{"id":21036,"date":"2021-08-03T17:13:21","date_gmt":"2021-08-03T11:43:21","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=21036"},"modified":"2024-06-30T13:09:38","modified_gmt":"2024-06-30T07:39:38","slug":"unit-i-algebra","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=21036","title":{"rendered":"MATRICES AND DETERMINANTS (Text)"},"content":{"rendered":"\n<p class=\"has-text-align-justify has-arial-font-family has-custom-font\" style=\"font-family:Arial\">Matrix and its applications are very important part of Mathematics.  Also it is one of the most powerful tools of Mathematics.<\/p>\n\n\n\n<p class=\"has-text-align-justify has-arial-font-family has-custom-font\" style=\"font-family:Arial\">Matrix notation and operation are used in electronic spread sheet programmes for personal computer, business budgeting, sales projection, cost estimation, analyzing the results of an experiment etc. Also many physical operations such as magnification, rotation and reflection through a plane can be represented mathematically by matrices. Also matrix used in Cryptography.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Definition\\ 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<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 {royalblue} {Type\\ of\\ Matrix}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {1.\\ Row\\ Matrix}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<p>A matrix having only one row and any number of columns is called a row matrix.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {Eg:}\\ A =\\begin{bmatrix}\n1 &amp; 2 &amp; -3 \\\\\n\\end{bmatrix}\\ is\\ a\\ matrix\\ of\\ order\\ 1\u00d73 \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {2.\\ Column\\ Matrix}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<p>A matrix having only one column and any number of rows is called a column matrix.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {Eg:}\\ A =\\begin{bmatrix}\n3 \\\\\n-1\\\\\n5\\\\\n\\end{bmatrix}\\ is\\ a\\ matrix\\ of\\ order\\ 3\u00d71 \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {3.\\ Null\\ or\\ zero\\ Matrix}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<p>If all the elements of a matrix are zero, the matrix is called zero or null matrix.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {Eg:}\\ 0 =\\begin{bmatrix}\n0 &amp; 0 &amp; 0 \\\\\n0 &amp; 0 &amp; 0 \\\\\n0 &amp; 0 &amp; 0 \\\\\n\\end{bmatrix}\\ is\\ a\\ zero\\ matrix\\ of\\ order\\ 3\u00d73 \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {4.\\ 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} {5.\\ Triangular\\ Matrix}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {(a)\\ Upper\\ Triangular\\ Matrix}:\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<p>In a square matrix if all the elements below the leading diagonal are zero is called Upper Triangular Matrix.<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {Eg:}\\  A =\\begin{bmatrix}\n1 &amp; 4 &amp; 2 \\\\\n0 &amp; 3 &amp; 4 \\\\\n0 &amp; 0 &amp; 1 \\\\\n\\end{bmatrix}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {(b)\\ Lower\\ Triangular\\ Matrix}:\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<p> In a square matrix if all the elements above the leading diagonal are zero is called Lower Triangular Matrix. <\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {Eg:}\\  B =\\begin{bmatrix}\n1 &amp; 0 &amp; 0 \\\\\n2 &amp; 8 &amp; 0 \\\\\n4 &amp; 9 &amp; 7 \\\\\n\\end{bmatrix}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {6.\\ 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} {7.\\ 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\"><\/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 class=\"has-text-align-justify\">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\\ 1:}\\ \\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\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 2:}\\ \\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\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 3:}\\ \\color {red}{If}\\ A =\\begin{bmatrix}\n2 &amp; -3 &amp; 8 \\\\\n21 &amp; 6 &amp; -6 \\\\\n4 &amp; -33 &amp; 19 \\\\\n\\end{bmatrix}\\ ,\\ B =\\begin{bmatrix}\n1 &amp; -29 &amp; -8 \\\\\n2 &amp; 0 &amp; 3 \\\\\n17 &amp; 15 &amp; 4 \\\\\n\\end{bmatrix}\\ \\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\\ 2023\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Given\\  A =\\begin{bmatrix}\n2 &amp; -3 &amp; 8 \\\\\n21 &amp; 6 &amp; -6 \\\\\n4 &amp; -33 &amp; 19 \\\\\n\\end{bmatrix}\\ ,\\ B =\\begin{bmatrix}\n1 &amp; -29 &amp; -8 \\\\\n2 &amp; 0 &amp; 3 \\\\\n17 &amp; 15 &amp; 4 \\\\\n\\end{bmatrix}\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A + B=\\begin{bmatrix}\n2 + 1 &amp; -3 &#8211; 29 &amp; 8 &#8211; 8 \\\\\n21 + 2 &amp; 6 + 0 &amp; -6 + 3 \\\\\n4 + 17 &amp; -33 + 15 &amp; 19 + 4 \\\\\n\\end{bmatrix}\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A + B=\\begin{bmatrix}\n3 &amp; -32 &amp; 0 \\\\\n23 &amp; 6 &amp; -3\\\\\n21 &amp; -18  &amp; 23\\\\\n\\end{bmatrix}\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(A + B)^T\\ =\\ \\begin{bmatrix}\n3 &amp; 23 &amp; 21 \\\\\n-32 &amp; 6 &amp; &#8211; 18 \\\\\n0 &amp; -3  &amp; 23\\\\\n\\end{bmatrix}\\ \\hspace{2cm}\\ &#8212;&#8212;&#8211; (1)\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^T =\\begin{bmatrix}\n2 &amp; 21 &amp; 4 \\\\\n-3 &amp; 6 &amp; -33 \\\\\n8 &amp; -6 &amp; 19 \\\\\n\\end{bmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[B^T =\\begin{bmatrix}\n1 &amp; 2 &amp; 17 \\\\\n-29 &amp; 0 &amp; 15 \\\\\n&#8211; 8 &amp; 3 &amp; 4 \\\\\n\\end{bmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^T\\ +\\ B^T\\ =\\begin{bmatrix}\n2 + 1 &amp; 21\\ +\\ 2 &amp;  4\\ +\\ 17\\\\\n-3\\ -\\ 29 &amp; 6 + 0 &amp; -33 + 15 \\\\\n8 &#8211; 8 &amp; -6 + 3 &amp; 19 + 4 \\\\\n\\end{bmatrix}\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[A^T\\ +\\ B^T\\ =\\begin{bmatrix}\n3 &amp; 23 &amp;  21\\\\\n-32 &amp; 6 &amp; &#8211; 18 \\\\\n0 &amp; -3 &amp; 23 \\\\\n\\end{bmatrix}\\ \\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\"><\/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\\ 4:}\\ \\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\">\\[\\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\\ 5:}\\ \\color {red}{If}\\ A =\\begin{bmatrix}\n4 &amp; 3 &amp; 2\\\\\n5 &amp; 1 &amp; 0 \\\\\n7 &amp; 2 &amp; 8 \\\\\n\\end{bmatrix}\\  and\\ B =\\begin{bmatrix}\n-3 &amp; 1 &amp; 0 \\\\\n2 &amp; 7 &amp; 1 \\\\\n4 &amp; 3 &amp; 5\\\\\n\\end{bmatrix}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red}{Then\\ Find}\\ 2A\\ and\\ 7B\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ Given\\  A =\\begin{bmatrix}\n4 &amp; 3 &amp; 2\\\\\n5 &amp; 1 &amp; 0 \\\\\n7 &amp; 2 &amp; 8 \\\\\n\\end{bmatrix}\\ ,\\ B =\\begin{bmatrix}\n-3 &amp; 1 &amp; 0 \\\\\n2 &amp; 7 &amp; 1 \\\\\n4 &amp; 3 &amp; 5\\\\\n\\end{bmatrix}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2A = 2 \\begin{bmatrix}\n4 &amp; 3 &amp; 2\\\\\n5 &amp; 1 &amp; 0 \\\\\n7 &amp; 2 &amp; 8 \\\\\n\\end{bmatrix}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\begin{bmatrix}\n2 \u00d7 4 &amp; 2 \u00d7 3 &amp; 2 \u00d7 2\\\\\n2 \u00d7 5 &amp; 2 \u00d7 1 &amp; 2 \u00d7 0 \\\\\n2 \u00d7 7 &amp; 2 \u00d7 2 &amp; 2 \u00d7 8 \\\\\n\\end{bmatrix}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ 2A= \\begin{bmatrix}\n8 &amp; 6 &amp; 4\\\\\n10 &amp; 2  &amp; 0 \\\\\n14 &amp; 4 &amp; 16 \\\\\n\\end{bmatrix}\\ \\hspace{12cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[7B = 7 \\begin{bmatrix}\n-3 &amp; 1 &amp; 0\\\\\n2 &amp; 7 &amp; 1 \\\\\n4 &amp; 3 &amp; 5 \\\\\n\\end{bmatrix}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= \\begin{bmatrix}\n7 \u00d7 -3 &amp; 7 \u00d7 1 &amp; 7 \u00d7 0\\\\\n7 \u00d7 2 &amp; 7 \u00d7 7 &amp; 7 \u00d7 1 \\\\\n7\u00d7 4 &amp; 7 \u00d7 3 &amp; 7 \u00d7 5 \\\\\n\\end{bmatrix}\\ \\hspace{10cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\"><\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ 7B= \\begin{bmatrix}\n-21 &amp; 7 &amp; 0\\\\\n14 &amp; 49  &amp; 7 \\\\\n28 &amp; 21 &amp; 35 \\\\\n\\end{bmatrix}\\ \\hspace{12cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 6:}\\ \\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\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\\ 7:}\\ \\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\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 8:}\\ \\color {Red}{If}\\ A =\\begin{pmatrix}\n6 &amp; 5 \\\\\n-3 &amp; 8 \\\\\n\\end{pmatrix}\\ and\\ B =\\begin{pmatrix}\n2 &amp; 3 \\\\\n4 &amp; -1 \\\\\n\\end{pmatrix}\\ ,\\ \\color {red} {Find\\ BA}\\ \\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\">\\[BA =\\begin{pmatrix}\n2 &amp; 3 \\\\\n4 &amp; -1 \\\\\n\\end{pmatrix}\\  \\begin{pmatrix}\n6 &amp; 5 \\\\\n-3 &amp; 8 \\\\\n\\end{pmatrix}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{pmatrix}\n2 \u00d7 6 + 3 \u00d7 -3 &amp; 2 \u00d7 5 + 3 \u00d7 8 \\\\\n4 \u00d7 6 + -1 \u00d7 -3 &amp; 4 \u00d7 5 + -1 \u00d7 8\\\\\n\\end{pmatrix}\\  \\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[  = \\begin{pmatrix}\n12 &#8211; 9&amp; 10 + 24\\\\\n24 + 3&amp; 20 &#8211; 8\\\\\n\\end{pmatrix}\\  \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{BA\\ =\\ \\begin{bmatrix}\n3 &amp;  34\\\\\n27&amp; 12\\\\\n\\end{bmatrix}}\\  \\hspace{12cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Sidebar ad -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:336px;height:280px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"3663062588\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} { DETERMINANTS}\\ \\hspace{20cm}\\] <\/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\\ 9:}\\ \\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\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 10:}\\ \\color {red}{Solve}\\ \\begin{vmatrix}\nx &amp; 2 \\\\\n2 &amp; x \\\\\n\\end{vmatrix}\\ = 0\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\begin{vmatrix}\nx &amp; 2 \\\\\n2 &amp; x \\\\\n\\end{vmatrix}\\ = 0 \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x(x) &#8211; 2 (2) = 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 {pruple} {Example\\ 11:}\\ \\color {red}{Evaluate}\\ \\begin{vmatrix}\nsin\\ \\theta &amp; -\\ cos\\ \\theta \\\\\ncos\\ \\theta &amp; sin\\ \\theta \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\ \\begin{vmatrix}\nsin\\ \\theta &amp; -\\ cos\\ \\theta \\\\\ncos\\ \\theta &amp; sin\\ \\theta \\\\\n\\end{vmatrix}\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ sin\\ \\theta(sin\\ \\theta)\\ -\\  cos\\ \\theta (-\\ cos\\ \\theta)\\  \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ sin^2\\ \\theta\\ +\\ cos^2\\ \\theta \\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[=\\ 1\\ \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\boxed{\\therefore \\begin{vmatrix}\nsin\\ \\theta &amp; -\\ cos\\ \\theta \\\\\ncos\\ \\theta &amp; sin\\ \\theta \\\\\n\\end{vmatrix}\\ =\\ 1}\\]<\/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\\ 12:}\\ \\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\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 13:}\\ \\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\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\"><\/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<div class=\"wp-block-mathml-mathmlblock\"><\/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\\ 14:}\\ \\color {red}{Prove\\ that}\\ A =\\begin{bmatrix}\n3 &amp; 6 \\\\\n2 &amp; 4 \\\\\n\\end{bmatrix}\\ \\color {red} {is\\ 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}\\ = 3(4) &#8211; 6(2) \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= 12 &#8211; 12 \\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\\ 15:}\\ Prove\\ that\\ the\\ matrix \\begin{pmatrix}\n8 &amp; 16 \\\\\n6 &amp; 12\\\\\n\\end{pmatrix}\\ \\color {red} {is\\ singular}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {blue}{Solution:}\\  Given\\ A =\\begin{pmatrix}\n8 &amp; 16 \\\\\n6 &amp; 12\\\\\n\\end{pmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ = 8(12) &#8211; 16(6) \\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= 96 &#8211; 96 \\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\\ 16:}\\ \\color {red}{Verify\\ the\\ matrix}\\ A =\\begin{pmatrix}\n2 &amp; 3 \\\\\n4 &amp; 5 \\\\\n\\end{pmatrix}\\ \\color {red} {is\\ Non\\ -\\ singular}\\ \\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{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\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{Example\\ 17:}\\ \\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\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{Example\\ 18:}\\ \\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<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\">\\[\\color {red} {1.\\ Solve:}\\ \\begin{vmatrix}\nx &amp; 2 \\\\\n5 &amp; 2 \\\\\n\\end{vmatrix}\\ = 0\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red} {2.\\ Find\\ &#8216;x&#8217;\\ if}\\ \\begin{vmatrix}\nx &amp; 3 \\\\\n1 &amp; 2 \\\\\n\\end{vmatrix}\\ = 0\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red} {3.\\ Find\\ &#8216;x&#8217;\\ if}\\ \\begin{vmatrix}\nx &amp; 4 \\\\\n9 &amp; x \\\\\n\\end{vmatrix}\\ = 0\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red} {4.\\ Find\\ &#8216;x&#8217;\\ if}\\ \\begin{vmatrix}\nx &amp; 4 \\\\\n16 &amp; x \\\\\n\\end{vmatrix}\\ = 0\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red} {5.\\ If}\\ A =\\begin{bmatrix}\n2 &amp; 3 \\\\\n-1 &amp; 4\\\\\n\\end{bmatrix}\\  and\\ B =\\begin{bmatrix}\n5 &amp; 0 \\\\\n3 &amp; 6\\\\\n\\end{bmatrix}\\ then\\  \\color {red} {find\\ A\\ +\\ B}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red}{6.\\ If}\\ A =\\begin{pmatrix}\n5 &amp; &#8211; 6 \\\\\n3 &amp; 7\\\\\n\\end{pmatrix},\\ \\color {red}{find\\ 3\\ A?}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red} {7.\\ 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}\\ then\\  \\color {red} {find\\ 4A\\ +\\ B}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[8.\\ \\color {red}{Define\\ Non-singular\\ matrix}\\ \\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<!-- 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\">\\[\\LARGE{\\color {purple} {PART- B}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[8.\\ \\color {red}{Prove\\ that\\ the\\ matrix}\\ \\begin{pmatrix}\n8 &amp; 16 \\\\\n6 &amp; 12\\\\\n\\end{pmatrix},\\ \\color {red} {is\\ singular.}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red} {9.\\ Prove\\ that}\\ \\begin{pmatrix}\n1 &amp; 3 &amp; 5 \\\\\n3 &amp; 5 &amp; 7 \\\\\n17 &amp; 9 &amp; 1 \\\\\n\\end{pmatrix}\\ \\color {red} {is\\ a\\ Singular\\ Matrix.} \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {red} {10.\\ Prove\\ that}\\ \\begin{pmatrix}\n1 &amp; 2 &amp; -3 \\\\\n3 &amp; 4 &amp; 5 \\\\\n4 &amp; 8 &amp; -12 \\\\\n\\end{pmatrix}\\ \\color {red} {is\\ a\\ Singular\\ Matrix.} \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[11.\\ \\color {red}{If}\\ A =\\begin{bmatrix}\n2 &amp; 3 \\\\\n-1 &amp; 2 \\\\\n\\end{bmatrix}\\ ,\\ B =\\begin{bmatrix}\n-1 &amp; 2 \\\\\n1 &amp; -5 \\\\\n\\end{bmatrix}\\ ,\\ \\color {red} {Find\\ the\\\n value\\ of\\ AB\\ and\\  BA}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[12.\\ \\color {red}{If}\\ A =\\begin{pmatrix}\n3 &amp; 6 \\\\\n1 &amp; -2 \\\\\n\\end{pmatrix}\\ and\\ \\ B =\\begin{pmatrix}\n5 &amp; 0 \\\\\n2 &amp; 3 \\\\\n\\end{pmatrix}\\ , \\color {red} {Show\\ that\\ (AB)^T\\ =\\ B^T\\ A^T}\\ \\hspace{12cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<!-- Leader board 1 -->\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"8769628924\" data-ad-format=\"auto\" data-full-width-responsive=\"true\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n","protected":false},"excerpt":{"rendered":"<p>Matrix and its applications are very important part of Mathematics. Also it is one of the most powerful tools of Mathematics. Matrix notation and operation are used in electronic spread sheet programmes for personal computer, business budgeting, sales projection, cost estimation, analyzing the results of an experiment etc. Also many physical operations such as magnification, [&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":[711788205,711788224,711787788,711788223,711788206],"tags":[],"class_list":["post-21036","post","type-post","status-publish","format-standard","hentry","category-academics","category-matrix-and-determinants-text","category-n-scheme","category-n-unit-i-algebra","category-polytechnic-mathematics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>MATRICES AND DETERMINANTS (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=21036\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"MATRICES AND DETERMINANTS (Text) - YANAMTAKSHASHILA\" \/>\n<meta property=\"og:description\" content=\"Matrix and its applications are very important part of Mathematics. Also it is one of the most powerful tools of Mathematics. Matrix notation and operation are used in electronic spread sheet programmes for personal computer, business budgeting, sales projection, cost estimation, analyzing the results of an experiment etc. Also many physical operations such as magnification, [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/yanamtakshashila.com\/?p=21036\" \/>\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=\"2021-08-03T11:43:21+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-06-30T07:39:38+00:00\" \/>\n<meta property=\"og:image\" content=\"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2025\/01\/yanamtakshashila.png?fit=1200%2C675&ssl=1\" \/>\n\t<meta property=\"og:image:width\" content=\"1200\" \/>\n\t<meta property=\"og:image:height\" content=\"675\" \/>\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=\"14 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=21036#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=21036\"},\"author\":{\"name\":\"rajuviswa\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"headline\":\"MATRICES AND DETERMINANTS (Text)\",\"datePublished\":\"2021-08-03T11:43:21+00:00\",\"dateModified\":\"2024-06-30T07:39:38+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=21036\"},\"wordCount\":2893,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"articleSection\":[\"ACADEMICS\",\"Matrix and Determinants (Text)\",\"N-Scheme\",\"N-Unit - I - ALGEBRA\",\"Polytechnic Mathematics\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/yanamtakshashila.com\\\/?p=21036#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=21036\",\"url\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=21036\",\"name\":\"MATRICES AND DETERMINANTS (Text) - YANAMTAKSHASHILA\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#website\"},\"datePublished\":\"2021-08-03T11:43:21+00:00\",\"dateModified\":\"2024-06-30T07:39:38+00:00\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=21036#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/yanamtakshashila.com\\\/?p=21036\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=21036#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/yanamtakshashila.com\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"MATRICES AND DETERMINANTS (Text)\"}]},{\"@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":"MATRICES AND DETERMINANTS (Text) - 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=21036","og_locale":"en_US","og_type":"article","og_title":"MATRICES AND DETERMINANTS (Text) - YANAMTAKSHASHILA","og_description":"Matrix and its applications are very important part of Mathematics. Also it is one of the most powerful tools of Mathematics. Matrix notation and operation are used in electronic spread sheet programmes for personal computer, business budgeting, sales projection, cost estimation, analyzing the results of an experiment etc. Also many physical operations such as magnification, [&hellip;]","og_url":"https:\/\/yanamtakshashila.com\/?p=21036","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":"2021-08-03T11:43:21+00:00","article_modified_time":"2024-06-30T07:39:38+00:00","og_image":[{"width":1200,"height":675,"url":"https:\/\/i0.wp.com\/yanamtakshashila.com\/wp-content\/uploads\/2025\/01\/yanamtakshashila.png?fit=1200%2C675&ssl=1","type":"image\/png"}],"author":"rajuviswa","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rajuviswa","Est. reading time":"14 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/yanamtakshashila.com\/?p=21036#article","isPartOf":{"@id":"https:\/\/yanamtakshashila.com\/?p=21036"},"author":{"name":"rajuviswa","@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"headline":"MATRICES AND DETERMINANTS (Text)","datePublished":"2021-08-03T11:43:21+00:00","dateModified":"2024-06-30T07:39:38+00:00","mainEntityOfPage":{"@id":"https:\/\/yanamtakshashila.com\/?p=21036"},"wordCount":2893,"commentCount":0,"publisher":{"@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"articleSection":["ACADEMICS","Matrix and Determinants (Text)","N-Scheme","N-Unit - I - ALGEBRA","Polytechnic Mathematics"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/yanamtakshashila.com\/?p=21036#respond"]}]},{"@type":"WebPage","@id":"https:\/\/yanamtakshashila.com\/?p=21036","url":"https:\/\/yanamtakshashila.com\/?p=21036","name":"MATRICES AND DETERMINANTS (Text) - YANAMTAKSHASHILA","isPartOf":{"@id":"https:\/\/yanamtakshashila.com\/#website"},"datePublished":"2021-08-03T11:43:21+00:00","dateModified":"2024-06-30T07:39:38+00:00","breadcrumb":{"@id":"https:\/\/yanamtakshashila.com\/?p=21036#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/yanamtakshashila.com\/?p=21036"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/yanamtakshashila.com\/?p=21036#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/yanamtakshashila.com\/"},{"@type":"ListItem","position":2,"name":"MATRICES AND DETERMINANTS (Text)"}]},{"@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":"","jetpack_likes_enabled":true,"jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/pc3kmt-5ti","_links":{"self":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/21036","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=21036"}],"version-history":[{"count":99,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/21036\/revisions"}],"predecessor-version":[{"id":47954,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/21036\/revisions\/47954"}],"wp:attachment":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=21036"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=21036"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=21036"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}