{"id":246511,"date":"2025-07-07T08:27:55","date_gmt":"2025-07-07T08:27:55","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=246511"},"modified":"2025-07-07T08:27:59","modified_gmt":"2025-07-07T08:27:59","slug":"prove-that-if-a-and-b-are-both-invertible-then-ab-must-also-be-invertible","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/07\/07\/prove-that-if-a-and-b-are-both-invertible-then-ab-must-also-be-invertible\/","title":{"rendered":"Prove that if A and B are both invertible then AB must also be invertible"},"content":{"rendered":"\n<p>Prove that if A and B are both invertible then AB must also be invertible.<\/p>\n\n\n\n<p><strong><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\">The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n<p>To prove that if matrices A and B are both invertible, then the matrix product AB is also invertible, we proceed as follows:<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Given:<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>A is an invertible matrix.<\/li>\n\n\n\n<li>B is an invertible matrix.<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Goal:<\/h3>\n\n\n\n<p>We want to show that AB is also invertible and find its inverse.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Proof:<\/h3>\n\n\n\n<p>Since A is invertible, there exists a matrix A\u22121A^{-1} such that: A\u22121A=IA^{-1}A = I<\/p>\n\n\n\n<p>where II is the identity matrix. Similarly, since B is invertible, there exists a matrix B\u22121B^{-1} such that: B\u22121B=IB^{-1}B = I<\/p>\n\n\n\n<p>Now, consider the product ABAB. We aim to find a matrix XX such that: (AB)X=IandX(AB)=I(AB)X = I \\quad \\text{and} \\quad X(AB) = I<\/p>\n\n\n\n<p>To find such a matrix XX, let&#8217;s test with X=B\u22121A\u22121X = B^{-1}A^{-1}, the product of the inverses of B and A. First, compute: (AB)(B\u22121A\u22121)=A(BB\u22121)A\u22121=AIA\u22121=AA\u22121=I(AB)(B^{-1}A^{-1}) = A(BB^{-1})A^{-1} = AIA^{-1} = AA^{-1} = I<\/p>\n\n\n\n<p>Thus, (AB)(B\u22121A\u22121)=I(AB)(B^{-1}A^{-1}) = I.<\/p>\n\n\n\n<p>Next, compute the other product: (B\u22121A\u22121)(AB)=B\u22121(A\u22121A)B=B\u22121IB=B\u22121B=I(B^{-1}A^{-1})(AB) = B^{-1}(A^{-1}A)B = B^{-1}IB = B^{-1}B = I<\/p>\n\n\n\n<p>Thus, (B\u22121A\u22121)(AB)=I(B^{-1}A^{-1})(AB) = I.<\/p>\n\n\n\n<p>Since both products result in the identity matrix, B\u22121A\u22121B^{-1}A^{-1} is the inverse of ABAB. Therefore, ABAB is invertible, and its inverse is: (AB)\u22121=B\u22121A\u22121(AB)^{-1} = B^{-1}A^{-1}<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Conclusion:<\/h3>\n\n\n\n<p>If both A and B are invertible, then the product ABAB is also invertible, and its inverse is B\u22121A\u22121B^{-1}A^{-1}.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Prove that if A and B are both invertible then AB must also be invertible. The correct answer and explanation is: To prove that if matrices A and B are both invertible, then the matrix product AB is also invertible, we proceed as follows: Given: Goal: We want to show that AB is also invertible [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","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":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-246511","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/246511","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=246511"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/246511\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=246511"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=246511"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=246511"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}