{"id":238307,"date":"2025-06-18T04:20:16","date_gmt":"2025-06-18T04:20:16","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=238307"},"modified":"2025-06-18T04:20:18","modified_gmt":"2025-06-18T04:20:18","slug":"let-a-be-a-3x2-matrix-and-b-a-2x3-matrix","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/18\/let-a-be-a-3x2-matrix-and-b-a-2x3-matrix\/","title":{"rendered":"let a be a 3×2 matrix and b a 2×3 matrix"},"content":{"rendered":"\n

let a be a 3×2 matrix and b a 2×3 matrix. show that c = a.b is a singular matrix.<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

Let matrix AAA be a 3\u00d723 \\times 23\u00d72 matrix and matrix BBB be a 2\u00d732 \\times 32\u00d73 matrix. Then, the product C=ABC = ABC=AB is defined and results in a 3\u00d733 \\times 33\u00d73 matrix.<\/p>\n\n\n\n

We aim to show that matrix CCC is singular<\/strong>, meaning it is not invertible<\/strong> or has determinant zero<\/strong>.<\/p>\n\n\n\n

\n\n\n\n

Step-by-step Analysis:<\/h3>\n\n\n\n
    \n
  1. Matrix dimensions<\/strong>:\n
      \n
    • AAA: 3 rows, 2 columns \u21d2A\u2208R3\u00d72\\Rightarrow A \\in \\mathbb{R}^{3 \\times 2}\u21d2A\u2208R3\u00d72<\/li>\n\n\n\n
    • BBB: 2 rows, 3 columns \u21d2B\u2208R2\u00d73\\Rightarrow B \\in \\mathbb{R}^{2 \\times 3}\u21d2B\u2208R2\u00d73<\/li>\n\n\n\n
    • C=AB\u21d2C\u2208R3\u00d73C = AB \\Rightarrow C \\in \\mathbb{R}^{3 \\times 3}C=AB\u21d2C\u2208R3\u00d73<\/li>\n<\/ul>\n<\/li>\n\n\n\n
    • Rank consideration<\/strong>:\n
        \n
      • The rank of a matrix product is at most the minimum of the ranks of the two matrices<\/strong> involved.<\/li>\n\n\n\n
      • Matrix AAA has at most rank 2 (since it has only 2 columns).<\/li>\n\n\n\n
      • Matrix BBB has at most rank 2 (since it has only 2 rows).<\/li>\n\n\n\n
      • Therefore, the rank of C=ABC = ABC=AB is at most 2.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
      • Conclusion<\/strong>:\n
          \n
        • Since CCC is a 3\u00d733 \\times 33\u00d73 matrix but has rank at most 2, its columns are linearly dependent.<\/li>\n\n\n\n
        • This means the determinant of CCC is zero.<\/li>\n\n\n\n
        • Hence, matrix CCC is singular<\/strong>.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n\n\n\n
          \n\n\n\n

          Explanation<\/h3>\n\n\n\n

          Let us consider a matrix AAA with 3 rows and 2 columns and a matrix BBB with 2 rows and 3 columns. The matrix multiplication ABABAB is defined, and the resulting matrix CCC will have dimensions 3 by 3. However, not every square matrix is invertible. To be invertible, a matrix must have full rank. For a 3\u00d733 \\times 33\u00d73 matrix, full rank means rank 3.<\/p>\n\n\n\n

          Matrix AAA maps vectors from R2\\mathbb{R}^2R2 to R3\\mathbb{R}^3R3, and matrix BBB maps vectors from R3\\mathbb{R}^3R3 to R2\\mathbb{R}^2R2. The composite transformation ABABAB thus maps vectors from R3\\mathbb{R}^3R3 back into R3\\mathbb{R}^3R3 through an intermediate step in R2\\mathbb{R}^2R2. However, because the image of BBB lies in a 2-dimensional space, any further mapping through AAA will result in a space of at most dimension 2. Consequently, the product ABABAB cannot span the full R3\\mathbb{R}^3R3 space.<\/p>\n\n\n\n

          This means that the columns of the resulting matrix CCC are linearly dependent, and the determinant of such a matrix must be zero. A zero determinant is the defining property of a singular matrix. Therefore, the product C=ABC = ABC=AB is necessarily a singular matrix due to the rank deficiency inherited from the dimensions of AAA and BBB.<\/p>\n\n\n\n

          \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

          let a be a 3×2 matrix and b a 2×3 matrix. show that c = a.b is a singular matrix. The Correct Answer and Explanation is: Let matrix AAA be a 3\u00d723 \\times 23\u00d72 matrix and matrix BBB be a 2\u00d732 \\times 32\u00d73 matrix. Then, the product C=ABC = ABC=AB is defined and results in […]<\/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-238307","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/238307","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=238307"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/238307\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=238307"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=238307"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=238307"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}