{"id":42360,"date":"2023-07-22T12:39:16","date_gmt":"2023-07-22T07:09:16","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=42360"},"modified":"2023-07-23T20:09:12","modified_gmt":"2023-07-23T14:39:12","slug":"inverse-of-a-matrix","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=42360","title":{"rendered":"INVERSE OF A MATRIX"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 1:}\\ \\color{red}{Find\\ the\\ inverse\\ of}\\ \\begin{bmatrix}\n1 &amp; -1 \\\\\n-2 &amp; 0 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {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\\ 2:}\\ \\color{red}{Find\\ the\\ inverse\\ of}\\ \\begin{bmatrix}\n2 &amp; -1 \\\\\n4 &amp; 5 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 3:}\\ \\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<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/eJ8q9ayyRkc\" title=\"Inverse of a Matrix - Example - 3\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 4:}\\ \\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\">\\[\\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-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-mathml-mathmlblock\">\\[= (-1) (-\\ 10)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 =\\ 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\n\n<p><iframe width=\"787\" height=\"443\" src=\"https:\/\/www.youtube.com\/embed\/3RFQMGRBQyU\" title=\"Inverse of a Matrix - Example - 4\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" allowfullscreen=\"\"><\/iframe><\/p>\n\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} {Example\\ 5:}\\ \\color{red}{Find\\ the\\ inverse\\ of\\ the\\ matrix}\\ \\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\">\\[\\color {blue}{Solution:}\\ Let\\ A\\ =\\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\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =2\\begin{vmatrix}\n3 &amp; 1 \\\\\n2 &amp; 4 \\\\\n\\end{vmatrix}\\ -\\ 3\\begin{vmatrix}\n4 &amp; 1 \\\\\n1 &amp; 4 \\\\\n\\end{vmatrix}\\ +\\ 4\\begin{vmatrix}\n4 &amp; 3\\\\\n1 &amp;  2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =2(12\\ -\\ 2)\\ &#8211; 3 (16\\ -\\ 1) + 4(8\\ -\\ 3)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =2(10)\\ &#8211; 3 (15) + 4(5)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ =20\\ -45 + 20\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\ =-5\\ \\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\\ 2 = (-1)^{1\\ +\\ 1}\\ \\begin{vmatrix}\n3 &amp; 1 \\\\\n2 &amp; 4 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^2 (12 &#8211; 2)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (10)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = 10\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3 = (-1)^{1\\ +\\ 2}\\ \\begin{vmatrix}\n4 &amp; 1 \\\\\n1 &amp; 4 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (16 &#8211; 1)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1) (15)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3 = -15\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 4 = (-1)^{1\\ +\\ 3}\\ \\begin{vmatrix}\n4 &amp; 3 \\\\\n1 &amp; 2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (8- 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\\ 4 = 5\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 4 = (-1)^{2\\ +\\ 1}\\ \\begin{vmatrix}\n3 &amp; 4 \\\\\n2 &amp; 4 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^3 (12- 8)\\ \\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\\ 4 = -4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3 = (-1)^{2\\ +\\ 2}\\ \\begin{vmatrix}\n2 &amp; 4 \\\\\n1 &amp; 4 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (8- 4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (1) (4)\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 3 = 4\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = (-1)^{2\\ +\\ 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)^5 (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\\ 1 = -1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 1 = (-1)^{3\\ +\\ 1}\\ \\begin{vmatrix}\n3 &amp; 4 \\\\\n3 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^4 (3- 12)\\ \\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\\ 1 = -9\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 2 = (-1)^{3\\ +\\ 2}\\ \\begin{vmatrix}\n2 &amp; 4 \\\\\n4 &amp; 1 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^5 (2- 16)\\ \\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\\ 2 = 14\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[cofactor\\ of\\ 4 = (-1)^{3\\ +\\ 3}\\ \\begin{vmatrix}\n2 &amp; 3 \\\\\n4 &amp; 3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[= (-1)^6 (6- 12)\\ \\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\\ 4 = -6\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cofactor\\ matrix=\\begin{bmatrix}\n10 &amp; -15 &amp; 5 \\\\\n-4 &amp; 4 &amp; -1 \\\\\n-9 &amp; 14 &amp; -6 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Adj.\\ A=\\begin{bmatrix}\n10 &amp; -4 &amp; -9 \\\\\n-15 &amp; 4 &amp; 14 \\\\\n5 &amp; -1 &amp; -6 \\\\\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}{-5}\\ \\begin{bmatrix}\n10 &amp; -4 &amp; -9 \\\\\n-15 &amp; 4 &amp; 14 \\\\\n5 &amp; -1 &amp; -6 \\\\\n\\end{bmatrix}\\ \\hspace{2cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Example\\ 5:}\\ \\color{red}{Find\\ the\\ inverse\\ of\\ the\\ matrix}\\ \\begin{bmatrix}\n2 &amp; 3 &amp; 4 \\\\\n4 &amp; 3 &amp; 1 \\\\\n1 &amp; 2 &amp; 4 \\\\\n\\end{bmatrix}\\ \\hspace{15cm}\\]<\/div>\n","protected":false},"excerpt":{"rendered":"","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":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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