{"id":23920,"date":"2021-12-14T20:23:41","date_gmt":"2021-12-14T14:53:41","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=23920"},"modified":"2023-07-23T11:06:59","modified_gmt":"2023-07-23T05:36:59","slug":"applications-of-matrices-and-determinantsrevision","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=23920","title":{"rendered":"APPLICATIONS\u00a0OF\u00a0MATRICES\u00a0AND\u00a0DETERMINANTS:(Revision)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black} {1.}\\ Solve\\ the\\ following\\ equations\\ using\\ Cramers\\ Rule\\ \\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\">\\[x\\ +\\ y\\ +\\  z\\ =\\ 3,\\ 2\\ x\\ -\\ y\\ +\\  z\\ =\\ 2\\ and\\ 3\\ x\\ +\\ 2\\ y\\ -\\ 2\\ z\\ =\\  3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x\\ +\\ y\\ +\\  z=3\\ &#8212;&#8212;&#8212;&#8212;&#8212;&#8212;-(1)\\ \\hspace{6cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2\\ x\\ -\\ y\\ +\\  z\\ =\\ 2\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[3\\ x\\ +\\ 2\\ y\\ -\\ 2\\ z\\ =\\  3\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta = \\begin{vmatrix}\n1 &amp; 1 &amp; 1 \\\\\n2 &amp; -1 &amp; 1 \\\\\n3 &amp; 2 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =1\\begin{vmatrix}\n-1 &amp; 1 \\\\\n2 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 1 \\\\\n3 &amp; -2 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; -1\\\\\n3 &amp; 2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =1(2\\ -\\ 2)\\ &#8211; 1 (-4\\ -\\  3)\\ +\\ 1(4\\ +\\ 3)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta\\ =\\ 1(0)\\ &#8211; 1 (-\\ 7)\\ +\\ 1(7)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =0\\ +\\ 7\\ +\\  7\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta\\ =\\ 14\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x = \\begin{vmatrix}\n3 &amp; 1 &amp; 1 \\\\\n2 &amp; -1 &amp; 1 \\\\\n3 &amp; 2 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x =3\\begin{vmatrix}\n-1 &amp; 1 \\\\\n2 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 1 \\\\\n3 &amp; -2 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; -1\\\\\n3 &amp; 2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 3(2\\ -\\ 2) &#8211; 1 (-\\ 4\\ -\\ (3))\\ +\\ 1(4\\ +\\ 3)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 3(0)\\ &#8211; 1 (-\\ 7)\\ +\\ 1(7)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x = 0\\ +\\ 7\\ +\\ 7\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_x\\ =\\ 14\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y = \\begin{vmatrix}\n1 &amp; 3 &amp; 1 \\\\\n2 &amp; 2 &amp; 1 \\\\\n3 &amp; 3 &amp; -2 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ 1\\begin{vmatrix}\n2 &amp;  1 \\\\\n3 &amp; -2 \\\\\n\\end{vmatrix}\\ -\\ 3\\begin{vmatrix}\n2 &amp; 1 \\\\\n3 &amp; -2 \\\\\n\\end{vmatrix}\\ +\\ 1\\begin{vmatrix}\n2 &amp; 2\\\\\n3 &amp;  3 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ 1(-\\ 4\\ -\\ 3)\\ -\\ 3 (-4\\ -\\ 3)\\ +\\ 1(6\\ -\\ 6)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ 1(- \\ 7)\\ -\\ 3 (-\\ 7)\\ +\\ 1(0)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ -\\ 7\\ +\\ 21\\ +\\ 0\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_y\\ =\\ 14\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z = \\begin{vmatrix}\n1 &amp; 1 &amp; 3 \\\\\n2 &amp; -1 &amp; 2 \\\\\n3 &amp; 2 &amp; 3 \\\\\n\\end{vmatrix}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 1\\begin{vmatrix}\n-1 &amp; 2 \\\\\n2 &amp; 3 \\\\\n\\end{vmatrix}\\ -\\ 1\\begin{vmatrix}\n2 &amp; 2 \\\\\n3 &amp; 3 \\\\\n\\end{vmatrix}\\ +\\ 3\\begin{vmatrix}\n2 &amp; &#8211; 1\\\\\n3 &amp;  2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 1(-\\ 3\\ -\\ 4)\\ -\\ 1 (6\\ -\\ 6)\\ +\\ 3(4\\ +\\ 3)\\ \n\\hspace{9cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 1(-\\ 7)\\ -\\ 1 (0)\\ +\\ 3(7)\\ \n\\hspace{13cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ -\\ 7\\  -\\  0\\ +\\  21\\ \n\\hspace{14cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta_z\\ =\\ 14\\ \n\\hspace{17cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[The\\ Solution\\ is\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[x=\\ \\frac{\\Delta_x}{\\Delta} =\\ \\frac{14}{14} =\\ 1\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[y=\\ \\frac{\\Delta_y}{\\Delta} =\\ \\frac{14}{14} =\\ 1\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[z=\\ \\frac{\\Delta_z}{\\Delta} =\\ \\frac{14}{14} =\\ 1\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[For\\ cross\\ verification\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Put\\ x\\ =\\ 1,\\ y\\ =\\ 2\\ and\\ z = 3\\ in\\ equation (1)\\ \\hspace{18cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[LHS\\ =\\ 1 + 2 &#8211; 3\\]\\[ = 3 + 2 &#8211; 3 = 2\\]\\[ = 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<!-- wide skyscraper -->\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:160px;height:600px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"9987820756\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ 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\">\\[\\color {black}{Solution:}\\ 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<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","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 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":[711788250],"tags":[],"class_list":["post-23920","post","type-post","status-publish","format-standard","hentry","category-applications-of-matrices-and-determinants-revision"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>APPLICATIONS\u00a0OF\u00a0MATRICES\u00a0AND\u00a0DETERMINANTS:(Revision) - 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=23920\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"APPLICATIONS\u00a0OF\u00a0MATRICES\u00a0AND\u00a0DETERMINANTS:(Revision) - YANAMTAKSHASHILA\" \/>\n<meta property=\"og:url\" content=\"https:\/\/yanamtakshashila.com\/?p=23920\" \/>\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=\"2021-12-14T14:53:41+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2023-07-23T05:36:59+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=\"4 minutes\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=23920#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=23920\"},\"author\":{\"name\":\"rajuviswa\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"headline\":\"APPLICATIONS\u00a0OF\u00a0MATRICES\u00a0AND\u00a0DETERMINANTS:(Revision)\",\"datePublished\":\"2021-12-14T14:53:41+00:00\",\"dateModified\":\"2023-07-23T05:36:59+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=23920\"},\"wordCount\":721,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#\\\/schema\\\/person\\\/a990a0af264ac2298c19fa61d2bda16e\"},\"articleSection\":[\"Applications of matrices and determinants (Revision)\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/yanamtakshashila.com\\\/?p=23920#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=23920\",\"url\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=23920\",\"name\":\"APPLICATIONS\u00a0OF\u00a0MATRICES\u00a0AND\u00a0DETERMINANTS:(Revision) - YANAMTAKSHASHILA\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/#website\"},\"datePublished\":\"2021-12-14T14:53:41+00:00\",\"dateModified\":\"2023-07-23T05:36:59+00:00\",\"breadcrumb\":{\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=23920#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/yanamtakshashila.com\\\/?p=23920\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/yanamtakshashila.com\\\/?p=23920#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/yanamtakshashila.com\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"APPLICATIONS\u00a0OF\u00a0MATRICES\u00a0AND\u00a0DETERMINANTS:(Revision)\"}]},{\"@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":"APPLICATIONS\u00a0OF\u00a0MATRICES\u00a0AND\u00a0DETERMINANTS:(Revision) - 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=23920","og_locale":"en_US","og_type":"article","og_title":"APPLICATIONS\u00a0OF\u00a0MATRICES\u00a0AND\u00a0DETERMINANTS:(Revision) - YANAMTAKSHASHILA","og_url":"https:\/\/yanamtakshashila.com\/?p=23920","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":"2021-12-14T14:53:41+00:00","article_modified_time":"2023-07-23T05:36:59+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":"4 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/yanamtakshashila.com\/?p=23920#article","isPartOf":{"@id":"https:\/\/yanamtakshashila.com\/?p=23920"},"author":{"name":"rajuviswa","@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"headline":"APPLICATIONS\u00a0OF\u00a0MATRICES\u00a0AND\u00a0DETERMINANTS:(Revision)","datePublished":"2021-12-14T14:53:41+00:00","dateModified":"2023-07-23T05:36:59+00:00","mainEntityOfPage":{"@id":"https:\/\/yanamtakshashila.com\/?p=23920"},"wordCount":721,"commentCount":0,"publisher":{"@id":"https:\/\/yanamtakshashila.com\/#\/schema\/person\/a990a0af264ac2298c19fa61d2bda16e"},"articleSection":["Applications of matrices and determinants (Revision)"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/yanamtakshashila.com\/?p=23920#respond"]}]},{"@type":"WebPage","@id":"https:\/\/yanamtakshashila.com\/?p=23920","url":"https:\/\/yanamtakshashila.com\/?p=23920","name":"APPLICATIONS\u00a0OF\u00a0MATRICES\u00a0AND\u00a0DETERMINANTS:(Revision) - YANAMTAKSHASHILA","isPartOf":{"@id":"https:\/\/yanamtakshashila.com\/#website"},"datePublished":"2021-12-14T14:53:41+00:00","dateModified":"2023-07-23T05:36:59+00:00","breadcrumb":{"@id":"https:\/\/yanamtakshashila.com\/?p=23920#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/yanamtakshashila.com\/?p=23920"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/yanamtakshashila.com\/?p=23920#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/yanamtakshashila.com\/"},{"@type":"ListItem","position":2,"name":"APPLICATIONS\u00a0OF\u00a0MATRICES\u00a0AND\u00a0DETERMINANTS:(Revision)"}]},{"@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-6dO","_links":{"self":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/23920","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=23920"}],"version-history":[{"count":35,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/23920\/revisions"}],"predecessor-version":[{"id":47960,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=\/wp\/v2\/posts\/23920\/revisions\/47960"}],"wp:attachment":[{"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=23920"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=23920"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/yanamtakshashila.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=23920"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}