d7\\ 2\\ +\\ 2\\ \u00d7\\ 1\\\\\n2\\ \u00d7\\ 1\\ +\\ 1 \u00d7\\ 2\\ +\\ 2\\ \u00d7\\ 2\\ &amp; 2\\ \u00d7\\ 2\\ +\\ 1 \u00d7\\ 1\\ +\\ 2\\ \u00d7\\ 2\\ &amp; 2\\ \u00d7\\ 2\\ +\\ 1 \u00d7\\ 2\\ +\\ 2\\ \u00d7\\ 1\\\\\n2\\ \u00d7\\ 1\\ +\\ 2 \u00d7\\ 2\\ +\\ 1\\ \u00d7\\ 2\\ &amp; 2\\ \u00d7\\ 2\\ +\\ 2 \u00d7\\ 1\\ +\\ 1\\ \u00d7\\ 2\\ &amp; 2\\ \u00d7\\ 2\\ +\\ 2 \u00d7\\ 2\\ +\\ 1\\ \u00d7\\ 1\\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n1\\ +\\ 4\\ +\\ 4\\ &amp; 2\\ +\\ 2\\ +\\ 4 &amp; 2\\ +\\ 4  +\\ 2\\\\\n2\\ +\\ 2\\ +\\ 4\\ &amp; 4\\ +\\ 1\\ +\\ 4\\ &amp; 4\\ +\\ 2\\ +\\ 2\\\\\n2\\ +\\ 4\\ +\\ 2\\ &amp; 4\\ +\\  2\\ +\\ 2\\ &amp; 4\\ +\\ 4\\ +\\ 1\\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^2 = \\begin{bmatrix}\n9 &amp; 8 &amp; 8 \\\\\n8 &amp; 9 &amp; 8 \\\\\n8 &amp; 8 &amp; 9 \\\\\n\\end{bmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4\\ A\\ =\\ 4\\ \\begin{bmatrix}\n1 &amp; 2 &amp; 2 \\\\\n2 &amp; 1 &amp;  2 \\\\\n2 &amp; 2 &amp; 1 \\\\\n\\end{bmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4\\ A\\ = \\begin{bmatrix}\n4 &amp; 8 &amp; 8 \\\\\n8 &amp; 4 &amp; 8 \\\\\n8 &amp; 8 &amp; 4 \\\\\n\\end{bmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5\\ I\\ =\\ 5\\ \\begin{bmatrix}\n1 &amp; 0 &amp; 0 \\\\\n0 &amp; 1 &amp;  0 \\\\\n0 &amp; 0 &amp; 1 \\\\\n\\end{bmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5\\ I\\ = \\begin{bmatrix}\n5 &amp; 0 &amp; 0 \\\\\n0 &amp; 5 &amp; 0 \\\\\n0 &amp; 0 &amp; 5 \\\\\n\\end{bmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^2\\ -\\ 4\\ A\\ -\\ 5I\\ =\\ \\begin{bmatrix}\n9 &amp; 8 &amp; 8 \\\\\n8 &amp; 9 &amp; 8 \\\\\n8 &amp; 8 &amp; 9 \\\\\n\\end{bmatrix}\\ -\\ \\begin{bmatrix}\n4 &amp; 8 &amp; 8 \\\\\n8 &amp; 4 &amp; 8 \\\\\n8 &amp; 8 &amp; 4 \\\\\n\\end{bmatrix}\\ -\\ \\begin{bmatrix}\n5 &amp; 0 &amp; 0 \\\\\n0 &amp; 5 &amp; 0 \\\\\n0 &amp; 0 &amp; 5 \\\\\n\\end{bmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{bmatrix}\n9\\ -\\ 4\\ -\\ 5\\ &amp; 8\\ -\\ 8\\ -\\ 0 &amp; 8\\ -\\ 8  -\\ 0\\\\\n8\\ -\\ 8\\ -\\ 0\\ &amp; 9\\ -\\ 4\\ -\\ 5\\ &amp; 8\\ -\\ 8\\ -\\ 0\\\\\n8\\ -\\ 8\\ -\\ 0\\ &amp; 8\\ -\\  8\\ -\\ 0\\ &amp; 9\\ -\\ 4\\ -\\ 5\\\\\n\\end{bmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{\\boxed{\\therefore\\ A^2\\ -\\ 4\\ A\\ -\\ 5I\\ =\\ 0}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {Singular\\ and\\ Non-Singular\\ Matrix}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">A square matrix A is called a singular matrix <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[if\\ \\begin{vmatrix} A \\\\ \\end{vmatrix}\\ = 0\\ and\\ non\\ \u2013\\ singular\\ matrix\\ if\\ \\begin{vmatrix} A \\\\ \\end{vmatrix}\\ \\neq {0}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 24:}\\ \\color {red}{Prove\\ that}\\ \\begin{pmatrix}\n1 &amp; 3 \\\\\n2 &amp; 6 \\\\\n\\end{pmatrix}\\ \\color {red} {is\\ a\\ singular\\ matrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution\\ :}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =\\ 1(6)\\ -\\ 2(3) \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 6\\ -\\ 6 \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= 0 \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ is\\ a\\ singular\\ matrix\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 25:}\\ Prove\\ that\\ the\\ matrix \\begin{pmatrix}\n1 &amp; &#8211; 2 \\\\\n&#8211; 2 &amp; 4\\\\\n\\end{pmatrix}\\ \\color {red} {is\\ a\\ singular\\ matrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  October\\ 2024\\ June\\ 2025\\ (Supp)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  Given\\ A =\\begin{pmatrix}\n1 &amp; &#8211; 2 \\\\\n&#8211; 2 &amp; 4\\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =\\ 1(4) &#8211; (-2)(- 2) \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 4\\ -\\ 4 \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= 0 \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ is\\ a\\ singular\\ matrix\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 26:}\\ \\color {red}{Prove\\ that}\\ A =\\begin{bmatrix}\n2 &amp; 3 \\\\\n4 &amp; 5 \\\\\n\\end{bmatrix}\\ \\color {red} {is\\ Non\\ -\\ singular}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution\\ :}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ = 2(5) &#8211; 3(4) \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= 10 &#8211; 12 \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix} A \\\\ \\end{vmatrix}\\ = -2\\ \\neq {0}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ is\\ a\\ Non\\ &#8211; \\ Singular\\ matrix\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"771\" height=\"434\" src=\"https:\/\/www.youtube.com\/embed\/MlSLt6ia5MI?list=PLQIom4Rz29vxcUyYeKo2Sw4cmEJ0RbBA2\" title=\"Non-singular matrix of order 2 - Example - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{Example\\ 27:}\\ \\color {red}{Prove\\ that}\\ \\begin{bmatrix}\n1 &amp; -1 &amp; 2 \\\\\n2 &amp; -2 &amp; 4 \\\\\n3 &amp; -3 &amp; 6 \\\\\n\\end{bmatrix}\\ \\color {red} {is\\ a\\ singular\\ matrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ A\\ =\\begin{bmatrix}\n1 &amp; -1 &amp; 2 \\\\\n2 &amp; -2 &amp; 4 \\\\\n3 &amp; -3 &amp; 6 \\\\\n\\end{bmatrix}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =1\\begin{vmatrix}\n-2 &amp; 4 \\\\\n-3 &amp; 6 \\\\\n\\end{vmatrix}\\ &#8211; \\ 1\\begin{vmatrix}\n2 &amp; 2 \\\\\n2 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1 \\begin{vmatrix}\n2 &amp; -1 \\\\\n2 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{4cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(-12\\ +\\ 12)\\ + 1 (12\\ -\\ 12) + 2(-6\\ +\\ 6)\\ \n\\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(0)\\ + 1 (0) + 2(0)\\ \n\\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix} A \\\\ \\end{vmatrix}\\ = 0\\  \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ is\\ a\\ Singular\\ matrix\\ \\hspace{10cm}\\]<\/div>\n\n\n<p><iframe width=\"771\" height=\"434\" src=\"https:\/\/www.youtube.com\/embed\/GkPnRFawjUM?list=PLQIom4Rz29vxcUyYeKo2Sw4cmEJ0RbBA2\" title=\"Singular matrix of order - 3 - Example 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{Example\\ 28:}\\ \\color {red}{Prove\\ that\\ the\\ matrix}\\ \\begin{pmatrix}\n2 &amp; 3 &amp; &#8211; 1 \\\\\n4 &amp; 6 &amp; 5 \\\\\n6 &amp; 2 &amp; 1 \\\\\n\\end{pmatrix}\\ \\color {red} {is\\ non\\ -\\ singular}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ A\\ =\\begin{pmatrix}\n2 &amp; 3 &amp; -1 \\\\\n4 &amp; 6 &amp; 5 \\\\\n6 &amp; 2 &amp; 1 \\\\\n\\end{pmatrix}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =\\ 2\\begin{vmatrix}\n6 &amp; 5 \\\\\n2 &amp; 1 \\\\\n\\end{vmatrix}\\ &#8211; \\ 3\\begin{vmatrix}\n4 &amp; 5 \\\\\n6 &amp; 1 \\\\\n\\end{vmatrix}\\ -\\ 1 \\begin{vmatrix}\n4 &amp; 6 \\\\\n6 &amp; 2 \\\\\n\\end{vmatrix}\\ \\hspace{4cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 2(6\\ -\\ 10)\\ -\\ 3 (4\\ -\\ 30)\\ -\\ 1(8\\ -\\ 36)\\ \n\\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 2(- 4)\\ -\\ 3 (-26)\\ -\\ 1(-28)\\ \n\\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ -\\ 8\\ +\\ 78\\ +\\ 28\\ \n\\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\begin{vmatrix} A \\\\ \\end{vmatrix}\\ = 98\\ \\neq {0}}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ is\\ a\\ Non\\ &#8211; \\ Singular\\ matrix\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Minor\\ of\\ an\\ element\\ of\\ a\\  Matrix}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Minor\\ of\\ an\\ element\\ is\\ a\\ determinant\\ obtained\\ by\\ deleting\\ the\\ row\\ and\\ column\\ in\\ which\\ the\\ element\\ occurs\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{purple}{Example:\\ 29}\\ \\color{red}{Find\\ the\\ Minor\\ of\\ a_1\\ in\\ the\\ matrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\nA \n \\ = \\begin{bmatrix}\na_1 &amp; b_1 &amp; c_1 \\\\\na_2 &amp; b_2 &amp; c_2 \\\\\na_3 &amp; b_3 &amp; c_3 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{22cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Minor\\ of\\ a_1 =\\begin{vmatrix}\nb_2 &amp; c_2 \\\\\nb_3 &amp; c_3 \\\\\n\\end{vmatrix}\\ = b_2c_3\\ -\\ b_3c_2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{Example\\ 30:}\\ \\color{red}{Find\\ the\\ Minor\\ of\\ 2\\ in\\ the\\ following\\ matrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\nA\n\\ = \\begin{bmatrix}\n1 &amp; 0 &amp; -1 \\\\\n2 &amp; 3 &amp; 4 \\\\\n7 &amp; 8 &amp; -2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{22cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ Minor\\ of\\ 2 =\\begin{vmatrix}\n0 &amp; -1 \\\\\n8 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= 0 + 8\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{Minor\\ of\\ 2= 8}\\ \\hspace{15cm}\\]<\/div>\n\n\n<p><iframe width=\"771\" height=\"434\" src=\"https:\/\/www.youtube.com\/embed\/wHqIXqATz_A?list=PLQIom4Rz29vxcUyYeKo2Sw4cmEJ0RbBA2\" title=\"Minor of element in the Matrix of order 3 - Example - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Cofactor\\ of\\ an\\ element of\\ a\\  Matrix}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cofactor\\ of\\ an\\ element\\ is\\ a\\ signed\\ minor\\ of\\ that\\ element\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ cofactor\\ of\\ a_{ij} = (-1)^{i\\ +\\ j}\\ minor\\ of\\ a_{ij}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{Example\\ 31 :}\\ \\color {red}{Find\\ the\\ cofactor\\ of\\ &#8216;2&#8217;\\ in}\\  \\begin{pmatrix}\n3 &amp; 0 &amp; 2 \\\\\n5 &amp; 1 &amp; 7 \\\\\n4 &amp; 5 &amp; -3 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{22cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2\\ =\\ (-1)^{1\\ +\\ 3}\\ \\begin{vmatrix}\n5 &amp; 1 \\\\\n4 &amp; 5 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (25\\  -\\ 4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1(21)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{\\boxed{cofactor\\ of\\ 2\\ =\\  21}}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{Example\\ 32 :}\\ \\color {red}{Find\\ the\\ cofactor\\ of\\ &#8216;2&#8217;\\ in}\\  \\begin{pmatrix}\n1 &amp; 0 &amp; &#8211; 1 \\\\\n2 &amp; 3 &amp; 4 \\\\\n7 &amp; 8 &amp; &#8211; 2 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  October\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{22cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2\\ =\\ (-1)^{2\\ +\\ 1}\\ \\begin{vmatrix}\n0 &amp; &#8211; 1 \\\\\n8 &amp; &#8211; 2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (0\\  -\\ (- 8)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ &#8211; 1(8)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{\\boxed{cofactor\\ of\\ 2\\ =\\  &#8211; 8}}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{Example\\ 33 :}\\ \\color {red}{Find\\ the\\ cofactor\\ of\\ &#8216;2&#8217;\\ in}\\  \\begin{pmatrix}\n1 &amp; &#8211; 1 &amp;  1 \\\\\n2 &amp; 3 &amp; &#8211; 3 \\\\\n6 &amp; &#8211; 2 &amp; &#8211; 1 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  April\\ 2025\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{22cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2\\ =\\ (-1)^{2\\ +\\ 1}\\ \\begin{vmatrix}\n&#8211; 1 &amp; 1 \\\\\n&#8211; 2 &amp; &#8211; 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (1 -\\ (- 2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ &#8211; 1(3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{green}{\\boxed{cofactor\\ of\\ 2\\ =\\  &#8211; 3}}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{Example\\ 34 :}\\ \\color {red}{Find\\ the\\ cofactor\\ of\\ -\\ 2\\ in}\\  \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm} \\begin{vmatrix}\n1 &amp; &#8211; 1 &amp; 1 \\\\\n2 &amp; 3 &amp; -\\ 3 \\\\\n6 &amp; -\\ 2 &amp; -\\ 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  October\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\hspace{22cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -\\ 2\\ =\\ (-1)^{3\\ +\\ 2}\\ \\begin{vmatrix}\n1 &amp; 1 \\\\\n2 &amp; -\\ 3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^5 (-\\ 3\\ -\\ 2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (-\\ 5)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{cofactor\\ of\\ -\\ 2 =\\  5}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 35:} \\color {red}{Find\\ the\\ cofactor\\ matrix\\ of}\\ \\begin{bmatrix}\n2 &amp; 3 &amp; 4 \\\\\n1 &amp; 2 &amp; 3 \\\\\n-1 &amp; 1 &amp; 2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ A\\ =\\begin{bmatrix}\n2 &amp; 3 &amp; 4 \\\\\n1 &amp; 2 &amp; 3 \\\\\n-1 &amp; 1 &amp; 2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = (-1)^{1\\ +\\ 1}\\ \\begin{vmatrix}\n2 &amp; 3 \\\\\n1 &amp; 2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^2 (4 -3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3 = (-1)^{1\\ +\\ 2}\\ \\begin{vmatrix}\n1 &amp; 3 \\\\\n-1 &amp; 2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (2 + 3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (5)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3 = -5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 4 = (-1)^{1\\ +\\ 3}\\ \\begin{vmatrix}\n1 &amp; 2 \\\\\n-1 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (1 + 2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 4 = 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = (-1)^{2\\ +\\ 1}\\ \\begin{vmatrix}\n3 &amp; 4 \\\\\n1 &amp; 2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (6 &#8211; 4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = -2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = (-1)^{2\\ +\\ 2}\\ \\begin{vmatrix}\n2 &amp; 4 \\\\\n-1 &amp; 2 \\\\ \\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (4 + 4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (8)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = 8\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3 = (-1)^{2\\ +\\ 3}\\ \\begin{vmatrix}\n2 &amp; 3 \\\\\n-1 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^5 (2 + 3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (5)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3 = -5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -1 = (-1)^{3\\ +\\ 1}\\ \\begin{vmatrix}\n3 &amp; 4 \\\\\n2 &amp; 3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (9 &#8211; 8)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -1 = 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = (-1)^{3\\ +\\ 2}\\ \\begin{vmatrix}\n2 &amp; 4 \\\\\n1 &amp; 3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^5 (6 &#8211; 4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = -2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = (-1)^{3\\ +\\ 3}\\ \\begin{vmatrix}\n2 &amp; 3 \\\\\n1 &amp; 2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^6 (4 &#8211; 3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cofactor\\ matrix=\\begin{bmatrix}\n1 &amp; -5 &amp; 3 \\\\\n-2 &amp; 8 &amp; -5 \\\\\n1 &amp; -2 &amp; 1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n<p><iframe width=\"771\" height=\"434\" src=\"https:\/\/www.youtube.com\/embed\/22kC4Cr-NTU?list=PLQIom4Rz29vxcUyYeKo2Sw4cmEJ0RbBA2\" title=\"Finding the cofactor matrix of order - 3 - Example - 1\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Adjoint\\ of\\ Matrix}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ Adjoint\\ of\\ a\\ square\\ matrix\\ A\\ is\\ the\\ transpose\\ of\\ the\\ matrix\\ which\\ is\\ formed\\ by\\]\\[the\\ elements\\ which\\ are\\ the\\  cofactors\\ of\\ the\\ corresponding\\ elements\\ of\\ the\\ determinant\\ of\\ the\\ matrix\\ A\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{Method\\ for\\ to\\ find\\ adjoint\\ of\\ Matrix\\ of\\ order\\ 3\\ (order 2)}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[i)\\ A\\ is\\ square\\ Matrix\\ of\\ order\\ 3\\ (order 2)\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ii)\\ Find\\ the\\ co-factor\\ of\\ all\\ the\\ elements\\ of\\ det\\ A\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[iii)\\ Form\\ the\\ matrix\\ by\\ replacing\\ all\\ the\\ elements\\ of\\ A\\ by\\ the\\ corresponding\\ cofactor\\ in\\ \\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[iv)\\ Then\\ take\\ the\\ Transpose\\ of\\ that\\ matrix,\\ then\\ we\\ get\\ adj. A.\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 36:}\\ \\color{red}{Find\\ the\\ Adjoint\\ of}\\ \\begin{pmatrix}\n1 &amp; 5 \\\\\n3 &amp; 6 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  October\\ 2024\\ June\\ 2025\\ (Supp)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ A\\ =\\begin{pmatrix}\n1 &amp; 5 \\\\\n3 &amp; 6 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = (-1)^{1\\ +\\ 1}\\ (6)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (6)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1\\ = \\ 6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 5 = (-1)^{1\\ +\\ 2}\\ (3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -1 =\\ -\\ 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3\\ = (-1)^{2\\ +\\ 1}\\ (5)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (5)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3\\ =\\ -\\  5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 6\\ = (-1)^{2\\ +\\ 2}\\ (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 6\\ = 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ cofactor\\ matrix\\ = \\begin{pmatrix}\n6 &amp; -\\ 3 \\\\\n-\\ 5 &amp; 1 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Adj.\\ A = \\begin{pmatrix}\n6 &amp; -\\ 5 \\\\\n-\\ 3 &amp; 1 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 37:}\\ \\color{red}{Find\\ the\\ Adjoint\\ Matrix\\ of}\\ \\begin{bmatrix}\n5 &amp; -6 \\\\\n3 &amp; 2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ A\\ =\\begin{bmatrix}\n5 &amp; -6 \\\\\n3 &amp; 2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 5 = (-1)^{1\\ +\\ 1}\\ (2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 5 = \\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -6 = (-1)^{1\\ +\\ 2}\\ (3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -6 = -3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3 = (-1)^{2\\ +\\ 1}\\ (-6)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (-6)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3 = 6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = (-1)^{2\\ +\\ 2}\\ (5)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (5)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = 5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ cofactor\\ matrix\\ = \\begin{bmatrix}\n2 &amp; -3 \\\\\n6 &amp; 5 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Adj.\\ A = \\begin{bmatrix}\n2 &amp; 6 \\\\\n-3 &amp; 5 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 38:}\\ \\color{red}{Find\\ the\\ Adjoint\\ of}\\ A\\ = \\begin{bmatrix}\n2 &amp; 3 &amp; 4 \\\\\n1 &amp; 2 &amp; 3 \\\\\n-1 &amp; 1 &amp; 2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ A\\ =\\begin{bmatrix}\n2 &amp; 3 &amp; 4 \\\\\n1 &amp; 2 &amp; 3 \\\\\n-1 &amp; 1 &amp; 2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cofactor\\ matrix=\\begin{bmatrix}\n1 &amp; -5 &amp; 3 \\\\\n-2 &amp; 8 &amp; -5 \\\\\n1 &amp; -2 &amp; 1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Adj.\\ A=\\ (Cofactor Matrix)^T\\ =\\ \\begin{bmatrix}\n1 &amp; -2 &amp; 1 \\\\\n-5 &amp; 8 &amp; -2 \\\\\n3 &amp; -5 &amp; 1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Inverse\\ of\\ Matrix}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ A\\ be\\ a\\ non\\ singular\\ matrix\\ if\\ there\\ exists\\ a\\ square\\ matrix\\ B\\, such\\ that\\ AB\\ =\\ BA\\ = I\\]\\[Where\\ I\\ is\\ the\\ unit\\  matrix\\ of\\ same\\ order\\ then\\ B\\ is\\ called\\ the\\ the\\ Inverse\\ of\\ A\\ and\\ it\\ is\\ denoted\\ by\\ A^{-1}.\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Note:}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[i)\\ inverse\\ of\\ matrix\\ is\\ unique\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ii)\\ AB=\\ BA\\ =\\ I\\  \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[iii)\\ (AB)^{-1} =\\ B^{-1}A^{-1}\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Formula\\ for\\ Inverse\\ of\\ Matrix:}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {\\boxed {A^{-1} = \\frac{1}{\\begin{vmatrix} A \\\\ \\end{vmatrix}}\\ adj.\\ A}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 39:}\\ \\color{red}{Find\\ the\\ inverse\\ of}\\ \\begin{bmatrix}\n1 &amp; -1 \\\\\n-2 &amp; 0 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Let\\ A\\ =\\begin{bmatrix}\n1 &amp; -1 \\\\\n-2 &amp; 0 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ = 1(0)\\ -\\ (-2) \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= 0 &#8211; 2 \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix} A \\\\ \\end{vmatrix}\\ = &#8211; 2\\ \\neq {0}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ A^{-1}\\ exist\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Cofactors\\ of\\ Matrix\\ A:}\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = (-1)^{1\\ +\\ 1}\\ (0)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^2 (0)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = 0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -1 = (-1)^{1\\ +\\ 2}\\ (-2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (-2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ &#8211; 1 = 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -2 = (-1)^{2\\ +\\ 1}\\ (-1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (-1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ &#8211; 2 = 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 0 = (-1)^{2\\ +\\ 2}\\ (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 0 = 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ cofactor\\ matrix\\ = \\begin{bmatrix}\n0 &amp; 2 \\\\\n1 &amp; 1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Adj.\\ A = \\begin{bmatrix}\n0 &amp; 1 \\\\\n2 &amp; 1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^{-1} = \\frac{1}{\\begin{vmatrix} A \\\\ \\end{vmatrix}}\\ adj.\\ A\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^{-1} = \\frac{1}{-2}\\ \\begin{bmatrix}\n0 &amp; 1 \\\\\n2 &amp; 1 \\\\\n\\end{bmatrix}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 40:}\\ \\color{red}{Find\\ the\\ inverse\\ matrix\\ of}\\ \\begin{pmatrix}\n5 &amp; 2 \\\\\n-4 &amp; 3 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  October\\ 2023\\ April\\ 2025\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Let\\ A\\ =\\begin{pmatrix}\n5 &amp; 2 \\\\\n-4 &amp; 3 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =\\ 5(3)\\ -\\ 2(-4) \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 15\\ +\\ 8 \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix} A \\\\ \\end{vmatrix}\\ =\\ 23\\ \\neq {0}\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ A^{-1}\\ exist\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Cofactors\\ of\\ Matrix\\ A:}\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 5 = (-1)^{1\\ +\\ 1}\\ (3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^2 (3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = (-1)^{1\\ +\\ 2}\\ (-4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (-4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = 4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -4 = (-1)^{2\\ +\\ 1}\\ (2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ &#8211; 4 =\\ -\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3\\ = (-1)^{2\\ +\\ 2}\\ (5)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (5)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3 =\\ 5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ cofactor\\ matrix\\ = \\begin{pmatrix}\n3 &amp; 4 \\\\\n-2 &amp; 5 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Adj.\\ A = \\begin{pmatrix}\n3 &amp; -2 \\\\\n4 &amp; 5 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^{-1} = \\frac{1}{\\begin{vmatrix} A \\\\ \\end{vmatrix}}\\ adj.\\ A\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^{-1} = \\frac{1}{23}\\ \\begin{pmatrix}\n3 &amp; -2 \\\\\n4 &amp; 5 \\\\\n\\end{pmatrix}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 41:}\\ \\color{red}{Find\\ the\\ inverse\\ of}\\ \\begin{bmatrix}\n1 &amp; -1 &amp; 1 \\\\\n2 &amp; -3 &amp; -3 \\\\\n6 &amp; -2 &amp;  -1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  October\\ 2023,\\ April\\ 2024\\ June\\ 2025(Supp)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Let\\ A\\ =\\begin{bmatrix}\n1 &amp; -1 &amp; 1 \\\\\n2 &amp; -3 &amp; -3 \\\\\n6 &amp; -2 &amp;  -1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =1\\begin{vmatrix}\n-3 &amp; -3 \\\\\n-2 &amp; -1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; -3 \\\\\n6 &amp; -1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; -3\\\\\n6 &amp;  -2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(3\\ -\\ 6)\\ + 1 (-2\\ +\\ 18) + 1(-4\\ +\\ 18)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(-3)\\ + 1 (16) + 1(14)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = -3\\ + 16 + 14\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ = 27\\ \\neq\\ 0\\\n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ Inverse\\ of\\ A\\ exist\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Cofactors\\ of\\ Matrix\\ A:}\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = (-1)^{1\\ +\\ 1}\\ \\begin{vmatrix}\n-3 &amp; -3 \\\\\n-2 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^2 (3 &#8211; 6)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (-3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = -3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -1 = (-1)^{1\\ +\\ 2}\\ \\begin{vmatrix}\n2 &amp; -3 \\\\\n6 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (-2 + 18)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (16)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -1 = -16\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = (-1)^{1\\ +\\ 3}\\ \\begin{vmatrix}\n2 &amp; -3 \\\\\n6 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (-4 + 18)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (14)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = 14\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = (-1)^{2\\ +\\ 1}\\ \\begin{vmatrix}\n-1 &amp; 1 \\\\\n-2 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (1+ 2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = -3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -3 = (-1)^{2\\ +\\ 2}\\ \\begin{vmatrix}\n1 &amp; 1 \\\\\n6 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (-1- 6)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (-7)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -3 = -7\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -3 = (-1)^{2\\ +\\ 3}\\ \\begin{vmatrix}\n1 &amp; -1 \\\\\n6 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^5 (-2+ 6)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -3 = -4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 6 = (-1)^{3\\ +\\ 1}\\ \\begin{vmatrix}\n-1 &amp; 1 \\\\\n-3 &amp; -3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (3 + 3)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (6)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 6 = 6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -2 = (-1)^{3\\ +\\ 2}\\ \\begin{vmatrix}\n1 &amp; 1 \\\\\n2 &amp; -3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^5 (-3- 2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (-5)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -2 = 5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -1 = (-1)^{3\\ +\\ 3}\\ \\begin{vmatrix}\n1 &amp; -1 \\\\\n2 &amp; -3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^6 (-3+ 2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (-1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -1 = -1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cofactor\\ matrix=\\begin{bmatrix}\n-3 &amp; -16 &amp; 14 \\\\\n-3 &amp; -7 &amp; -4 \\\\\n6 &amp; 5 &amp; -1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Adj.\\ A=\\begin{bmatrix}\n-3 &amp; -3 &amp; 6 \\\\\n-16 &amp; -7 &amp; 5 \\\\\n14 &amp; -4 &amp; -1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^{-1} = \\frac{1}{\\begin{vmatrix} A \\\\ \\end{vmatrix}}\\ adj.\\ A\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^{-1} = \\frac{1}{27}\\ \\begin{bmatrix}\n-3 &amp; -3 &amp; 6 \\\\\n-16 &amp; -7 &amp; 5 \\\\\n14 &amp; -4 &amp; -1 \\\\\n\\end{bmatrix}\\ \\hspace{2cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/eJ8q9ayyRkc\" title=\"Inverse of a Matrix - Example - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 42:}\\ \\color{red}{Find\\ the\\ inverse\\ of\\ the\\ matrix}\\ \\begin{bmatrix}\n1 &amp; -1 &amp; 2 \\\\\n4 &amp; 0 &amp; 6 \\\\\n0 &amp; 1 &amp;  -1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\  April\\ 2025\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Let\\ A\\ =\\begin{bmatrix}\n1 &amp; -1 &amp; 2 \\\\\n4 &amp; 0 &amp; 6 \\\\\n0 &amp; 1 &amp;  -1 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =\\ 1\\begin{vmatrix}\n0 &amp; 6 \\\\\n1 &amp; &#8211; 1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n4 &amp; 6 \\\\\n0 &amp; -1 \\\\\n\\end{vmatrix}\\ +\\ 2\\begin{vmatrix}\n4 &amp; 0\\\\\n0 &amp;  1 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 1(0\\ -\\ 6)\\ +\\ 1 (-4\\ -\\ 0)\\ +\\ 2(4\\ -\\ 0)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 1(-6)\\ +\\ 1 (-\\ 4)\\ +\\ 2(4)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ -\\ 6\\ -\\ 4\\ +\\  8\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =\\ -\\ 2\\ \\neq\\ 0\\\n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ Inverse\\ of\\ A\\ exist\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Cofactors\\ of\\ Matrix\\ A:}\\ \\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1\\ = (-1)^{1\\ +\\ 1}\\ \\begin{vmatrix}\n4 &amp; 6 \\\\\n0 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1\\ = (-1)^2\\ (-\\ 4\\ -\\ 0)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (-\\ 4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1\\ =\\ -\\ 4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -\\ 1\\ = (-1)^{1\\ +\\ 2}\\ \\begin{vmatrix}\n4 &amp; 6 \\\\\n0 &amp; -\\ 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (-\\ 4\\ -\\ 0)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (-\\ 4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\  -\\ 1\\ =\\ 4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = (-1)^{1\\ +\\ 3}\\ \\begin{vmatrix}\n4 &amp; 0 \\\\\n0 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (4\\ -\\ 0)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 =\\ 4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 4\\ =\\ (-1)^{2\\ +\\ 1}\\ \\begin{vmatrix}\n-\\ 1 &amp; 2 \\\\\n1 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (1\\ -\\ 2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (-\\ 1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 4\\ =\\  1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 0\\ =\\ (-1)^{2\\ +\\ 2}\\ \\begin{vmatrix}\n1 &amp; 2 \\\\\n0 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (-\\ 1\\ -\\ 0)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (-\\ 1)\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 0\\ =\\ -\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 6\\ = (-1)^{2\\ +\\ 3}\\ \\begin{vmatrix}\n1 &amp; &#8211;  1 \\\\\n0 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^5 (1 +\\ 0)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 6\\ =\\ -\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 0\\ =\\ (-1)^{3\\ +\\ 1}\\ \\begin{vmatrix}\n-\\ 1 &amp; 2 \\\\\n0 &amp; 6 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (-\\ 6\\ -\\ 0)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (-\\ 6)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 0\\ =\\ -\\ 6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1\\ =\\ (-1)^{3\\ +\\ 2}\\ \\begin{vmatrix}\n1 &amp; 4 \\\\\n2 &amp; 6 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^5 (6\\ -\\ 8)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (-2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1\\ =\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -\\ 1\\ =\\ (-1)^{3\\ +\\ 3}\\ \\begin{vmatrix}\n1 &amp; -\\ 1 \\\\\n4 &amp; 0 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^6 (0\\ +\\ 4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -\\ 1\\ =\\ 4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cofactor\\ matrix=\\begin{bmatrix}\n-4 &amp; 4 &amp; 4 \\\\\n1 &amp; &#8211; 1 &amp; &#8211; 1 \\\\\n&#8211; 6 &amp; 2 &amp; 4 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Adj.\\ A=\\begin{bmatrix}\n-4 &amp;1 &amp; &#8211; 6 \\\\\n4 &amp; &#8211; 1 &amp; 2 \\\\\n4 &amp; &#8211; 1 &amp; 4 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^{-1} = \\frac{1}{\\begin{vmatrix} A \\\\ \\end{vmatrix}}\\ adj.\\ A\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^{-1} = \\frac{1}{-2}\\ \\begin{bmatrix}\n-4 &amp;1 &amp; &#8211; 6 \\\\\n4 &amp; &#8211; 1 &amp; 2 \\\\\n4 &amp; &#8211; 1 &amp; 4 \\\\\n\\end{bmatrix}\\ \\hspace{2cm}\\]<\/div>\n\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/3RFQMGRBQyU\" title=\"Inverse of a Matrix - Example - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n","protected":false},"excerpt":{"rendered":"<p>SYLLABUS Definition and expansion of second and third order determinants \u2013 Solution of simultaneous equations using Cramer\u2019s rule for 2 and 3 unknowns \u2013 Types of matrices &#8211; Algebra of matrices \u2013 Equality, addition, subtraction, scalar multiplication and multiplication of matrices\u2013 Cofactor matrix \u2013 Adjoint matrix \u2013 Singular and non-singular matrices \u2013 Inverse of a [&hellip;]<\/p>\n","protected":false},"author":187055548,"featured_media":52299,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"set","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2},"_wpas_customize_per_network":false,"jetpack_post_was_ever_published":false},"categories":[711788674,711788673],"tags":[],"class_list":["post-41787","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-basic-mathematics","category-o-scheme"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>DETERMINANTS AND MATRICES (UNIT &#8211; I) - YANAMTAKSHASHILA<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/yanamtakshashila.com\/?p=41787\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"DETERMINANTS AND MATRICES (UNIT &#8211; I) - YANAMTAKSHASHILA\" \/>\n<meta property=\"og:description\" content=\"SYLLABUS Definition and expansion of second and third order determinants \u2013 Solution of simultaneous equations using Cramer\u2019s rule for 2 and 3 unknowns \u2013 Types of matrices &#8211; Algebra of matrices \u2013 Equality, addition, subtraction, scalar multiplication and multiplication of matrices\u2013 Cofactor matrix \u2013 Adjoint matrix \u2013 Singular and non-singular matrices \u2013 Inverse of a [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/yanamtakshashila.com\/?p=41787\" \/>\n<meta property=\"og:site_name\" content=\"YANAMTAKSHASHILA\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/profile.php?id=100063680185552\" \/>\n<meta property=\"article:author\" content=\"https:\/\/www.facebook.com\/profile.php?id=100063680185552\" \/>\n<meta property=\"article:published_time\" content=\"2023-07-06T10:31:56+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-11-19T13:34:33+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/yanamtakshashila.com\/wp-content\/uploads\/2023\/07\/yanamtakshashila.png\" \/>\n\t<meta property=\"og:image:width\" content=\"1545\" \/>\n\t<meta property=\"og:image:height\" content=\"2000\" \/>\n\t<meta property=\"og:image:type\" content=\"image\/png\" \/>\n<meta name=\"author\" content=\"rajuviswa\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rajuviswa\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"36 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=41787#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=41787\"},\"author\":{\"name\":\"rajuviswa\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"headline\":\"DETERMINANTS AND MATRICES (UNIT &#8211; I)\",\"datePublished\":\"2023-07-06T10:31:56+00:00\",\"dateModified\":\"2025-11-19T13:34:33+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=41787\"},\"wordCount\":7172,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"image\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=41787#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2023\\\/07\\\/yanamtakshashila.png?fit=1545%2C2000&ssl=1\",\"articleSection\":[\"Basic Mathematics\",\"O-Scheme\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/yanamtakshashila.com\\\/?p=41787#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=41787\",\"url\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=41787\",\"name\":\"DETERMINANTS AND MATRICES (UNIT &#8211; I) - YANAMTAKSHASHILA\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=41787#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=41787#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2023\\\/07\\\/yanamtakshashila.png?fit=1545%2C2000&ssl=1\",\"datePublished\":\"2023-07-06T10:31:56+00:00\",\"dateModified\":\"2025-11-19T13:34:33+00:00\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=41787#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/yanamtakshashila.com\\\/?p=41787\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=41787#primaryimage\",\"url\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2023\\\/07\\\/yanamtakshashila.png?fit=1545%2C2000&ssl=1\",\"contentUrl\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2023\\\/07\\\/yanamtakshashila.png?fit=1545%2C2000&ssl=1\",\"width\":1545,\"height\":2000},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=41787#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/yanamtakshashila.com\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"DETERMINANTS AND MATRICES (UNIT &#8211; I)\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#website\",\"url\":\"https:\\\/\\\/yanamtakshashila.com\\\/\",\"name\":\"yanamtakshashila.com\",\"description\":\"one stop solutions\",\"publisher\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/yanamtakshashila.com\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":[\"Person\",\"Organization\"],\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\",\"name\":\"rajuviswa\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/LOGO-PNG.png?fit=3600%2C3600&ssl=1\",\"url\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/LOGO-PNG.png?fit=3600%2C3600&ssl=1\",\"contentUrl\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/LOGO-PNG.png?fit=3600%2C3600&ssl=1\",\"width\":3600,\"height\":3600,\"caption\":\"rajuviswa\"},\"logo\":{\"@id\":\"https:\\\/\\\/i0.wp.com\\\/yanamtakshashila.com\\\/wp-content\\\/uploads\\\/2024\\\/12\\\/LOGO-PNG.png?fit=3600%2C3600&ssl=1\"},\"sameAs\":[\"http:\\\/\\\/yanamtakshashila.wordpress.com\",\"https:\\\/\\\/www.facebook.com\\\/profile.php?id=100063680185552\",\"https:\\\/\\\/www.instagram.com\\\/rajuviswa\\\/?hl=en\",\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UCjJ2KWWvsFm6F42UtMdbxzw\"],\"url\":\"https:\\\/\\\/yanamtakshashila.com\\\/?author=187055548\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"DETERMINANTS AND MATRICES (UNIT &#8211; I) - YANAMTAKSHASHILA","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/yanamtakshashila.com\/?p=41787","og_locale":"en_US","og_type":"article","og_title":"DETERMINANTS AND MATRICES (UNIT &#8211; I) - YANAMTAKSHASHILA","og_description":"SYLLABUS Definition and expansion of second and third order determinants \u2013 Solution of simultaneous equations using Cramer\u2019s rule for 2 and 3 unknowns \u2013 Types of matrices &#8211; Algebra of matrices \u2013 Equality, addition, subtraction, scalar multiplication and multiplication of matrices\u2013 Cofactor matrix \u2013 Adjoint matrix \u2013 Singular and non-singular matrices \u2013 Inverse of a [&hellip;]","og_url":"https:\/\/yanamtakshashila.com\/?p=41787","og_site_name":"YANAMTAKSHASHILA","article_publisher":"https:\/\/www.facebook.com\/profile.php?id=100063680185552","article_author":"https:\/\/www.facebook.com\/profile.php?id=100063680185552","article_published_time":"2023-07-06T10:31:56+00:00","article_modified_time":"2025-11-19T13:34:33+00:00","og_image":[{"width":1545,"height":2000,"url":"https:\/\/yanamtakshashila.com\/wp-content\/uploads\/2023\/07\/yanamtakshashila.png","type":"image\/png"}],"author":"rajuviswa","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rajuviswa","Est. reading time":"36 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/yanamtakshashila.com\/?p=41787#article","isPartOf":{"@id":"https:\/\/yanamtakshashila.com\/?p=41787"},"author":{"name":"rajuviswa","@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"headline":"DETERMINANTS AND MATRICES (UNIT &#8211; I)","datePublished":"2023-07-06T10:31:56+00:00","dateModified":"2025-11-19T13:34:33+00:00","mainEntityOfPage":{"@id":"https:\/\/yanamtakshashila.com\/?p=41787"},"wordCount":7172,"commentCount":0,"publisher":{"@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"image":{"@id":"https:\/\/yanamtakshashila.com\/?p=41787#primaryimage"},"thumbnailUrl":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/07\/yanamtakshashila.png?fit=1545%2C2000&ssl=1","articleSection":["Basic Mathematics","O-Scheme"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/yanamtakshashila.com\/?p=41787#respond"]}]},{"@type":"WebPage","@id":"https:\/\/yanamtakshashila.com\/?p=41787","url":"https:\/\/yanamtakshashila.com\/?p=41787","name":"DETERMINANTS AND MATRICES (UNIT &#8211; I) - YANAMTAKSHASHILA","isPartOf":{"@id":"https:\/\/yanamtakshashila.com\/#website"},"primaryImageOfPage":{"@id":"https:\/\/yanamtakshashila.com\/?p=41787#primaryimage"},"image":{"@id":"https:\/\/yanamtakshashila.com\/?p=41787#primaryimage"},"thumbnailUrl":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/07\/yanamtakshashila.png?fit=1545%2C2000&ssl=1","datePublished":"2023-07-06T10:31:56+00:00","dateModified":"2025-11-19T13:34:33+00:00","breadcrumb":{"@id":"https:\/\/yanamtakshashila.com\/?p=41787#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/yanamtakshashila.com\/?p=41787"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/yanamtakshashila.com\/?p=41787#primaryimage","url":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/07\/yanamtakshashila.png?fit=1545%2C2000&ssl=1","contentUrl":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/07\/yanamtakshashila.png?fit=1545%2C2000&ssl=1","width":1545,"height":2000},{"@type":"BreadcrumbList","@id":"https:\/\/yanamtakshashila.com\/?p=41787#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/yanamtakshashila.com\/"},{"@type":"ListItem","position":2,"name":"DETERMINANTS AND MATRICES (UNIT &#8211; I)"}]},{"@type":"WebSite","@id":"https:\/\/yanamtakshashila.com\/#website","url":"https:\/\/yanamtakshashila.com\/","name":"yanamtakshashila.com","description":"one stop solutions","publisher":{"@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/yanamtakshashila.com\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":["Person","Organization"],"@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e","name":"rajuviswa","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/12\/LOGO-PNG.png?fit=3600%2C3600&ssl=1","url":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/12\/LOGO-PNG.png?fit=3600%2C3600&ssl=1","contentUrl":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/12\/LOGO-PNG.png?fit=3600%2C3600&ssl=1","width":3600,"height":3600,"caption":"rajuviswa"},"logo":{"@id":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2024\/12\/LOGO-PNG.png?fit=3600%2C3600&ssl=1"},"sameAs":["http:\/\/yanamtakshashila.wordpress.com","https:\/\/www.facebook.com\/profile.php?id=100063680185552","https:\/\/www.instagram.com\/rajuviswa\/?hl=en","https:\/\/www.youtube.com\/channel\/UCjJ2KWWvsFm6F42UtMdbxzw"],"url":"https:\/\/yanamtakshashila.com\/?author=187055548"}]}},"jetpack_publicize_connections":[],"jetpack_featured_media_url":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2023\/07\/yanamtakshashila.png?fit=1545%2C2000&ssl=1","jetpack_likes_enabled":true,"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/pc3kmt-aRZ","_links":{"self":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/41787","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/users\/187055548"}],"replies":[{"embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=41787"}],"version-history":[{"count":99,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/41787\/revisions"}],"predecessor-version":[{"id":62841,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/41787\/revisions\/62841"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/media\/52299"}],"wp:attachment":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=41787"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=41787"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=41787"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}