{"id":5878,"date":"2025-08-08T10:56:32","date_gmt":"2025-08-08T05:26:32","guid":{"rendered":"https:\/\/study.madeeasy.in\/?p=5878"},"modified":"2025-08-08T10:56:32","modified_gmt":"2025-08-08T05:26:32","slug":"inverse-of-matrix","status":"publish","type":"post","link":"https:\/\/www.madeeasy.in\/study\/ee\/linear-algebra\/inverse-of-matrix","title":{"rendered":"Inverse of Matrix"},"content":{"rendered":"<h2>Inverse of Matrix<\/h2>\n<p>The inverse of a matrix A, exists if A is non-singular (i.e., \u23d0A\u23d0 \u2260 0) and is given by the formula<\/p>\n<p style=\"text-align: center;\">A<sup>\u20131<\/sup> = Adj (A) \/ |A|<\/p>\n<p>The inverse of a matrix is always unique.<\/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\/inverse-of-matrix\/#Adjoint-of-a-Square-Matrix\" >Adjoint of a Square Matrix<\/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\/inverse-of-matrix\/#Rank-of-A-Matrix\" >Rank of A Matrix<\/a><\/li><\/ul><\/nav><\/div>\n<h3><span class=\"ez-toc-section\" id=\"Adjoint-of-a-Square-Matrix\"><\/span>Adjoint of a Square Matrix<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p style=\"text-align: justify;\">Let A=[a<sub>ij<\/sub>] be any n \u00d7 n matrix. The transpose B of the matrix B = [A<sub>ij<\/sub>]n x n, where Aij denotes the cofactor of element aij is called the adjoint of matrix A and is denoted by the symbol Adj A.<br \/>\n\u2234 Adj (A) = [cof (A)]<sup>T<\/sup><\/p>\n<p><strong> Properties of Adjoint:<\/strong><\/p>\n<p>If A be any n-rowed square matrix, then (Adj A) A = A (Adj A) = \u23d0A\u23d0 I<sub>n<\/sub><\/p>\n<p>where I<sub>n<\/sub> is the n \u00d7 n identity matrix.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Rank-of-A-Matrix\"><\/span>Rank of A Matrix<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Rank is defined for any matrix A<sub>m \u00d7 n<\/sub> (need not be square)<\/p>\n<p><strong> Some important concepts:<\/strong><\/p>\n<p style=\"text-align: justify;\"><strong> 1. Submatrix of a Matrix:<\/strong> Suppose A is any matrix of the type m \u00d7 n. Then a matrix obtained by leaving some rows and some columns from A is called a submatrix of A.<\/p>\n<p style=\"text-align: justify;\"><strong>2. Rank of a Matrix:<\/strong> A number r is said to be the rank of a matrix A, if it possesses the following properties:<\/p>\n<p style=\"text-align: justify;\">(a) There is at least one square sub-matrix of A of order r whose determinant is not equal to zero.<\/p>\n<p style=\"text-align: justify;\">(b) If the matrix A contains any square sub-matrix of order (r + 1) and above, then the determinant of such a matrix should be zero.<\/p>\n<p style=\"text-align: justify;\">Put together, property (a) and (b) give the definition of the rank of a matrix as the \u201corder of the largest non-zero minor.\u201d<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Inverse of Matrix The inverse of a matrix A, exists if A is non-singular (i.e., \u23d0A\u23d0 \u2260 0) and is<\/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":[1731,1730,1732],"class_list":["post-5878","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","category-ee","tag-properties-of-adjoint","tag-rank-of-a-matrix","tag-submatrix-of-a-matrix"],"_links":{"self":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/5878","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=5878"}],"version-history":[{"count":0,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/posts\/5878\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/media?parent=5878"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/categories?post=5878"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.madeeasy.in\/study\/wp-json\/wp\/v2\/tags?post=5878"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}