{"id":191914,"date":"2025-02-17T03:33:53","date_gmt":"2025-02-17T03:33:53","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=191914"},"modified":"2025-02-17T03:33:57","modified_gmt":"2025-02-17T03:33:57","slug":"calculate-the-characteristic-polynomial-px-det-aai-of-the-matrix-a-for-aer-and","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/02\/17\/calculate-the-characteristic-polynomial-px-det-aai-of-the-matrix-a-for-aer-and\/","title":{"rendered":"Calculate the characteristic polynomial p(x) = det (AAI) of the matrix A for AER and"},"content":{"rendered":"\n<p>Calculate the characteristic polynomial p(x) = det (AAI) of the matrix A for AER and<\/p>\n\n\n\n<p>b) Determine the eigenvalues 21,12,13 of the matrix A, ie. the zeros of the polynomial p(\u03bb).<br>c) Give the eigenvectors for the eigenvalues, i.e. the &amp; solution set of the homogeneous linear system of equations (A-Ajl)x = 0 f for j = 1,2,3.<\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img decoding=\"async\" src=\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/02\/image-695.png\" alt=\"\" class=\"wp-image-191915\"\/><\/figure>\n\n\n\n<p><mark style=\"background-color:rgba(0, 0, 0, 0)\" class=\"has-inline-color has-ast-global-color-6-color\"><strong>The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n<p>Let&#8217;s break this down step by step for the problem involving the matrix ( A ), its characteristic polynomial, eigenvalues, and eigenvectors.<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">1. <strong>Characteristic Polynomial ( p(x) )<\/strong><\/h3>\n\n\n\n<p>The characteristic polynomial of a matrix ( A ) is calculated as the determinant of ( A &#8211; \\lambda I ), where ( I ) is the identity matrix and ( \\lambda ) represents the eigenvalues of ( A ). It\u2019s written as:<\/p>\n\n\n\n<p>[<br>p(\\lambda) = \\text{det}(A &#8211; \\lambda I)<br>]<\/p>\n\n\n\n<p>Here, ( A ) is the given matrix. To compute ( p(\\lambda) ), we substitute ( \\lambda ) into the equation and solve for the determinant of the matrix ( A &#8211; \\lambda I ). This determinant will give us the characteristic polynomial. After expanding this determinant, the roots of the polynomial are the eigenvalues of ( A ).<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">2. <strong>Eigenvalues<\/strong><\/h3>\n\n\n\n<p>The problem mentions the eigenvalues ( 21, 12, 13 ), meaning they are the roots of the characteristic polynomial ( p(\\lambda) ). These are the values of ( \\lambda ) for which the determinant ( \\text{det}(A &#8211; \\lambda I) = 0 ).<\/p>\n\n\n\n<p>Thus, the eigenvalues are:<br>[<br>\\lambda_1 = 21, \\quad \\lambda_2 = 12, \\quad \\lambda_3 = 13<br>]<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">3. <strong>Eigenvectors<\/strong><\/h3>\n\n\n\n<p>For each eigenvalue ( \\lambda ), the corresponding eigenvector ( v ) satisfies the equation:<\/p>\n\n\n\n<p>[<br>(A &#8211; \\lambda I)v = 0<br>]<\/p>\n\n\n\n<p>This is a system of linear equations. To find the eigenvectors corresponding to each eigenvalue, we substitute each eigenvalue ( \\lambda = 21, 12, 13 ) into the equation ( (A &#8211; \\lambda I)v = 0 ), and solve for the eigenvectors ( v ).<\/p>\n\n\n\n<p>The general process for solving this system involves:<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>Substituting the eigenvalue into the matrix equation ( (A &#8211; \\lambda I)v = 0 ).<\/li>\n\n\n\n<li>Row reducing the augmented matrix to find the solution set for ( v ).<\/li>\n<\/ol>\n\n\n\n<h3 class=\"wp-block-heading\">4. <strong>Conclusion<\/strong><\/h3>\n\n\n\n<p>The eigenvalues ( 21, 12, 13 ) give us the solutions to the characteristic polynomial, and solving the system ( (A &#8211; \\lambda I)v = 0 ) for each ( \\lambda ) provides the corresponding eigenvectors. The solution set of the linear system for each eigenvalue will give us the corresponding eigenvectors for the matrix.<\/p>\n\n\n\n<p>I&#8217;ll now generate the matrix ( A ) and corresponding characteristic polynomial image.<\/p>\n\n\n\n<p>It seems I couldn&#8217;t fetch the matrix image from the link provided. Could you upload the image directly here, or describe the matrix ( A ) so that I can assist you with the calculation and further steps?<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Calculate the characteristic polynomial p(x) = det (AAI) of the matrix A for AER and b) Determine the eigenvalues 21,12,13 of the matrix A, ie. the zeros of the polynomial p(\u03bb).c) Give the eigenvectors for the eigenvalues, i.e. the &amp; solution set of the homogeneous linear system of equations (A-Ajl)x = 0 f for j [&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-191914","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/191914","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=191914"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/191914\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=191914"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=191914"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=191914"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}