{"id":155244,"date":"2022-03-26T00:25:51","date_gmt":"2022-03-25T18:55:51","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/inverse-of-a-matrix-using-minors-cofactors-and-adjugate\/"},"modified":"2022-12-31T15:17:06","modified_gmt":"2022-12-31T09:47:06","slug":"inverse-of-a-matrix-using-minors-cofactors-and-adjugate","status":"publish","type":"post","link":"https:\/\/infinitylearn.com\/surge\/maths\/inverse-of-a-matrix-using-minors-cofactors-and-adjugate\/","title":{"rendered":"Inverse of a Matrix &#8211; Using Minors, Cofactors and Adjugate"},"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'><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/inverse-of-a-matrix-using-minors-cofactors-and-adjugate\/#Inverse_of_Matrix\" title=\"Inverse of Matrix\">Inverse of Matrix<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/inverse-of-a-matrix-using-minors-cofactors-and-adjugate\/#What_is_the_Matrix_of_Cofactors\" title=\"What is the Matrix of Cofactors?\">What is the Matrix of Cofactors?<\/a><\/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\/maths\/inverse-of-a-matrix-using-minors-cofactors-and-adjugate\/#Matrix_of_Cofactors_Formula\" title=\"Matrix of Cofactors Formula\">Matrix of Cofactors Formula<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/inverse-of-a-matrix-using-minors-cofactors-and-adjugate\/#Matrix_of_Cofactors_Example\" title=\"Matrix of Cofactors Example\">Matrix of Cofactors Example<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/inverse-of-a-matrix-using-minors-cofactors-and-adjugate\/#Adjugate_Matrix\" title=\"Adjugate Matrix\">Adjugate Matrix<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-6\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/inverse-of-a-matrix-using-minors-cofactors-and-adjugate\/#Minor_of_Matrices\" title=\"Minor of Matrices\">Minor of Matrices<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-7\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/inverse-of-a-matrix-using-minors-cofactors-and-adjugate\/#Minors_of_Matrix_Example\" title=\"Minors of Matrix Example\">Minors of Matrix Example<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-8\" href=\"https:\/\/infinitylearn.com\/surge\/maths\/inverse-of-a-matrix-using-minors-cofactors-and-adjugate\/#Finding_Inverse_of_Matrix_Using_Minors_Cofactors_and_Adjugate\" title=\"Finding Inverse of Matrix Using Minors, Cofactors, and Adjugate?\">Finding Inverse of Matrix Using Minors, Cofactors, and Adjugate?<\/a><\/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\/maths\/inverse-of-a-matrix-using-minors-cofactors-and-adjugate\/#Inverse_of_Matrix_Using_Minors_Cofactors_and_Adjugate_Example\" title=\"Inverse of Matrix Using Minors, Cofactors, and Adjugate Example\">Inverse of Matrix Using Minors, Cofactors, and Adjugate Example<\/a><\/li><\/ul><\/nav><\/div>\n<h2><span class=\"ez-toc-section\" id=\"Inverse_of_Matrix\"><\/span>Inverse of Matrix<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The inverse of a matrix is a matrix that &#8220;undoes&#8221; the original matrix. To find the inverse of a matrix, you must first determine its determinant. If the determinant is not zero, the inverse matrix can be found using the inverse matrix formula.<\/p>\n<p><img loading=\"lazy\" class=\"aligncenter wp-image-155243 size-full\" src=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/inverse-of-a-matrix-using-minors-cofactors-and-adjugate.jpg\" alt=\"\" width=\"606\" height=\"428\" srcset=\"https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/inverse-of-a-matrix-using-minors-cofactors-and-adjugate.jpg?v=1648234546 606w, https:\/\/infinitylearn.com\/surge\/wp-content\/uploads\/2022\/03\/inverse-of-a-matrix-using-minors-cofactors-and-adjugate-300x212.jpg?v=1648234546 300w\" sizes=\"(max-width: 606px) 100vw, 606px\" \/><\/p>\n<h2><span class=\"ez-toc-section\" id=\"What_is_the_Matrix_of_Cofactors\"><\/span>What is the Matrix of Cofactors?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A matrix of cofactors is a square matrix that is used to calculate the determinant of a larger matrix. The matrix of cofactors is created by taking the cofactor of each element in a larger matrix. The cofactor of an element is the determinant of the matrix that is created by taking the element out of the larger matrix.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Matrix_of_Cofactors_Formula\"><\/span>Matrix of Cofactors Formula<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The matrix of cofactors formula calculates the determinant of a square matrix using the cofactors of the matrix. The cofactors of a matrix are the elements of the matrix that are located on the main diagonal. The matrix of cofactors formula is:<\/p>\n<p>D = det(A) = (a ij )<\/p>\n<p>Where:<\/p>\n<p>D is the determinant of the matrix.<\/p>\n<p>A is the square matrix.<\/p>\n<p>a ij is the cofactor of the element in the ith row and the jth column of the matrix.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Matrix_of_Cofactors_Example\"><\/span>Matrix of Cofactors Example<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The following is a matrix of cofactors for a 4&#215;4 matrix:<\/p>\n<p>1 0 0 0<br \/>\n0 1 0 0<br \/>\n0 0 1 0<br \/>\n0 0 0 1<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Adjugate_Matrix\"><\/span>Adjugate Matrix<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The adjugate matrix is a matrix that is created by adding the transpose of a matrix to the matrix itself. This matrix is used to solve systems of linear equations.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Minor_of_Matrices\"><\/span>Minor of Matrices<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>A minor of a matrix is a matrix that is obtained by deleting one or more rows and\/or columns from the original matrix.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Minors_of_Matrix_Example\"><\/span>Minors of Matrix Example<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Let&#8217;s take a look at the minors of the matrix A =<\/p>\n<p>A =<\/p>\n[1 2 3]\n[4 5 6]\n<p>The minors of A are the matrices formed by deleting any row or column from A.<\/p>\n<p>For example, the minor of A with respect to the first row is<\/p>\n<p>A1 =<\/p>\n[1 2 3]\n[4 5 6]\n<p>The minor of A with respect to the first column is<\/p>\n<p>A1 =<\/p>\n[1 2 3]\n[4 5 6]\n<p>The minor of A with respect to the second row is<\/p>\n<p>A2 =<\/p>\n[2 3]\n[4 5 6]\n<p>The minor of A with respect to the second column is<\/p>\n<p>A2 =<\/p>\n[2 3]\n[4 5 6]\n<h2><span class=\"ez-toc-section\" id=\"Finding_Inverse_of_Matrix_Using_Minors_Cofactors_and_Adjugate\"><\/span>Finding Inverse of Matrix Using Minors, Cofactors, and Adjugate?<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>The inverse of a matrix can be found using minors, cofactors, and the adjugate. First, the minors of the matrix are found. Next, the cofactors of the matrix are found. Finally, the adjugate is used to find the inverse of the matrix.<\/p>\n<h2><span class=\"ez-toc-section\" id=\"Inverse_of_Matrix_Using_Minors_Cofactors_and_Adjugate_Example\"><\/span>Inverse of Matrix Using Minors, Cofactors, and Adjugate Example<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Given the matrix:<\/p>\n<p>A =<\/p>\n<p>We can use minors, cofactors, and the adjugate to find the inverse matrix.<\/p>\n<p>First, we find the minors of A.<\/p>\n<p>The minors of A are:<\/p>\n<p>A1 =<\/p>\n<p>A2 =<\/p>\n<p>A3 =<\/p>\n<p>A4 =<\/p>\n<p>A5 =<\/p>\n<p>A6 =<\/p>\n<p>A7 =<\/p>\n<p>A8 =<\/p>\n<p>A9 =<\/p>\n<p>A10 =<\/p>\n<p>Next, we find the cofactors of A.<\/p>\n<p>The cofactors of A are:<\/p>\n<p>A1 =<\/p>\n<p>A2 =<\/p>\n<p>A3 =<\/p>\n<p>A4 =<\/p>\n<p>A5 =<\/p>\n<p>A6 =<\/p>\n<p>A7 =<\/p>\n<p>A8 =<\/p>\n<p>A9 =<\/p>\n<p>A10 =<\/p>\n<p>Finally, we find the adjugate of A.<\/p>\n<p>The adjugate of A is:<\/p>\n<p>A1 =<\/p>\n<p>A2 =<\/p>\n<p>A3 =<\/p>\n<p>A4 =<\/p>\n<p>A5 =<\/p>\n<p>A6 =<\/p>\n<p>A7 =<\/p>\n<p>A8 =<\/p>\n<p>A9 =<\/p>\n<p>A10 =<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Inverse of Matrix The inverse of a matrix is a matrix that &#8220;undoes&#8221; the original matrix. To find the inverse [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_yoast_wpseo_focuskw":"Inverse of a Matrix","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Learn about inverse of a matrix using minors, cofactors and adjugate in detail explained by subject experts on infinitylearn.com.","custom_permalink":"maths\/inverse-of-a-matrix-using-minors-cofactors-and-adjugate\/"},"categories":[13],"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 a Matrix - Using Minors, Cofactors and Adjugate - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"Learn about inverse of a matrix using minors, cofactors and adjugate in detail explained by subject experts on infinitylearn.com.\" \/>\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\/maths\/inverse-of-a-matrix-using-minors-cofactors-and-adjugate\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Inverse of a Matrix - 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