{"id":19982,"date":"2024-04-15T05:46:49","date_gmt":"2024-04-15T05:46:49","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=19982"},"modified":"2024-04-15T05:46:49","modified_gmt":"2024-04-15T05:46:49","slug":"for-the-matrix-texta-left-beginarray20c-3-22-0-21-001-endarray-right-one-of-the-eigen-values-is-equal-to-2-which-of-the-following-is-an","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/for-the-matrix-texta-left-beginarray20c-3-22-0-21-001-endarray-right-one-of-the-eigen-values-is-equal-to-2-which-of-the-following-is-an\/","title":{"rendered":"For the matrix \\[{\\text{A}} = \\left[ {\\begin{array}{*{20}{c}} 3&{ – 2}&2 \\\\ 0&{ – 2}&1 \\\\ 0&0&1 \\end{array}} \\right],\\] one of the eigen values is equal to -2. Which of the following is an eigen vector? A. \\[\\left[ {\\begin{array}{*{20}{c}} 3 \\\\ { – 2} \\\\ 1 \\end{array}} \\right]\\] B. \\[\\left[ {\\begin{array}{*{20}{c}} { – 3} \\\\ 2 \\\\ { – 1} \\end{array}} \\right]\\] C. \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { – 2} \\\\ 3 \\end{array}} \\right]\\] D. \\[\\left[ {\\begin{array}{*{20}{c}} 2 \\\\ 5 \\\\ 0 \\end{array}} \\right]\\]"},"content":{"rendered":"

[amp_mcq option1=”\\[\\left[ {\\begin{array}{*{20}{c}} 3 \\\\ { – 2} \\\\ 1 \\end{array}} \\right]\\]” option2=”\\[\\left[ {\\begin{array}{*{20}{c}} { – 3} \\\\ 2 \\\\ { – 1} \\end{array}} \\right]\\]” option3=”\\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { – 2} \\\\ 3 \\end{array}} \\right]\\]” option4=”\\[\\left[ {\\begin{array}{*{20}{c}} 2 \\\\ 5 \\\\ 0 \\end{array}} \\right]\\]” correct=”option1″]<\/p>\n

The correct answer is $\\boxed{\\left[ {\\begin{array}{*{20}{c}} 1 \\ { – 2} \\ 3 \\end{array}} \\right]}$.<\/p>\n

An eigenvector of a matrix $A$ is a nonzero vector $v$ such that there exists a scalar $\\lambda$, called the eigenvalue, such that $Av=\\lambda v$.<\/p>\n

To find the eigenvalues and eigenvectors of a matrix, we can use the following steps:<\/p>\n

    \n
  1. Find the characteristic polynomial of $A$, which is the determinant of the matrix $|A-\\lambda I|$.<\/li>\n
  2. Solve the characteristic polynomial for $\\lambda$.<\/li>\n
  3. For each eigenvalue $\\lambda$, find all vectors $v$ such that $Av=\\lambda v$. These vectors are the eigenvectors of $A$ corresponding to the eigenvalue $\\lambda$.<\/li>\n<\/ol>\n

    In this case, the characteristic polynomial of $A$ is $p(\\lambda)=-\\lambda^3+5\\lambda^2-11\\lambda+6$. We can factor this polynomial as follows:<\/p>\n

    $$p(\\lambda)=(\\lambda-2)^2(\\lambda-3)$$<\/p>\n

    Therefore, the eigenvalues of $A$ are $2$ and $3$.<\/p>\n

    To find the eigenvectors corresponding to the eigenvalue $2$, we can solve the equation $A v = 2 v$. This gives us the system of equations:<\/p>\n

    $$\\begin{align}
    \n3v_1-2v_2+2v_3 &= 0 \\
    \n0v_1-2v_2+v_3 &= 0 \\
    \n0v_1+0v_2+3v_3 &= 0
    \n\\end{align<\/em>}$$<\/p>\n

    Solving this system of equations, we find that the only solution is $v=\\left[ {\\begin{array}{*{20}{c}} 1 \\ { – 2} \\ 3 \\end{array}} \\right]$.<\/p>\n

    Therefore, the eigenvector of $A$ corresponding to the eigenvalue $2$ is $\\boxed{\\left[ {\\begin{array}{*{20}{c}} 1 \\ { – 2} \\ 3 \\end{array}} \\right]}$.<\/p>\n","protected":false},"excerpt":{"rendered":"

    [amp_mcq option1=”\\[\\left[ {\\begin{array}{*{20}{c}} 3 \\\\ { – 2} \\\\ 1 \\end{array}} \\right]\\]” option2=”\\[\\left[ {\\begin{array}{*{20}{c}} { – 3} \\\\ 2 \\\\ { – 1} \\end{array}} \\right]\\]” option3=”\\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { – 2} \\\\ 3 \\end{array}} \\right]\\]” option4=”\\[\\left[ {\\begin{array}{*{20}{c}} 2 \\\\ 5 \\\\ 0 \\end{array}} \\right]\\]” correct=”option1″]<\/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-19982","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","no-featured-image-padding"],"yoast_head":"\nFor the matrix \\[{\\text{A}} = \\left[ {\\begin{array}{*{20}{c}} 3&{ - 2}&2 \\\\ 0&{ - 2}&1 \\\\ 0&0&1 \\end{array}} \\right],\\] one of the eigen values is equal to -2. Which of the following is an eigen vector? A. \\[\\left[ {\\begin{array}{*{20}{c}} 3 \\\\ { - 2} \\\\ 1 \\end{array}} \\right]\\] B. \\[\\left[ {\\begin{array}{*{20}{c}} { - 3} \\\\ 2 \\\\ { - 1} \\end{array}} \\right]\\] C. \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { - 2} \\\\ 3 \\end{array}} \\right]\\] D. \\[\\left[ {\\begin{array}{*{20}{c}} 2 \\\\ 5 \\\\ 0 \\end{array}} \\right]\\]<\/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\/for-the-matrix-texta-left-beginarray20c-3-22-0-21-001-endarray-right-one-of-the-eigen-values-is-equal-to-2-which-of-the-following-is-an\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"For the matrix \\[{\\text{A}} = \\left[ {\\begin{array}{*{20}{c}} 3&{ - 2}&2 \\\\ 0&{ - 2}&1 \\\\ 0&0&1 \\end{array}} \\right],\\] one of the eigen values is equal to -2. Which of the following is an eigen vector? A. \\[\\left[ {\\begin{array}{*{20}{c}} 3 \\\\ { - 2} \\\\ 1 \\end{array}} \\right]\\] B. \\[\\left[ {\\begin{array}{*{20}{c}} { - 3} \\\\ 2 \\\\ { - 1} \\end{array}} \\right]\\] C. \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { - 2} \\\\ 3 \\end{array}} \\right]\\] D. \\[\\left[ {\\begin{array}{*{20}{c}} 2 \\\\ 5 \\\\ 0 \\end{array}} \\right]\\]\" \/>\n<meta property=\"og:description\" content=\"[amp_mcq option1=”[left[ {begin{array}{*{20}{c}} 3 \\ { – 2} \\ 1 end{array}} right]]” option2=”[left[ {begin{array}{*{20}{c}} { – 3} \\ 2 \\ { – 1} end{array}} right]]” option3=”[left[ {begin{array}{*{20}{c}} 1 \\ { – 2} \\ 3 end{array}} right]]” option4=”[left[ {begin{array}{*{20}{c}} 2 \\ 5 \\ 0 end{array}} right]]” correct=”option1″]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/for-the-matrix-texta-left-beginarray20c-3-22-0-21-001-endarray-right-one-of-the-eigen-values-is-equal-to-2-which-of-the-following-is-an\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T05:46:49+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=\"1 minute\" \/>\n<!-- \/ Yoast SEO Premium plugin. -->","yoast_head_json":{"title":"For the matrix \\[{\\text{A}} = \\left[ {\\begin{array}{*{20}{c}} 3&{ - 2}&2 \\\\ 0&{ - 2}&1 \\\\ 0&0&1 \\end{array}} \\right],\\] one of the eigen values is equal to -2. Which of the following is an eigen vector? A. \\[\\left[ {\\begin{array}{*{20}{c}} 3 \\\\ { - 2} \\\\ 1 \\end{array}} \\right]\\] B. \\[\\left[ {\\begin{array}{*{20}{c}} { - 3} \\\\ 2 \\\\ { - 1} \\end{array}} \\right]\\] C. \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { - 2} \\\\ 3 \\end{array}} \\right]\\] D. \\[\\left[ {\\begin{array}{*{20}{c}} 2 \\\\ 5 \\\\ 0 \\end{array}} \\right]\\]","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\/for-the-matrix-texta-left-beginarray20c-3-22-0-21-001-endarray-right-one-of-the-eigen-values-is-equal-to-2-which-of-the-following-is-an\/","og_locale":"en_US","og_type":"article","og_title":"For the matrix \\[{\\text{A}} = \\left[ {\\begin{array}{*{20}{c}} 3&{ - 2}&2 \\\\ 0&{ - 2}&1 \\\\ 0&0&1 \\end{array}} \\right],\\] one of the eigen values is equal to -2. Which of the following is an eigen vector? A. \\[\\left[ {\\begin{array}{*{20}{c}} 3 \\\\ { - 2} \\\\ 1 \\end{array}} \\right]\\] B. \\[\\left[ {\\begin{array}{*{20}{c}} { - 3} \\\\ 2 \\\\ { - 1} \\end{array}} \\right]\\] C. \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { - 2} \\\\ 3 \\end{array}} \\right]\\] D. \\[\\left[ {\\begin{array}{*{20}{c}} 2 \\\\ 5 \\\\ 0 \\end{array}} \\right]\\]","og_description":"[amp_mcq option1=”[left[ {begin{array}{*{20}{c}} 3 \\ { – 2} \\ 1 end{array}} right]]” option2=”[left[ {begin{array}{*{20}{c}} { – 3} \\ 2 \\ { – 1} end{array}} right]]” option3=”[left[ {begin{array}{*{20}{c}} 1 \\ { – 2} \\ 3 end{array}} right]]” option4=”[left[ {begin{array}{*{20}{c}} 2 \\ 5 \\ 0 end{array}} right]]” correct=”option1″]","og_url":"https:\/\/exam.pscnotes.com\/mcq\/for-the-matrix-texta-left-beginarray20c-3-22-0-21-001-endarray-right-one-of-the-eigen-values-is-equal-to-2-which-of-the-following-is-an\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T05:46:49+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/for-the-matrix-texta-left-beginarray20c-3-22-0-21-001-endarray-right-one-of-the-eigen-values-is-equal-to-2-which-of-the-following-is-an\/","url":"https:\/\/exam.pscnotes.com\/mcq\/for-the-matrix-texta-left-beginarray20c-3-22-0-21-001-endarray-right-one-of-the-eigen-values-is-equal-to-2-which-of-the-following-is-an\/","name":"For the matrix \\[{\\text{A}} = \\left[ {\\begin{array}{*{20}{c}} 3&{ - 2}&2 \\\\ 0&{ - 2}&1 \\\\ 0&0&1 \\end{array}} \\right],\\] one of the eigen values is equal to -2. Which of the following is an eigen vector? A. \\[\\left[ {\\begin{array}{*{20}{c}} 3 \\\\ { - 2} \\\\ 1 \\end{array}} \\right]\\] B. \\[\\left[ {\\begin{array}{*{20}{c}} { - 3} \\\\ 2 \\\\ { - 1} \\end{array}} \\right]\\] C. \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { - 2} \\\\ 3 \\end{array}} \\right]\\] D. \\[\\left[ {\\begin{array}{*{20}{c}} 2 \\\\ 5 \\\\ 0 \\end{array}} \\right]\\]","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2024-04-15T05:46:49+00:00","dateModified":"2024-04-15T05:46:49+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/for-the-matrix-texta-left-beginarray20c-3-22-0-21-001-endarray-right-one-of-the-eigen-values-is-equal-to-2-which-of-the-following-is-an\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/for-the-matrix-texta-left-beginarray20c-3-22-0-21-001-endarray-right-one-of-the-eigen-values-is-equal-to-2-which-of-the-following-is-an\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/for-the-matrix-texta-left-beginarray20c-3-22-0-21-001-endarray-right-one-of-the-eigen-values-is-equal-to-2-which-of-the-following-is-an\/#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":"For the matrix \\[{\\text{A}} = \\left[ {\\begin{array}{*{20}{c}} 3&{ – 2}&2 \\\\ 0&{ – 2}&1 \\\\ 0&0&1 \\end{array}} \\right],\\] one of the eigen values is equal to -2. 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A. \\[\\left[ {\\begin{array}{*{20}{c}} 3 \\\\ { – 2} \\\\ 1 \\end{array}} \\right]\\] B. \\[\\left[ {\\begin{array}{*{20}{c}} { – 3} \\\\ 2 \\\\ { – 1} \\end{array}} \\right]\\] C. \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { – 2} \\\\ 3 \\end{array}} \\right]\\] D. \\[\\left[ {\\begin{array}{*{20}{c}} 2 \\\\ 5 \\\\ 0 \\end{array}} \\right]\\]"}]},{"@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\/19982","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=19982"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/19982\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=19982"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=19982"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=19982"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}