{"id":20024,"date":"2024-04-15T05:47:21","date_gmt":"2024-04-15T05:47:21","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20024"},"modified":"2024-04-15T05:47:21","modified_gmt":"2024-04-15T05:47:21","slug":"let-a-be-n-a%c2%97-n-real-valued-square-symmetric-matrix-of-rank-2-with-sumlimits_texti-1textn-sumlimits_textj-1textn-texta_textij2-5","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/let-a-be-n-a%c2%97-n-real-valued-square-symmetric-matrix-of-rank-2-with-sumlimits_texti-1textn-sumlimits_textj-1textn-texta_textij2-5\/","title":{"rendered":"Let A be n \u00c3\u0097 n real valued square symmetric matrix of rank 2 with \\[\\sum\\limits_{{\\text{i}} = 1}^{\\text{n}} {\\sum\\limits_{{\\text{j}} = 1}^{\\text{n}} {{\\text{A}}_{{\\text{ij}}}^2} } = 50.\\] Consider the following statements. I. One eigen value must be in [-5, 5] II. The eigen value with the largest magnitude must be strictly greater than 5. Which of the above statements about eigen values of A is\/are necessarily CORRECT? A. Both I and II B. I only C. II only D. Neither I nor II"},"content":{"rendered":"<p>[amp_mcq option1=&#8221;Both I and II&#8221; option2=&#8221;I only&#8221; option3=&#8221;II only&#8221; option4=&#8221;Neither I nor II&#8221; correct=&#8221;option2&#8243;]<!--more--><\/p>\n<p>The correct answer is: B. I only<\/p>\n<p>A real symmetric matrix has real eigenvalues. The sum of the eigenvalues of a real symmetric matrix is equal to the trace of the matrix, which is equal to the sum of the diagonal elements. In this case, the trace of $A$ is $50$. Since $A$ has rank 2, it has two nonzero eigenvalues. Therefore, the sum of the two nonzero eigenvalues of $A$ is $50$. This means that one of the eigenvalues must be in the range $[-50, 50]$. However, it is not necessarily the case that the eigenvalue with the largest magnitude must be strictly greater than 5. For example, if $A$ is the matrix $$A = \\begin{bmatrix} 10 &amp; 0 \\\\ 0 &amp; -10 \\end{bmatrix},$$ then the eigenvalues of $A$ are $10$ and $-10$, and the eigenvalue with the largest magnitude is $10$.<\/p>\n<p>Here is a more detailed explanation of each option:<\/p>\n<ul>\n<li>Option A: Both I and II. This is not necessarily true. For example, if $A$ is the matrix $$A = \\begin{bmatrix} 10 &amp; 0 \\\\ 0 &amp; -10 \\end{bmatrix},$$ then the eigenvalues of $A$ are $10$ and $-10$, and the eigenvalue with the largest magnitude is $10$. However, the eigenvalue with the smallest magnitude is not in the range $[-5, 5]$.<\/li>\n<li>Option B: I only. This is true. As explained above, the sum of the eigenvalues of a real symmetric matrix is equal to the trace of the matrix, which is equal to the sum of the diagonal elements. In this case, the trace of $A$ is $50$. Since $A$ has rank 2, it has two nonzero eigenvalues. Therefore, the sum of the two nonzero eigenvalues of $A$ is $50$. This means that one of the eigenvalues must be in the range $[-50, 50]$.<\/li>\n<li>Option C: II only. This is not necessarily true. For example, if $A$ is the matrix $$A = \\begin{bmatrix} 10 &amp; 0 \\\\ 0 &amp; -10 \\end{bmatrix},$$ then the eigenvalues of $A$ are $10$ and $-10$, and the eigenvalue with the largest magnitude is $10$. However, the eigenvalue with the largest magnitude is not strictly greater than 5.<\/li>\n<li>Option D: Neither I nor II. This is not necessarily true. For example, if $A$ is the matrix $$A = \\begin{bmatrix} 10 &amp; 0 \\\\ 0 &amp; -10 \\end{bmatrix},$$ then the eigenvalues of $A$ are $10$ and $-10$, and neither eigenvalue is in the range $[-5, 5]$.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>[amp_mcq option1=&#8221;Both I and II&#8221; option2=&#8221;I only&#8221; option3=&#8221;II only&#8221; option4=&#8221;Neither I nor II&#8221; correct=&#8221;option2&#8243;]<\/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-20024","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>Let A be n \u00c3\u0097 n real valued square symmetric matrix of rank 2 with \\[\\sum\\limits_{{\\text{i}} = 1}^{\\text{n}} {\\sum\\limits_{{\\text{j}} = 1}^{\\text{n}} {{\\text{A}}_{{\\text{ij}}}^2} } = 50.\\] Consider the following statements. I. One eigen value must be in [-5, 5] II. The eigen value with the largest magnitude must be strictly greater than 5. Which of the above statements about eigen values of A is\/are necessarily CORRECT? A. Both I and II B. I only C. II only D. Neither I nor II<\/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\/let-a-be-n-a\u0097-n-real-valued-square-symmetric-matrix-of-rank-2-with-sumlimits_texti-1textn-sumlimits_textj-1textn-texta_textij2-5\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Let A be n \u00c3\u0097 n real valued square symmetric matrix of rank 2 with \\[\\sum\\limits_{{\\text{i}} = 1}^{\\text{n}} {\\sum\\limits_{{\\text{j}} = 1}^{\\text{n}} {{\\text{A}}_{{\\text{ij}}}^2} } = 50.\\] Consider the following statements. I. One eigen value must be in [-5, 5] II. The eigen value with the largest magnitude must be strictly greater than 5. Which of the above statements about eigen values of A is\/are necessarily CORRECT? A. Both I and II B. I only C. II only D. 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I. One eigen value must be in [-5, 5] II. The eigen value with the largest magnitude must be strictly greater than 5. Which of the above statements about eigen values of A is\/are necessarily CORRECT? A. Both I and II B. I only C. II only D. Neither I nor II","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\/let-a-be-n-a\u0097-n-real-valued-square-symmetric-matrix-of-rank-2-with-sumlimits_texti-1textn-sumlimits_textj-1textn-texta_textij2-5\/","og_locale":"en_US","og_type":"article","og_title":"Let A be n \u00c3\u0097 n real valued square symmetric matrix of rank 2 with \\[\\sum\\limits_{{\\text{i}} = 1}^{\\text{n}} {\\sum\\limits_{{\\text{j}} = 1}^{\\text{n}} {{\\text{A}}_{{\\text{ij}}}^2} } = 50.\\] Consider the following statements. I. One eigen value must be in [-5, 5] II. The eigen value with the largest magnitude must be strictly greater than 5. Which of the above statements about eigen values of A is\/are necessarily CORRECT? A. Both I and II B. I only C. II only D. Neither I nor II","og_description":"[amp_mcq option1=&#8221;Both I and II&#8221; option2=&#8221;I only&#8221; option3=&#8221;II only&#8221; option4=&#8221;Neither I nor II&#8221; correct=&#8221;option2&#8243;]","og_url":"https:\/\/exam.pscnotes.com\/mcq\/let-a-be-n-a\u0097-n-real-valued-square-symmetric-matrix-of-rank-2-with-sumlimits_texti-1textn-sumlimits_textj-1textn-texta_textij2-5\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T05:47:21+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\/let-a-be-n-a%c2%97-n-real-valued-square-symmetric-matrix-of-rank-2-with-sumlimits_texti-1textn-sumlimits_textj-1textn-texta_textij2-5\/","url":"https:\/\/exam.pscnotes.com\/mcq\/let-a-be-n-a%c2%97-n-real-valued-square-symmetric-matrix-of-rank-2-with-sumlimits_texti-1textn-sumlimits_textj-1textn-texta_textij2-5\/","name":"Let A be n \u00c3\u0097 n real valued square symmetric matrix of rank 2 with \\[\\sum\\limits_{{\\text{i}} = 1}^{\\text{n}} {\\sum\\limits_{{\\text{j}} = 1}^{\\text{n}} {{\\text{A}}_{{\\text{ij}}}^2} } = 50.\\] Consider the following statements. I. One eigen value must be in [-5, 5] II. The eigen value with the largest magnitude must be strictly greater than 5. Which of the above statements about eigen values of A is\/are necessarily CORRECT? A. Both I and II B. I only C. II only D. 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