{"id":5867,"date":"2025-08-07T11:19:16","date_gmt":"2025-08-07T05:49:16","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=5867"},"modified":"2025-08-07T11:19:16","modified_gmt":"2025-08-07T05:49:16","slug":"determinants","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/ee\/linear-algebra\/determinants","title":{"rendered":"Determinants"},"content":{"rendered":"<h2><strong>Definition of Determinants:<\/strong><\/h2>\n<p>Let a11, a12, a21, a22 be any four numbers. The symbol\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5868 alignnone\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2025\/08\/determinants.jpg\" alt=\"Determinants\" width=\"85\" height=\"53\" \/>represents the number a11a22 \u2013 a12a21 and is called determinants of order 2. The number a11, a12, a21, a22 are called elements of the determinant and the number a11a22 \u2013 a21a12 is called the value of determinant.<\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_79_1 ez-toc-wrap-left counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #999;color:#999\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #999;color:#999\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 ' ><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/www.madeeasy.in\/study\/ee\/linear-algebra\/determinants\/#Minors-Cofactors-and-Adjoint\" >Minors, Cofactors and Adjoint<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/www.madeeasy.in\/study\/ee\/linear-algebra\/determinants\/#Cofactors\" >Cofactors<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/www.madeeasy.in\/study\/ee\/linear-algebra\/determinants\/#Adjoint\" >Adjoint<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/www.madeeasy.in\/study\/ee\/linear-algebra\/determinants\/#Determinant-of-order-n\" >Determinant of order n<\/a><\/li><\/ul><\/nav><\/div>\n<h3><span class=\"ez-toc-section\" id=\"Minors-Cofactors-and-Adjoint\"><\/span>Minors, Cofactors and Adjoint<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5869 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2025\/08\/cofactors.jpg\" alt=\"Cofactors\" width=\"711\" height=\"223\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/08\/cofactors.jpg 711w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/08\/cofactors-300x94.jpg 300w\" sizes=\"auto, (max-width: 711px) 100vw, 711px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Cofactors\"><\/span>Cofactors<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The minor Mij multiplied by (\u20131)i + j is called the cofactor of element aij . We shall denote the cofactor of an element by corresponding capital letter.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5870 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2025\/08\/cofactors-1.jpg\" alt=\"Cofactors\" width=\"709\" height=\"390\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/08\/cofactors-1.jpg 709w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/08\/cofactors-1-300x165.jpg 300w\" sizes=\"auto, (max-width: 709px) 100vw, 709px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Adjoint\"><\/span>Adjoint<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>When all the elements of a matrix \u2018A\u2019 are replaced by its co-factor, then the transpose of that matrix is known as adjoint of matrix \u2018A\u2019.<\/p>\n<p>a<sub>ij<\/sub> \u2192C<sub>ij<\/sub><\/p>\n<p>AdjA= [C<sub>ij<\/sub> ]T<\/p>\n<p>Properties of adjoint matrix A(Adj A)<\/p>\n<p>1. A x AdjA = |A| x I<\/p>\n<p>2. A<sup>-1<\/sup> = 1\/ |A| x (AdjA )<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Determinant-of-order-n\"><\/span>Determinant of order n<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<ul>\n<li>A determinant of order n has n-row and n-columns. It has n \u00d7 n elements.<\/li>\n<li>A determinant of order n is a square array of n \u00d7 n quantities enclosed between vertical bars.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-5871 size-full\" src=\"https:\/\/study.madeeasy.in\/wp-content\/uploads\/2025\/08\/determinant-of-order.jpg\" alt=\"Determinant of order\" width=\"715\" height=\"202\" srcset=\"https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/08\/determinant-of-order.jpg 715w, https:\/\/www.madeeasy.in\/study\/wp-content\/uploads\/2025\/08\/determinant-of-order-300x85.jpg 300w\" sizes=\"auto, (max-width: 715px) 100vw, 715px\" \/><\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Definition of Determinants: Let a11, a12, a21, a22 be any four numbers. The symbol\u00a0represents the number a11a22 \u2013 a12a21 and<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1711,5],"tags":[1726,1727,1725],"class_list":["post-5867","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","category-ee","tag-adjoint","tag-cofactors","tag-determinant-of-order-n"],"_links":{"self":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/5867","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/comments?post=5867"}],"version-history":[{"count":0,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/5867\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/media?parent=5867"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/categories?post=5867"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/tags?post=5867"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}