{"id":19995,"date":"2024-04-15T05:46:58","date_gmt":"2024-04-15T05:46:58","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=19995"},"modified":"2024-04-15T05:46:58","modified_gmt":"2024-04-15T05:46:58","slug":"which-one-of-the-following-statements-is-true-about-every-n-a%c2%97-n-matrix-with-only-real-eigen-values-a-if-the-trace-of-the-matrix-is-positive-and-the-determinant-of-the-matrix-is-negative-at-le","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/which-one-of-the-following-statements-is-true-about-every-n-a%c2%97-n-matrix-with-only-real-eigen-values-a-if-the-trace-of-the-matrix-is-positive-and-the-determinant-of-the-matrix-is-negative-at-le\/","title":{"rendered":"Which one of the following statements is TRUE about every n \u00c3\u0097 n matrix with only real eigen values? A. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative B. If the trace of the matrix is positive, all its eigen values are positive C. If the determinant of the matrix is positive, all its eigen values are positive D. If the product of the trace and determinant of the matrix is positive, all its eigen values are positive"},"content":{"rendered":"<p>[amp_mcq option1=&#8221;If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative&#8221; option2=&#8221;If the trace of the matrix is positive, all its eigen values are positive&#8221; option3=&#8221;If the determinant of the matrix is positive, all its eigen values are positive&#8221; option4=&#8221;If the product of the trace and determinant of the matrix is positive, all its eigen values are positive&#8221; correct=&#8221;option2&#8243;]<!--more--><\/p>\n<p>The correct answer is $\\boxed{\\text{A}}$.<\/p>\n<p>The trace of a matrix is the sum of its eigenvalues, and the determinant of a matrix is the product of its eigenvalues. If the trace of a matrix is positive and the determinant of the matrix is negative, then at least one of its eigenvalues must be negative. This is because the trace and determinant of a matrix are both invariants of the matrix, meaning that they do not change when the matrix is transformed by a similarity transformation. Therefore, if the trace and determinant of a matrix have different signs, then the matrix must have at least one eigenvalue with a negative real part.<\/p>\n<p>The other options are incorrect. Option $\\text{B}$ is incorrect because it is possible for a matrix to have a positive trace and negative eigenvalues. For example, the matrix $$A = \\begin{bmatrix} 1 &amp; 2 \\\\ 0 &amp; -1 \\end{bmatrix}$$ has a trace of 1 and a determinant of -1, but its eigenvalues are both negative. Option $\\text{C}$ is incorrect because it is possible for a matrix to have a positive determinant and negative eigenvalues. For example, the matrix $$B = \\begin{bmatrix} 1 &amp; 0 \\\\ 0 &amp; -2 \\end{bmatrix}$$ has a determinant of 2 and a negative eigenvalue. Option $\\text{D}$ is incorrect because it is possible for a matrix to have a positive product of the trace and determinant and negative eigenvalues. For example, the matrix $$C = \\begin{bmatrix} 1 &amp; 2 \\\\ 0 &amp; -3 \\end{bmatrix}$$ has a product of the trace and determinant of 3 and a negative eigenvalue.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>[amp_mcq option1=&#8221;If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative&#8221; option2=&#8221;If the trace of the matrix is positive, all its eigen values are positive&#8221; option3=&#8221;If the determinant of the matrix is positive, all its eigen values are positive&#8221; option4=&#8221;If &#8230; <\/p>\n<p class=\"read-more-container\"><a title=\"Which one of the following statements is TRUE about every n \u00c3\u0097 n matrix with only real eigen values? A. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative B. If the trace of the matrix is positive, all its eigen values are positive C. If the determinant of the matrix is positive, all its eigen values are positive D. If the product of the trace and determinant of the matrix is positive, all its eigen values are positive\" class=\"read-more button\" href=\"https:\/\/exam.pscnotes.com\/mcq\/which-one-of-the-following-statements-is-true-about-every-n-a%c2%97-n-matrix-with-only-real-eigen-values-a-if-the-trace-of-the-matrix-is-positive-and-the-determinant-of-the-matrix-is-negative-at-le\/#more-19995\">Detailed Solution<span class=\"screen-reader-text\">Which one of the following statements is TRUE about every n \u00c3\u0097 n matrix with only real eigen values? A. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative B. If the trace of the matrix is positive, all its eigen values are positive C. If the determinant of the matrix is positive, all its eigen values are positive D. If the product of the trace and determinant of the matrix is positive, all its eigen values are positive<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[489],"tags":[],"class_list":["post-19995","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","no-featured-image-padding"],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Which one of the following statements is TRUE about every n \u00c3\u0097 n matrix with only real eigen values? A. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative B. If the trace of the matrix is positive, all its eigen values are positive C. If the determinant of the matrix is positive, all its eigen values are positive D. If the product of the trace and determinant of the matrix is positive, all its eigen values are positive<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/exam.pscnotes.com\/mcq\/which-one-of-the-following-statements-is-true-about-every-n-a\u0097-n-matrix-with-only-real-eigen-values-a-if-the-trace-of-the-matrix-is-positive-and-the-determinant-of-the-matrix-is-negative-at-le\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Which one of the following statements is TRUE about every n \u00c3\u0097 n matrix with only real eigen values? A. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative B. If the trace of the matrix is positive, all its eigen values are positive C. If the determinant of the matrix is positive, all its eigen values are positive D. If the product of the trace and determinant of the matrix is positive, all its eigen values are positive\" \/>\n<meta property=\"og:description\" content=\"[amp_mcq option1=&#8221;If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative&#8221; option2=&#8221;If the trace of the matrix is positive, all its eigen values are positive&#8221; option3=&#8221;If the determinant of the matrix is positive, all its eigen values are positive&#8221; option4=&#8221;If ... Detailed SolutionWhich one of the following statements is TRUE about every n \u00c3\u0097 n matrix with only real eigen values? A. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative B. If the trace of the matrix is positive, all its eigen values are positive C. If the determinant of the matrix is positive, all its eigen values are positive D. If the product of the trace and determinant of the matrix is positive, all its eigen values are positive\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/which-one-of-the-following-statements-is-true-about-every-n-a\u0097-n-matrix-with-only-real-eigen-values-a-if-the-trace-of-the-matrix-is-positive-and-the-determinant-of-the-matrix-is-negative-at-le\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T05:46:58+00:00\" \/>\n<meta name=\"author\" content=\"rawan239\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"rawan239\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"2 minutes\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"Which one of the following statements is TRUE about every n \u00c3\u0097 n matrix with only real eigen values? A. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative B. If the trace of the matrix is positive, all its eigen values are positive C. If the determinant of the matrix is positive, all its eigen values are positive D. If the product of the trace and determinant of the matrix is positive, all its eigen values are positive","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/which-one-of-the-following-statements-is-true-about-every-n-a\u0097-n-matrix-with-only-real-eigen-values-a-if-the-trace-of-the-matrix-is-positive-and-the-determinant-of-the-matrix-is-negative-at-le\/","og_locale":"en_US","og_type":"article","og_title":"Which one of the following statements is TRUE about every n \u00c3\u0097 n matrix with only real eigen values? A. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative B. If the trace of the matrix is positive, all its eigen values are positive C. If the determinant of the matrix is positive, all its eigen values are positive D. If the product of the trace and determinant of the matrix is positive, all its eigen values are positive","og_description":"[amp_mcq option1=&#8221;If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative&#8221; option2=&#8221;If the trace of the matrix is positive, all its eigen values are positive&#8221; option3=&#8221;If the determinant of the matrix is positive, all its eigen values are positive&#8221; option4=&#8221;If ... Detailed SolutionWhich one of the following statements is TRUE about every n \u00c3\u0097 n matrix with only real eigen values? A. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative B. If the trace of the matrix is positive, all its eigen values are positive C. If the determinant of the matrix is positive, all its eigen values are positive D. If the product of the trace and determinant of the matrix is positive, all its eigen values are positive","og_url":"https:\/\/exam.pscnotes.com\/mcq\/which-one-of-the-following-statements-is-true-about-every-n-a\u0097-n-matrix-with-only-real-eigen-values-a-if-the-trace-of-the-matrix-is-positive-and-the-determinant-of-the-matrix-is-negative-at-le\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T05:46:58+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"2 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/which-one-of-the-following-statements-is-true-about-every-n-a%c2%97-n-matrix-with-only-real-eigen-values-a-if-the-trace-of-the-matrix-is-positive-and-the-determinant-of-the-matrix-is-negative-at-le\/","url":"https:\/\/exam.pscnotes.com\/mcq\/which-one-of-the-following-statements-is-true-about-every-n-a%c2%97-n-matrix-with-only-real-eigen-values-a-if-the-trace-of-the-matrix-is-positive-and-the-determinant-of-the-matrix-is-negative-at-le\/","name":"Which one of the following statements is TRUE about every n \u00c3\u0097 n matrix with only real eigen values? A. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative B. If the trace of the matrix is positive, all its eigen values are positive C. If the determinant of the matrix is positive, all its eigen values are positive D. If the product of the trace and determinant of the matrix is positive, all its eigen values are positive","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2024-04-15T05:46:58+00:00","dateModified":"2024-04-15T05:46:58+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/which-one-of-the-following-statements-is-true-about-every-n-a%c2%97-n-matrix-with-only-real-eigen-values-a-if-the-trace-of-the-matrix-is-positive-and-the-determinant-of-the-matrix-is-negative-at-le\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/which-one-of-the-following-statements-is-true-about-every-n-a%c2%97-n-matrix-with-only-real-eigen-values-a-if-the-trace-of-the-matrix-is-positive-and-the-determinant-of-the-matrix-is-negative-at-le\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/which-one-of-the-following-statements-is-true-about-every-n-a%c2%97-n-matrix-with-only-real-eigen-values-a-if-the-trace-of-the-matrix-is-positive-and-the-determinant-of-the-matrix-is-negative-at-le\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"mcq","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/"},{"@type":"ListItem","position":3,"name":"Linear Algebra","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/linear-algebra\/"},{"@type":"ListItem","position":4,"name":"Which one of the following statements is TRUE about every n \u00c3\u0097 n matrix with only real eigen values? A. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative B. If the trace of the matrix is positive, all its eigen values are positive C. If the determinant of the matrix is positive, all its eigen values are positive D. If the product of the trace and determinant of the matrix is positive, all its eigen values are positive"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/19995","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=19995"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/19995\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=19995"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=19995"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=19995"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}