{"id":21725,"date":"2021-09-05T11:30:11","date_gmt":"2021-09-05T06:00:11","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=21725"},"modified":"2024-06-30T18:35:35","modified_gmt":"2024-06-30T13:05:35","slug":"largecolor-redchapter-1-2-applications-of-matrices-and-determinants-text","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=21725","title":{"rendered":"CHAPTER\u00a01.2:\u00a0APPLICATIONS\u00a0OF\u00a0MATRICES\u00a0AND\u00a0DETERMINANTS:(Text)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Minor\\ of\\ an\\ element\\ of\\ a\\  Matrix}\\ \\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\">\\[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:\\ 1}\\ \\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\\ 2:}\\ \\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\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\\ 3 :}\\ \\color {red}{Find\\ the\\ cofactor\\ of\\ 4\\ in\\ the\\ following\\ matrix}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ = \\begin{vmatrix}\n1 &amp; 0 &amp; -1 \\\\\n2 &amp; 3 &amp; 4 \\\\\n7 &amp; 8 &amp; -2 \\\\\n\\end{vmatrix}\\ \\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\">\\[cofactor\\ of\\ 4 = (-1)^{2\\ +\\ 3}\\ \\begin{vmatrix}\n1 &amp; 0 \\\\\n7 &amp; 8 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^5 (8 -0)\\ \\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\">\\[\\boxed{cofactor\\ of\\ 4 = &#8211; 8}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 4:} \\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\">\\[\\hspace{5cm}\\ October\\ 2023\\]<\/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\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"4869133702\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\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\\ 5:}\\ \\color{red}{Find\\ the\\ Adjoint\\ Matrix\\ of}\\ \\begin{bmatrix}\n2 &amp; 1 \\\\\n-5 &amp; 3 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ A\\ =\\begin{bmatrix}\n2 &amp; 1 \\\\\n-5 &amp; 3 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = (-1)^{1\\ +\\ 1}\\ (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\\ 2 = \\ 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = (-1)^{1\\ +\\ 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\\ -1 = 5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -5 = (-1)^{2\\ +\\ 1}\\ (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\\ -5 =\\ -\\  1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3 = (-1)^{2\\ +\\ 2}\\ (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\\ 3 = 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ cofactor\\ matrix\\ = \\begin{bmatrix}\n3 &amp; 5 \\\\\n-\\ 1 &amp; 2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Adj.\\ A = \\begin{bmatrix}\n3 &amp; -\\ 1 \\\\\n5 &amp; 2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 6:}\\ \\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\">\\[\\hspace{5cm}\\ Feb\\ 2022\\ ,\\ April\\ 2024\\]<\/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\\ 7:}\\ \\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<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- billboard -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:970px;height:250px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9933277607\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\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\\ 8:}\\ \\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\\ 9:}\\ \\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\">\\[\\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\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 10:}\\ \\color{red}{Find\\ the\\ inverse\\ of\\ the\\ matrix}\\ \\begin{bmatrix}\n3 &amp; 4 &amp; 1 \\\\\n0 &amp; -1 &amp; 2 \\\\\n5 &amp; -2 &amp;  6 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{5cm}\\ June\\ 2022\\ ,\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Let\\ A\\ =\\begin{bmatrix}\n3 &amp; 4 &amp; 1 \\\\\n0 &amp; -1 &amp; 2 \\\\\n5 &amp; -2 &amp;  6 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =\\ 3\\begin{vmatrix}\n-1 &amp; 2 \\\\\n-2 &amp; 6 \\\\\n\\end{vmatrix}\\ -\\ 4\\begin{vmatrix}\n0 &amp; 2 \\\\\n5 &amp; 6 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n0 &amp; -1\\\\\n5 &amp;  -2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 3(- 6\\ +\\ 4)\\ -\\ 4 (0\\ -\\ 10)\\ +\\ 1(0\\ +\\ 5)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 3(-2)\\ -\\ 4 (-10)\\ +\\ 1(5)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-group is-layout-constrained wp-block-group-is-layout-constrained\"><div class=\"wp-block-group__inner-container\">\n<div class=\"wp-block-group is-layout-constrained wp-block-group-is-layout-constrained\"><div class=\"wp-block-group__inner-container\">\n<div class=\"wp-block-group is-layout-constrained wp-block-group-is-layout-constrained\"><div class=\"wp-block-group__inner-container\">\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ -6\\ +\\ 40\\ + 5\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =\\ 39\\ \\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\\ 3\\ = (-1)^{1\\ +\\ 1}\\ \\begin{vmatrix}\n-1 &amp; 2 \\\\\n-2 &amp; 6 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3\\ = (-1)^2\\ (-6\\ +\\ 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\\ 3\\ =\\ -\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 4\\ = (-1)^{1\\ +\\ 2}\\ \\begin{vmatrix}\n0 &amp; 2 \\\\\n5 &amp; 6 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (0\\ -\\ 10)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-group is-layout-constrained wp-block-group-is-layout-constrained\"><div class=\"wp-block-group__inner-container\">\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (-\\ 10)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 4\\ =\\ 10\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = (-1)^{1\\ +\\ 3}\\ \\begin{vmatrix}\n0 &amp; -1 \\\\\n5 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (0\\ +\\ 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\\ 1 =\\ 5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 0\\ =\\ (-1)^{2\\ +\\ 1}\\ \\begin{vmatrix}\n4 &amp; 1 \\\\\n-2 &amp; 6 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (24\\ +\\ 2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (26)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 0\\ =\\ -\\ 26\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -\\ 1\\ =\\ (-1)^{2\\ +\\ 2}\\ \\begin{vmatrix}\n3 &amp; 1 \\\\\n5 &amp; 6 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (18\\ -\\ 5)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (13)\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -\\ 1\\ =\\ 13\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = (-1)^{2\\ +\\ 3}\\ \\begin{vmatrix}\n3 &amp; 4 \\\\\n5 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^5 (-\\ 6 -\\ 20)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (-\\ 26)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 =\\ 26\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 5\\ =\\ (-1)^{3\\ +\\ 1}\\ \\begin{vmatrix}\n4 &amp; 1 \\\\\n-1 &amp; 2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (8\\ +\\ 1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (9)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 5\\ =\\ 9\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ -\\ 2\\ =\\ (-1)^{3\\ +\\ 2}\\ \\begin{vmatrix}\n3 &amp; 1 \\\\\n0 &amp; 2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^5 (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\\ -\\ 2\\ =\\ -\\ 6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 6\\ =\\ (-1)^{3\\ +\\ 3}\\ \\begin{vmatrix}\n3 &amp; 4 \\\\\n0 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^6 (-3\\ -\\ 0)\\ \\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\\ matrix=\\begin{bmatrix}\n-2 &amp; 10 &amp; 5 \\\\\n-26 &amp; 13 &amp; 26 \\\\\n9 &amp; -6 &amp; -3 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Adj.\\ A=\\begin{bmatrix}\n-2 &amp; -26 &amp; 9 \\\\\n10 &amp; 13 &amp; &#8211; 6 \\\\\n5 &amp; 26 &amp; -3 \\\\\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}{39}\\ \\begin{bmatrix}\n-2 &amp; -26 &amp; 9 \\\\\n10 &amp; 13 &amp; -6 \\\\\n5 &amp; 26 &amp; -3 \\\\\n\\end{bmatrix}\\ \\hspace{2cm}\\]<\/div>\n<\/div><\/div>\n<\/div><\/div>\n<\/div><\/div>\n<\/div><\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Ad1 -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:300px;height:600px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8240817448\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Rank\\ of\\ Matrix:}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ A\\ be\\ any\\ m\u00d7n\\  matrix.\\ The\\ order\\ of\\ the\\ largest\\ square\\ sub\\ matrix\\ of\\ A\\ whose\\ determinant\\]\\[ has\\ a\\ non\\ -\\ zero\\ value\\ is\\ known\\ as\\ the\\ rank\\ of\\ the\\ matrix\\ A\\]\\[and\\ is\\ denoted\\ by\\ \\rho(A)\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 11:}\\ \n\\color{red}{Find\\ the\\ rank\\ of\\ the\\ matrix}\\ \\begin{bmatrix}\n5 &amp; 2 \\\\\n6 &amp; 3 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ A\\ =\\begin{bmatrix}\n5 &amp; 2 \\\\\n6 &amp; 3 \\\\\n\\end{bmatrix}\\ \\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Order\\ of\\ A = 2 \u00d7 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore \\ \\rho(A)\\ \\leq  2 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ higher\\ order\\ of\\ minor\\ of\\ A = 2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ minor\\ is\\ \\begin{vmatrix}\n5 &amp; 2 \\\\\n6 &amp; 3 \\\\\n\\end{vmatrix}\\ =\\ 15 &#8211; 12\\ =\\ 3\\ \\neq\\ 0\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\therefore\\ Rank\\ of\\ A\\ =\\ 2}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 12:}\\ \\color {red}{Find\\ the\\ rank\\ of\\ the\\ matrix}\\ \\begin{bmatrix}\n1 &amp; 2 \\\\\n1 &amp; 2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ A\\ =\\begin{bmatrix}\n1 &amp; 2 \\\\\n1 &amp; 2 \\\\\n\\end{bmatrix}\\ \\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Order\\ of\\ A = 2 \u00d7 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore \\ \\rho(A)\\ \\leq  2 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ higher\\ order\\ of\\ minor\\ of\\ A = 2\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ minor\\ is\\ \\begin{vmatrix}\n1 &amp; 2 \\\\\n1 &amp; 2 \\\\\n\\end{vmatrix}\\ =\\ 6- 6\\ =\\ 0\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore \\ \\rho(A)\\ \\neq  2 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[To\\ find\\ at\\ least\\ one\\ non\\ zero\\ first\\ order\\ minor\\ i.e.\\ to\\ find\\ at\\ least\\ one\\ non\\ zero\\ element\\]\\[of\\ A\\ ,\\ non\\ zero\\ element\\ exist\\ in\\ A\\ .\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Rank\\ =\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\therefore \\ \\rho(A)\\ =\\  1} \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 13:}\\ \\color {red}{Find\\ the\\ rank\\ of\\ the\\ matrix}\\ \\begin{bmatrix}\n-1 &amp; 2 &amp; 3 \\\\\n0 &amp; 3 &amp; 1 \\\\\n4 &amp; 5 &amp; 2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ A\\ =\\begin{bmatrix}\n-1 &amp; 2 &amp; 3 \\\\\n0 &amp; 3 &amp; 1 \\\\\n4 &amp; 5 &amp; 2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Order\\ of\\ A = 3 \u00d7 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore \\ \\rho(A)\\ \\leq  3 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ higher\\ order\\ of\\ minor\\ of\\ A = 3\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ minor\\ is\\ =\\ \\begin{vmatrix}\n-1 &amp; 2 &amp; 3 \\\\\n0 &amp; 3 &amp; 1 \\\\\n4 &amp; 5 &amp; 2 \\\\\n\\end{vmatrix}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=-1\\begin{vmatrix}\n3 &amp; 1 \\\\\n5 &amp; 2 \\\\\n\\end{vmatrix}\\ -\\ 2\\begin{vmatrix}\n0 &amp; 1 \\\\\n4 &amp; 2 \\\\\n\\end{vmatrix}\\ +\\ 3\\begin{vmatrix}\n0 &amp; 3\\\\\n4 &amp;  5 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =-1(6\\ -\\ 5)\\ &#8211; 2 (0\\ -\\ 4) + 3(0\\ -\\ 12)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =-1(1)\\ &#8211; 2 (-4) + 3(-12)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = -1\\ + 8 &#8211; 36\\ =\\ -29\\ \\neq\\ 0\\\n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\therefore\\ rank\\ of\\ A=\\ \\rho(A)\\ =\\ 3}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 14:}\\ \\color {red}{Find\\ the\\ rank\\ of\\ the\\ matrix}\\ \\begin{pmatrix}\n1 &amp; 2 &amp; -1  &amp; 3 \\\\\n2 &amp; 4 &amp; -4 &amp; 7 \\\\\n-1 &amp; -2 &amp; -2 &amp; -2 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\hspace{18cm}\\ June\\ 2022\\ ,\\ April\\ 2024\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ A\\ =\\begin{pmatrix}\n1 &amp; 2 &amp; -1  &amp; 3 \\\\\n2 &amp; 4 &amp; -4 &amp; 7 \\\\\n-1 &amp; -2 &amp; -2 &amp; -2 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Order\\ of\\ A = 3 \u00d7 4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore \\ rank\\ of\\ A=\\rho(A)\\ \\leq\\ Min{3,4} =3\\ . \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ highest\\ order\\ of\\ minors\\ of\\ A = 3.\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ has\\ the\\ following\\ minors\\ of\\ order 3.\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ A_1\\ =\\begin{vmatrix}\n1 &amp; 2 &amp; -1 \\\\\n2 &amp; 4 &amp; -4 \\\\\n-1 &amp; -2 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=1\\begin{vmatrix}\n4 &amp; -4 \\\\\n-2 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 2\\begin{vmatrix}\n2 &amp; -4 \\\\\n-1 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 4\\\\\n-1 &amp;  -2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(-8\\ -\\ 8)\\ &#8211; 2 (-4\\ -\\ 4) &#8211; 1(-4\\ +\\ 4)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(-16)\\ &#8211; 2 (-8) -1(0)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = -16\\ +\\ 16\\ -\\ 0\\ =\\ 0\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ A_2\\ =\\begin{vmatrix}\n1 &amp; 2 &amp; 3 \\\\\n2 &amp; 4 &amp; 7 \\\\\n-1 &amp; -2 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=1\\begin{vmatrix}\n4 &amp; 7 \\\\\n-2 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 2\\begin{vmatrix}\n2 &amp; 7 \\\\\n-1 &amp; -2 \\\\\n\\end{vmatrix}\\ +\\ 2\\begin{vmatrix}\n2 &amp; 3\\\\\n1 &amp;  3 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(-8\\ +\\ 14)\\ &#8211; 2 (-4\\ +\\ 7) +\\ 3(-4\\ +\\ 4)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(6)\\ &#8211; 2 (3)\\ +\\ 3(0)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 6\\ -\\ 6\\ + 0\\ =\\ 0\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ A_3\\ =\\begin{bmatrix}\n1 &amp; -1 &amp; 3 \\\\\n2 &amp; -4 &amp; 7 \\\\\n-1 &amp; -2 &amp; -2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1\\begin{vmatrix}\n-4 &amp; 7 \\\\\n-2 &amp; -2 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; 7 \\\\\n-1 &amp; -2 \\\\\n\\end{vmatrix}\\ +\\ 3\\begin{vmatrix}\n2 &amp; -4\\\\\n-1 &amp;  -2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 1(8\\ +\\ 14)\\ +\\ 1 (-4\\ +\\ 7)\\ + 3(-4\\ -\\ 4)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 1(22)\\ +\\ 1 (3)\\  +\\  3(-8)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 22\\ +\\ 3\\ -\\ 24\\ =\\ 1\\ \\neq\\ 0\\\n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ A_4\\ =\\begin{bmatrix}\n2 &amp; -1 &amp; 3 \\\\\n4 &amp; -4 &amp; 7 \\\\\n-2 &amp; -2 &amp; -2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 2\\begin{vmatrix}\n-4 &amp; 7 \\\\\n-2 &amp; -2 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n4 &amp; 7 \\\\\n-2 &amp; -2 \\\\\n\\end{vmatrix}\\ +\\ 3\\begin{vmatrix}\n4 &amp; -4\\\\\n-2 &amp;  -2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 2(8\\ +\\ 14)\\ + 1 (-8\\ +\\ 14)\\ +\\ 3(-8\\ -\\ 8)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 2(22)\\ +\\ 1 (6)\\ +\\ 3(-16)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =\\ 44\\ +\\ 6\\ -\\  48\\ =\\ 2\\ \\neq\\ 0\\\n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\therefore\\ rank\\ of\\ A=\\ \\rho(A)\\ =\\ 3}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 15:}\\ \\color {red}{Find\\ the\\ rank\\ of\\ the\\ matrix}\\ \\begin{bmatrix}\n1 &amp; 2 &amp; 3  &amp; 2 \\\\\n2 &amp; 3 &amp; 5 &amp; 1 \\\\\n1 &amp; 3 &amp; 4 &amp; 5 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ A\\ =\\begin{bmatrix}\n1 &amp; 2 &amp; 3  &amp; 2 \\\\\n2 &amp; 3 &amp; 5 &amp; 1 \\\\\n1 &amp; 3 &amp; 4 &amp; 5 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Order\\ of\\ A = 3 \u00d7 4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore \\ rank\\ of\\ A=\\rho(A)\\ \\leq\\ Min{3,4} =3\\ . \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ highest\\ order\\ of\\ minors\\ of\\ A = 3.\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A\\ has\\ the\\ following\\ minors\\ of\\ order 3.\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ A_1\\ =\\begin{bmatrix}\n1 &amp; 2 &amp; 3 \\\\\n2 &amp; 3 &amp; 5 \\\\\n1 &amp; 3 &amp; 4 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=1\\begin{vmatrix}\n3 &amp; 5 \\\\\n3 &amp; 4 \\\\\n\\end{vmatrix}\\ -\\ 2\\begin{vmatrix}\n2 &amp; 5 \\\\\n1 &amp; 4 \\\\\n\\end{vmatrix}\\ +\\ 3\\begin{vmatrix}\n2 &amp; 3\\\\\n1 &amp;  3 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(12\\ -\\ 15)\\ &#8211; 2 (8\\ -\\ 5) + 3(6\\ -\\ 3)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(-3)\\ &#8211; 2 (3) + 3(3)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = -3\\ &#8211; 6 + 9\\ =\\ 0\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ A_2\\ =\\begin{bmatrix}\n1 &amp; 2 &amp; 2 \\\\\n2 &amp; 3 &amp; 1 \\\\\n1 &amp; 3 &amp; 5 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=1\\begin{vmatrix}\n3 &amp; 1 \\\\\n3 &amp; 5 \\\\\n\\end{vmatrix}\\ -\\ 2\\begin{vmatrix}\n2 &amp; 1 \\\\\n1 &amp; 5 \\\\\n\\end{vmatrix}\\ +\\ 2\\begin{vmatrix}\n2 &amp; 3\\\\\n1 &amp;  3 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(15\\ -\\ 3)\\ &#8211; 2 (10\\ -\\ 1) + 2(6\\ -\\ 3)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(12)\\ &#8211; 2 (9) + 2(3)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 12\\ &#8211; 18 + 6\\ =\\ 0\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ A_3\\ =\\begin{bmatrix}\n2 &amp; 3 &amp; 2 \\\\\n3 &amp; 5 &amp; 1 \\\\\n3 &amp; 4 &amp; 5 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=2\\begin{vmatrix}\n5 &amp; 1 \\\\\n4 &amp; 5 \\\\\n\\end{vmatrix}\\ -\\ 3\\begin{vmatrix}\n3 &amp; 1 \\\\\n3 &amp; 5 \\\\\n\\end{vmatrix}\\ +\\ 2\\begin{vmatrix}\n3 &amp; 5\\\\\n3 &amp;  4 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =2(25\\ -\\ 4)\\ &#8211; 3 (15\\ -\\ 3) + 2(12\\ -\\ 15)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =2(21)\\ &#8211; 3 (12) + 2(-3)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 42\\ &#8211; 36 &#8211; 6\\ =\\ 0\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ A_4\\ =\\begin{bmatrix}\n1 &amp; 3 &amp; 2 \\\\\n2 &amp; 5 &amp; 1 \\\\\n1 &amp; 4 &amp; 5 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=1\\begin{vmatrix}\n5 &amp; 1 \\\\\n4 &amp; 5 \\\\\n\\end{vmatrix}\\ -\\ 3\\begin{vmatrix}\n2 &amp; 1 \\\\\n1 &amp; 5 \\\\\n\\end{vmatrix}\\ +\\ 2\\begin{vmatrix}\n2 &amp; 5\\\\\n1 &amp;  4 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(25\\ -\\ 4)\\ &#8211; 3 (10\\ -\\ 1) + 2(8\\ -\\ 5)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =1(21)\\ &#8211; 3 (9) + 2(3)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ = 21\\ &#8211; 27 + 6\\ =\\ 0\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[All\\ third\\ order\\ minors\\ vanish\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ \\rho(A)\\ \\lt 3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[To\\ find\\ at\\ least\\ a\\ non\\ zero\\ of\\ order\\ 2 \u00d7  2\\ \\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\n1 &amp; 3 \\\\\n2 &amp; 5\\\\\n\\end{vmatrix}\\ =\\ 5- 6\\ =\\ -1\\ \\neq\\ 0\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ A\\ has\\ at\\ least\\ one\\ non\\ zero\\ minor\\ of\\ order\\ 2\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\therefore\\ \\rho(A)\\  =2}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {eigen\\ value\\ of\\ a\\ Matrix:}\\ \\hspace{15cm}\\] <\/div>\n\n\n\n<p>An eigenvalue of a matrix is a special number associated with the matrix that provides significant insights into its properties and behaviors. Formally, for a square matrix&nbsp;A, a scalar&nbsp;\u03bb&nbsp;is called an eigenvalue if there exists a nonzero vector&nbsp;v&nbsp;(known as an eigenvector) such that:<\/p>\n\n\n\n<p>Av=\u03bbv<\/p>\n\n\n\n<p>In other words, the transformation described by the matrix&nbsp;AA&nbsp;scales the vector&nbsp;vv&nbsp;by the factor&nbsp;\u03bb\u03bb. The equation above can be rearranged to:<\/p>\n\n\n\n<p>(A\u2212\u03bbI)v=0<\/p>\n\n\n\n<p>where&nbsp;II&nbsp;is the identity matrix of the same dimension as&nbsp;A. For nontrivial solutions (where&nbsp;v\u22600), the determinant of&nbsp;(A\u2212\u03bbI)(A\u2212\u03bbI)&nbsp;must be zero:<\/p>\n\n\n\n<p>det(A\u2212\u03bbI)=0<\/p>\n\n\n\n<p>This equation is known as the characteristic equation, and solving it yields the eigenvalues&nbsp;\u03bb&nbsp;of the matrix&nbsp;A. The eigenvalues are important in various applications such as stability analysis, vibration analysis, facial recognition systems, and quantum mechanics.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 16:}\\ \\color {red}{Define\\ an\\ eigen\\ value\\ of\\ a\\ matrix}\\  \\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\">\\[\\text{For a square matrix&nbsp;A, a scalar&nbsp;\u03bb&nbsp;is called an eigenvalue if there exists a nonzero Eigen vector&nbsp;v}\\\\ \\text&nbsp;{such that: Av = \u03bbv}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- display h -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"3583972194\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\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\\ 15:}\\ \\color{red}{Solve\\ the\\ following\\ equations\\ using\\ Cramers\\ Rule}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x + 2y &#8211; z=-1,\\ 3x + 8y + 2z = 28\\ and\\ 4x + 9y &#8211; z = 14\\ \\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\">\\[x + 2y &#8211; z = &#8211; 1\\ &#8212;&#8212;&#8212;&#8212;&#8212; (1) \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3x + 8y + 2z = 28\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4x + 9y &#8211; z = 14\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta = \\begin{vmatrix}\n1 &amp; 2 &amp; -1 \\\\\n3 &amp; 8 &amp; 2 \\\\\n4 &amp; 9 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =1\\begin{vmatrix}\n8 &amp; 2 \\\\\n9 &amp; &#8211; 1 \\\\\n\\end{vmatrix}\\ -\\ 2\\begin{vmatrix}\n3 &amp; 2 \\\\\n4 &amp; -1 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n3 &amp; 8\\\\\n4 &amp;  9 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =1(-8\\ -\\ 18)\\ &#8211; 2 (-3\\ -\\ 8) &#8211; 1(27\\ -\\ 32)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =1(-26)\\ &#8211; 2 (-11) &#8211; 1(-5)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =-26\\ +22 + 5\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta =1}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x = \\begin{vmatrix}\n-1 &amp; 2 &amp; -1 \\\\\n28 &amp; 8 &amp; 2 \\\\\n14 &amp; 9 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =-1\\begin{vmatrix}\n8 &amp; 2 \\\\\n9 &amp; -1 \\\\\n\\end{vmatrix}\\ -\\ 2\\begin{vmatrix}\n28 &amp; 2 \\\\\n14 &amp;  -1 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n28 &amp; 8\\\\\n14 &amp; 9 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =-1(-8\\ -\\ 18)\\ &#8211; 2 (-28\\ -\\ 28) &#8211; 1(252\\ -\\ 112)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =-1(-26)\\ &#8211; 2 (-56) -1 (140)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =26\\ + 112 &#8211; 140\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_x = -2}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y = \\begin{vmatrix}\n1 &amp; -1 &amp; -1 \\\\\n3 &amp; 28 &amp; 2 \\\\\n4 &amp; 14 &amp; -1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y =1\\begin{vmatrix}\n28 &amp; 2 \\\\\n14 &amp; -1 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n3 &amp; 2 \\\\\n4 &amp; -1 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n3 &amp; 28\\\\\n4 &amp;  14 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y =1(-28\\ -\\ 28)\\ + 1 (-3\\ -\\ 8) &#8211; 1(42\\ -\\ 112)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y =1(-56)\\ + 1 (-11) &#8211; 1(-70)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y = -56\\ -11\\ + 70\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_y =3}\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z = \\begin{vmatrix}\n1 &amp; 2 &amp; &#8211; 1 \\\\\n3 &amp; 8 &amp; 28 \\\\\n4 &amp; 9 &amp; 14 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z =1\\begin{vmatrix}\n8 &amp; 28 \\\\\n9 &amp; 14 \\\\\n\\end{vmatrix}\\ -\\ 2\\begin{vmatrix}\n3 &amp; 28 \\\\\n4 &amp; 14 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n3 &amp; 8\\\\\n4 &amp;  9 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z =1(112\\ -\\ 252)\\ &#8211; 2 (42\\ -\\ 112) &#8211; 1(27\\ -\\ 32)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z =1(-140)\\ &#8211; 2 (-70) &#8211; 1(-5)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z =-140\\ + 140 + 5\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\Delta_z =5}\\ \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{-2}{1} =\\ -2\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y=\\ \\frac{\\Delta_y}{\\Delta} =\\ \\frac{3}{1} =\\ 3\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[z=\\ \\frac{\\Delta_z}{\\Delta} =\\ \\frac{5}{1} =\\ 5\\ \\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 =-2\\ y = 3\\ z = 5\\ in\\ equation (1)\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[LHS = -2 + 2(3) &#8211; 5\\]\\[ = -2 + 6 &#8211; 5 = -1\\]\\[ = RHS\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {violet}{Example\\ 16:}\\ \\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\\ 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\">\\[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\"><\/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\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block; text-align:center;\" data-ad-layout=\"in-article\" data-ad-format=\"fluid\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"6565358754\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<h3 class=\"wp-block-heading has-text-align-center has-vivid-purple-color has-text-color\">Exercise Problems<\/h3>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- A}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ \\color {red}{Find\\ the\\ cofactor\\ of\\ 3\\ in\\ the\\ matrix}\\ \\begin{bmatrix}\n1 &amp; 2 &amp; 0 \\\\\n-1 &amp; 3 &amp; 4 \\\\\n5 &amp; 6 &amp; 7 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ \\color {red}{Find\\ the\\ Adjoint\\ matrix\\ of}\\ \\begin{bmatrix}\n2 &amp; 2 \\\\\n3 &amp; -5 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3.\\ \\color {red}{Find\\ the\\ Adjoint\\ matrix\\ of}\\ \\begin{bmatrix}\n3 &amp; -4 \\\\\n2 &amp; 5 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[4.\\ \\color {red}{Find\\ the\\ Adjoint\\ matrix\\ of}\\ \\begin{bmatrix}\n1 &amp; 4 \\\\\n5 &amp; -2 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[5.\\ \\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\">\\[6.\\ \\color {red}{Find\\ the\\ rank\\ of}\\ \\begin{bmatrix}\n3 &amp; -4 \\\\\n-6 &amp; 8 \\\\\n\\end{bmatrix}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- display s -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"4398623749\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- B}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7.\\  \\color {red}{Find\\ the\\ inverse\\ of}\\ \\begin{bmatrix}\n2 &amp; -1 \\\\\n4 &amp; 5 \\\\\n\\end{bmatrix}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[8.\\  \\color {red}{Find\\ the\\ inverse\\ of}\\ \\begin{bmatrix}\n2 &amp; 1 \\\\\n1 &amp; 2 \\\\\n\\end{bmatrix}\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} { 9\\ .}\\ \\color {red}{Find\\ the\\ rank\\ of}\\ \\ \\begin{bmatrix}\n1 &amp; 2 &amp; 3 \\\\\n2 &amp; 4 &amp; 6 \\\\\n3 &amp; -1 &amp; 4 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"fluid\" data-ad-layout-key=\"-6x+db-3g-5s+t1\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"3292591486\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {PART- C}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red} {10.\\ By\\ using\\ Cramers\\ Rule\\,\\ Solve\\ the\\ equations}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x\\ +\\  y\\ +\\  z\\ =\\ 2,\\ 2x\\ -\\ y\\ -\\ 2z\\ =\\ -1\\ and\\ x\\ -\\  2y\\ -\\ z\\ =\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color{red}{11.\\ Solve\\ the\\ following\\ equations\\ using\\ Cramers\\ Rule}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3x\\ -\\ y\\ +\\ 2z\\ =\\ 8,\\ x\\ +\\ y\\ +\\ z\\ =\\ 2\\ and\\ 2x\\ +\\ y\\ -\\ z\\ =\\ -\\ 1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[12.\\ \\color {red}{Find\\ the\\ inverse\\ of}\\ \\begin{bmatrix}\n2 &amp; 3 &amp; 4 \\\\\n4 &amp; 3 &amp; 1 \\\\\n1 &amp; 2 &amp; 4 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[13.\\ \\color {red}{Find\\ the\\ inverse\\ of}\\ \\begin{bmatrix}\n1 &amp; 2 &amp; -1\\\\ \n3 &amp; 8 &amp; 2 \\\\\n4 &amp; 9 &amp; 1 \\\\ \\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[14.\\ \\color{red}{Find\\ the\\ inverse\\ of}\\ \\begin{bmatrix}\n1 &amp; 1 &amp; -\\ 1 \\\\\n2 &amp; 1 &amp; 0 \\\\\n-1 &amp; 2 &amp;  3 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {15}\\ \\color {red}{Find\\ the\\ rank\\ of\\ the\\ matrix}\\ \\begin{pmatrix}\n1 &amp; 2 &amp; 3  &amp; 1 \\\\\n2 &amp; 3 &amp; 4 &amp; 2 \\\\\n3 &amp; 4 &amp; 5 &amp; 1 \\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/cse.google.com\/cse.js?cx=2f7b32630c6dfd4fb\"><\/script>\n<div class=\"gcse-search\"><\/div>\n","protected":false},"excerpt":{"rendered":"<p>An eigenvalue of a matrix is a special number associated with the matrix that provides significant insights into its properties and behaviors. Formally, for a square matrix&nbsp;A, a scalar&nbsp;\u03bb&nbsp;is called an eigenvalue if there exists a nonzero vector&nbsp;v&nbsp;(known as an eigenvector) such that: Av=\u03bbv In other words, the transformation described by the matrix&nbsp;AA&nbsp;scales the vector&nbsp;vv&nbsp;by [&hellip;]<\/p>\n","protected":false},"author":187055548,"featured_media":0,"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":[711788205,711788227,711787788,711788223],"tags":[],"class_list":["post-21725","post","type-post","status-publish","format-standard","hentry","category-academics","category-application-of-matrices-determinants","category-n-scheme","category-n-unit-i-algebra"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>CHAPTER\u00a01.2:\u00a0APPLICATIONS\u00a0OF\u00a0MATRICES\u00a0AND\u00a0DETERMINANTS:(Text) - 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=21725\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"CHAPTER\u00a01.2:\u00a0APPLICATIONS\u00a0OF\u00a0MATRICES\u00a0AND\u00a0DETERMINANTS:(Text) - YANAMTAKSHASHILA\" \/>\n<meta property=\"og:description\" content=\"An eigenvalue of a matrix is a special number associated with the matrix that provides significant insights into its properties and behaviors. 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