{"id":664346,"date":"2023-07-10T17:45:28","date_gmt":"2023-07-10T12:15:28","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/?p=664346"},"modified":"2025-05-29T13:54:10","modified_gmt":"2025-05-29T08:24:10","slug":"inverse-of-matrices","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/topics\/inverse-of-matrices\/","title":{"rendered":"Inverse of Matrices"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_37 counter-hierarchy ez-toc-counter ez-toc-grey ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\">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\" style=\"display: none;\"><label for=\"item\" aria-label=\"Table of Content\"><span style=\"display: flex;align-items: center;width: 35px;height: 30px;justify-content: center;\"><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><\/label><input type=\"checkbox\" id=\"item\"><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1' style='display:block'><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/inverse-of-matrices\/#Inverse_of_a_Matrix\" title=\"Inverse of a Matrix\">Inverse of a 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:\/\/infinitylearn.com\/surge\/topics\/inverse-of-matrices\/#Matrix_Inverse\" title=\"Matrix Inverse\">Matrix Inverse<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/inverse-of-matrices\/#Inverse_Matrix_Method\" title=\"Inverse Matrix Method\">Inverse Matrix Method<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/inverse-of-matrices\/#Method_1_Inverse-matrix-method\" title=\"Method 1: Inverse-matrix-method\">Method 1: Inverse-matrix-method<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/inverse-of-matrices\/#Method_2_minors_and_cofactors_methods\" title=\"Method 2: minors and cofactors methods\">Method 2: minors and cofactors methods<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/inverse-of-matrices\/#Method_3_Elementary_Transformation\" title=\"Method 3: Elementary Transformation\">Method 3: Elementary Transformation<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/inverse-of-matrices\/#Inverse_Matrix_2_x_2_Example\" title=\"Inverse Matrix 2 x 2 Example\">Inverse Matrix 2 x 2 Example<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/inverse-of-matrices\/#Properties_of_the_inverse_of_a_matrix\" title=\"Properties of the inverse of a matrix\">Properties of the inverse of a matrix<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-9\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/inverse-of-matrices\/#FAQs_on_Inverse_of_Matrices\" title=\"FAQs on Inverse of Matrices\">FAQs on Inverse of Matrices<\/a><ul class='ez-toc-list-level-3'><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-10\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/inverse-of-matrices\/#What_is_concept_inverse_of_a_matrix\" title=\"What is concept inverse of a matrix?\">What is concept inverse of a matrix?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-11\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/inverse-of-matrices\/#How_do_you_find_the_inverse_of_a_3%C3%973_matrix\" title=\"How do you find the inverse of a 3\u00d73 matrix?\">How do you find the inverse of a 3\u00d73 matrix?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-12\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/inverse-of-matrices\/#Is_adjoint_and_inverse_the_same\" title=\"Is adjoint and inverse the same?\">Is adjoint and inverse the same?<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/inverse-of-matrices\/#How_to_do_you_know_whether_the_given_matrix_has_an_inverse\" title=\"How to do you know whether the given matrix has an inverse?\">How to do you know whether the given matrix has an inverse?<\/a><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<p>To understand the Inverse of Matrices, one initially needs to understand Matrices.<br \/>\nA matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It is a fundamental mathematical concept used in various fields, including linear algebra, computer science, physics, and engineering.<\/p>\n<p>In matrix notation, a matrix is typically represented by a capital letter and enclosed in brackets or parentheses. The size of a matrix is determined by the number of rows and columns it contains. For example, an &#8220;m x n&#8221; matrix has m rows and n columns.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Inverse_of_a_Matrix\"><\/span>Inverse of a Matrix<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>The <strong>inverse of matrix<\/strong> is obtained by dividing the adjoint of the given matrix by the determinant of the given matrix. Students must note that matrix inverse could be found only for square matrices.<\/p>\n<p>This article discusses about the inverse of a matrix, steps to find the inverse of a matrix, the properties of the inverse matrix along with the examples.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Matrix_Inverse\"><\/span>Matrix Inverse<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>If A is a non-singular square matrix, then there exists a n x n matrix A-1 which is called the inverse matrix of A, such that it satisfies the property:<\/p>\n<p>AA-1 = A-1A = I, where I is the Identity matrix.<\/p>\n<p>The identity matrix for the 2 x 2 matrix is given by:<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-664348\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/07\/2-by-2-matrix-1.png\" alt=\"2 by 2 matrix\" width=\"115\" height=\"93\" \/><\/p>\n<p>It is noted that to find the inverse of a matrix, the square matrix should be a non-singular matrix whose determinant value is not equal to zero.<\/p>\n<p>Let us take the square matrix A<\/p>\n<p><img loading=\"lazy\" class=\"alignnone size-full wp-image-664349\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/07\/square-matrix-1.png\" alt=\"square matrix\" width=\"121\" height=\"87\" \/><\/p>\n<p>Where a, b, c, and d are the numbers.<\/p>\n<ul>\n<li>The determinant of matrix A is written as ad-bc.<\/li>\n<li>For the existence of the inverse of the matrices, the determinant should not equal zero.<\/li>\n<li>The inverse matrix can be found for 2\u00d7 2, 3\u00d7 3, \u2026n \u00d7 n matrices.<\/li>\n<li>As the value of n increases, finding the inverse of the matrix becomes difficult.<\/li>\n<\/ul>\n<p><a href=\"https:\/\/infinitylearn.com\/surge\/topics\/maths-topics\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 80px;\" type=\"button\">Maths Topics<\/button><\/a> <a href=\"https:\/\/infinitylearn.com\/surge\/articles\/math-articles\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 80px;\" type=\"button\">Math Articles<\/button><\/a> <a href=\"https:\/\/infinitylearn.com\/surge\/topics\/transpose-of-a-matrix\/\"><button class=\"btn btn-dark mx-2 my-2 px-4\" style=\"border-radius: 80px;\" type=\"button\">Transpose of Matrices<\/button><\/a><\/p>\n<h2><span class=\"ez-toc-section\" id=\"Inverse_Matrix_Method\"><\/span>Inverse Matrix Method<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The inverse of a matrix can be found using three different methods. However, any of these three methods will produce the same result.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Method_1_Inverse-matrix-method\"><\/span>Method 1: Inverse-matrix-method<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Let the matrix A be:<\/p>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-664350 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/07\/Inverse-matrix-method.png\" alt=\"Inverse matrix method\" width=\"673\" height=\"310\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/07\/Inverse-matrix-method.png 673w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/07\/Inverse-matrix-method-300x138.png 300w\" sizes=\"(max-width: 673px) 100vw, 673px\" \/><\/p>\n<p>We can find the inverse of a matrix using this method.<\/p>\n<h3><span class=\"ez-toc-section\" id=\"Method_2_minors_and_cofactors_methods\"><\/span>Method 2: minors and cofactors methods<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>Second method of finding the matrix inverse is by using the minors and cofactors of elements of the given matrix.<\/p>\n<p>The inverse matrix can be found using the following equation:<\/p>\n<p><strong>A<sup>-1 <\/sup>= adj(A)\/det(A),<\/strong><\/p>\n<p>where adj(A) refers to the adjoint of a matrix A,<br \/>\nAnd det(A) refers to the determinant of a matrix A.<\/p>\n<p>To know more about the adjoint and co-factor of a matrix,<br \/>\nCheck :<\/p>\n<ul>\n<li><strong>Adjoint of a matrix A<\/strong><\/li>\n<li><strong>Cofactor of a matrix A<\/strong><\/li>\n<\/ul>\n<h3><span class=\"ez-toc-section\" id=\"Method_3_Elementary_Transformation\"><\/span>Method 3: Elementary Transformation<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>To find an Inverse of a Matrix by Elementary Transformation we follow the below-mentioned method.<\/p>\n<p>Let us consider three matrices X, A and B such that X = AB.<br \/>\nTo find the inverse of a matrix using elementary transformation, we convert the given matrix into an identity matrix.<\/p>\n<p>Check: <strong>how to do elementary transformations of matrices<\/strong><\/p>\n<p>If, for a matrix A, A-1 exists, then to determine A-1 using elementary row operations, we follow the following steps.<\/p>\n<ul>\n<li>Let A = IA, where I is the identity matrix of the same order as A.<\/li>\n<li>Apply a sequence of row operations on LHS and RHS till we get an identity matrix on the LHS. while performing the operations we will get I = BA. The matrix B obtained on the RHS is the inverse of matrix A.<\/li>\n<li>To find the inverse of A using column operations, write A = IA and apply column operations in a similar manner as the above-mentioned step sequentially till I = AB is obtained, where B is the inverse matrix of A.<\/li>\n<\/ul>\n<p>Also check: <strong>how to find the inverse of a matrix using elementary operations<\/strong><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Inverse_Matrix_2_x_2_Example\"><\/span>Inverse Matrix 2 x 2 Example<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>To understand in a better way, let us take a look at the following example.<\/p>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-664351 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/07\/Inverse-Matrix-2-x-2-example.png\" alt=\"Inverse Matrix 2 x 2 example\" width=\"426\" height=\"793\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/07\/Inverse-Matrix-2-x-2-example.png 426w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2023\/07\/Inverse-Matrix-2-x-2-example-161x300.png 161w\" sizes=\"(max-width: 426px) 100vw, 426px\" \/><\/p>\n<h3><span class=\"ez-toc-section\" id=\"Properties_of_the_inverse_of_a_matrix\"><\/span>Properties of the inverse of a matrix<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<p>A few important properties of the inverse matrix are mentioned below.<\/p>\n<ul>\n<li>If matrix A is nonsingular, then (A-1)-1 = A<\/li>\n<li>If A and B are nonsingular matrices, then AB is also nonsingular.<br \/>\nThus, (AB)-1 = B-1A-1<\/li>\n<li>If A is nonsingular matrix, then (AT)-1 = (A-1)T<\/li>\n<li>If A is any matrix and A-1 is its inverse, then AA-1 = In = A-1A , where n is the order of matrices<\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"FAQs_on_Inverse_of_Matrices\"><\/span>FAQs on Inverse of Matrices<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"What_is_concept_inverse_of_a_matrix\"><\/span>What is concept inverse of a matrix?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tMatrices also have reciprocals just like numbers. In the case of matrices, this reciprocal is called an inverse matrix. If A is a square matrix and B is its inverse, then the product of matrices A and B is equal to the unit matrix. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"How_do_you_find_the_inverse_of_a_3%C3%973_matrix\"><\/span>How do you find the inverse of a 3\u00d73 matrix?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tThe steps required to find the inverse of a 3\u00d73 matrix are:<br \/>\nCompute the determinant of the given matrix and check whether the matrix is invertible.<br \/>\nCalculate the determinant of 2\u00d72 minor matrices.<br \/>\nFormulate the matrix of cofactors.<br \/>\nTake the transpose of the cofactor matrix to get the adjugate matrix.<br \/>\nFinally, divide each term of the adjugate matrix by the determinant \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"Is_adjoint_and_inverse_the_same\"><\/span>Is adjoint and inverse the same?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tNo, the adjoint matrix and inverse matrix are not the same. However, by dividing the each term of the adjoint of the matrix by the determinant of the original matrix, we get an inverse matrix. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"sc_fs_faq sc_card \">\n\t\t\t<div>\n\t\t\t\t<h3><span class=\"ez-toc-section\" id=\"How_to_do_you_know_whether_the_given_matrix_has_an_inverse\"><\/span>How to do you know whether the given matrix has an inverse?<span class=\"ez-toc-section-end\"><\/span><\/h3>\t\t\t\t<div>\n\t\t\t\t\t\t\t\t\t\t<p>\n\t\t\t\t\t\tIf the determinant of a given matrix is not equal to 0, i.e. it is non-singular, then the matrix is invertible. \t\t\t\t\t<\/p>\n\t\t\t\t<\/div>\n\t\t\t<\/div>\n\t\t<\/section>\n\t\t\n<script type=\"application\/ld+json\">\n\t{\n\t\t\"@context\": \"https:\/\/schema.org\",\n\t\t\"@type\": \"FAQPage\",\n\t\t\"mainEntity\": [\n\t\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"What is concept inverse of a matrix?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"Matrices also have reciprocals just like numbers. In the case of matrices, this reciprocal is called an inverse matrix. If A is a square matrix and B is its inverse, then the product of matrices A and B is equal to the unit matrix.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"How do you find the inverse of a 3\u00d73 matrix?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"The steps required to find the inverse of a 3\u00d73 matrix are:<br\/>\nCompute the determinant of the given matrix and check whether the matrix is invertible.<br\/>\nCalculate the determinant of 2\u00d72 minor matrices.<br\/>\nFormulate the matrix of cofactors.<br\/>\nTake the transpose of the cofactor matrix to get the adjugate matrix.<br\/>\nFinally, divide each term of the adjugate matrix by the determinant\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"Is adjoint and inverse the same?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"No, the adjoint matrix and inverse matrix are not the same. However, by dividing the each term of the adjoint of the matrix by the determinant of the original matrix, we get an inverse matrix.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t,\t\t\t\t{\n\t\t\t\t\"@type\": \"Question\",\n\t\t\t\t\"name\": \"How to do you know whether the given matrix has an inverse?\",\n\t\t\t\t\"acceptedAnswer\": {\n\t\t\t\t\t\"@type\": \"Answer\",\n\t\t\t\t\t\"text\": \"If the determinant of a given matrix is not equal to 0, i.e. it is non-singular, then the matrix is invertible.\"\n\t\t\t\t\t\t\t\t\t}\n\t\t\t}\n\t\t\t\t\t\t]\n\t}\n<\/script>\n\n","protected":false},"excerpt":{"rendered":"<p>To understand the Inverse of Matrices, one initially needs to understand Matrices. A matrix is a rectangular array of numbers, [&hellip;]<\/p>\n","protected":false},"author":53,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Inverse of Matrices","_yoast_wpseo_title":"Inverse of Matrices - Methods, Properties and Examples","_yoast_wpseo_metadesc":"The inverse of matrix is obtained by dividing the adjoint of the given matrix by the determinant of the given matrix.","custom_permalink":"topics\/inverse-of-matrices\/"},"categories":[8594,8591],"tags":[],"table_tags":[],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v17.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Inverse of Matrices - Methods, Properties and Examples<\/title>\n<meta name=\"description\" content=\"The inverse of matrix is obtained by dividing the adjoint of the given matrix by the determinant of the given matrix.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/infinitylearn.com\/surge\/topics\/inverse-of-matrices\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Inverse of Matrices - 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