{"id":20006,"date":"2024-04-15T05:47:06","date_gmt":"2024-04-15T05:47:06","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20006"},"modified":"2024-04-15T05:47:06","modified_gmt":"2024-04-15T05:47:06","slug":"a-matrix-has-eigen-values-1-and-2-the-corresponding-eigen-vectors-are-left-beginarray20c-1-1-endarray-right-and-left-beginarray20c-1","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/a-matrix-has-eigen-values-1-and-2-the-corresponding-eigen-vectors-are-left-beginarray20c-1-1-endarray-right-and-left-beginarray20c-1\/","title":{"rendered":"A matrix has eigen values -1 and -2. The corresponding eigen vectors are \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { – 1} \\end{array}} \\right]\\] and \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { – 2} \\end{array}} \\right]\\] respectively. The matrix is A. \\[\\left[ {\\begin{array}{*{20}{c}} 1&1 \\\\ { – 1}&{ – 2} \\end{array}} \\right]\\] B. \\[\\left[ {\\begin{array}{*{20}{c}} 1&2 \\\\ { – 2}&{ – 4} \\end{array}} \\right]\\] C. \\[\\left[ {\\begin{array}{*{20}{c}} { – 1}&0 \\\\ 0&{ – 2} \\end{array}} \\right]\\] D. \\[\\left[ {\\begin{array}{*{20}{c}} 0&1 \\\\ { – 2}&{ – 3} \\end{array}} \\right]\\]"},"content":{"rendered":"

[amp_mcq option1=”\\[\\left[ {\\begin{array}{*{20}{c}} 1&1 \\\\ { – 1}&{ – 2} \\end{array}} \\right]\\]” option2=”\\[\\left[ {\\begin{array}{*{20}{c}} 1&2 \\\\ { – 2}&{ – 4} \\end{array}} \\right]\\]” option3=”\\[\\left[ {\\begin{array}{*{20}{c}} { – 1}&0 \\\\ 0&{ – 2} \\end{array}} \\right]\\]” option4=”\\[\\left[ {\\begin{array}{*{20}{c}} 0&1 \\\\ { – 2}&{ – 3} \\end{array}} \\right]\\]” correct=”option3″]<\/p>\n

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

To find the matrix, we can use the following formula:<\/p>\n

$$A = \\frac{v_1 v_2^T + v_2 v_1^T}{v_1^T v_2}$$<\/p>\n

where $v_1$ and $v_2$ are the eigenvectors corresponding to the eigenvalues $-1$ and $-2$, respectively.<\/p>\n

In this case, $v_1 = \\left[ {\\begin{array}{{20}{c}} 1 \\ { – 1} \\end{array}} \\right]$ and $v_2 = \\left[ {\\begin{array}{<\/em>{20}{c}} 1 \\ { – 2} \\end{array}} \\right]$. Therefore,<\/p>\n

$$A = \\frac{v_1 v_2^T + v_2 v_1^T}{v_1^T v_2} = \\frac{\\left[ {\\begin{array}{{20}{c}} 1 \\ { – 1} \\end{array}} \\right] \\left[ {\\begin{array}{<\/em>{20}{c}} 1 & { – 2} \\end{array}} \\right] + \\left[ {\\begin{array}{{20}{c}} 1 & { – 2} \\end{array}} \\right] \\left[ {\\begin{array}{<\/em>{20}{c}} 1 \\ { – 1} \\end{array}} \\right]}{(1, -1)^T \\cdot (1, -1)} = \\left[ {\\begin{array}{*{20}{c}} 1&1 \\ { – 1}&{ – 2} \\end{array}} \\right]$$<\/p>\n","protected":false},"excerpt":{"rendered":"

[amp_mcq option1=”\\[\\left[ {\\begin{array}{*{20}{c}} 1&1 \\\\ { – 1}&{ – 2} \\end{array}} \\right]\\]” option2=”\\[\\left[ {\\begin{array}{*{20}{c}} 1&2 \\\\ { – 2}&{ – 4} \\end{array}} \\right]\\]” option3=”\\[\\left[ {\\begin{array}{*{20}{c}} { – 1}&0 \\\\ 0&{ – 2} \\end{array}} \\right]\\]” option4=”\\[\\left[ {\\begin{array}{*{20}{c}} 0&1 \\\\ { – 2}&{ – 3} \\end{array}} \\right]\\]” correct=”option3″]<\/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-20006","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","no-featured-image-padding"],"yoast_head":"\nA matrix has eigen values -1 and -2. The corresponding eigen vectors are \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { - 1} \\end{array}} \\right]\\] and \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { - 2} \\end{array}} \\right]\\] respectively. The matrix is A. \\[\\left[ {\\begin{array}{*{20}{c}} 1&1 \\\\ { - 1}&{ - 2} \\end{array}} \\right]\\] B. \\[\\left[ {\\begin{array}{*{20}{c}} 1&2 \\\\ { - 2}&{ - 4} \\end{array}} \\right]\\] C. \\[\\left[ {\\begin{array}{*{20}{c}} { - 1}&0 \\\\ 0&{ - 2} \\end{array}} \\right]\\] D. \\[\\left[ {\\begin{array}{*{20}{c}} 0&1 \\\\ { - 2}&{ - 3} \\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\/a-matrix-has-eigen-values-1-and-2-the-corresponding-eigen-vectors-are-left-beginarray20c-1-1-endarray-right-and-left-beginarray20c-1\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"A matrix has eigen values -1 and -2. The corresponding eigen vectors are \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { - 1} \\end{array}} \\right]\\] and \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { - 2} \\end{array}} \\right]\\] respectively. The matrix is A. \\[\\left[ {\\begin{array}{*{20}{c}} 1&1 \\\\ { - 1}&{ - 2} \\end{array}} \\right]\\] B. \\[\\left[ {\\begin{array}{*{20}{c}} 1&2 \\\\ { - 2}&{ - 4} \\end{array}} \\right]\\] C. \\[\\left[ {\\begin{array}{*{20}{c}} { - 1}&0 \\\\ 0&{ - 2} \\end{array}} \\right]\\] D. \\[\\left[ {\\begin{array}{*{20}{c}} 0&1 \\\\ { - 2}&{ - 3} \\end{array}} \\right]\\]\" \/>\n<meta property=\"og:description\" content=\"[amp_mcq option1=”[left[ {begin{array}{*{20}{c}} 1&1 \\ { – 1}&{ – 2} end{array}} right]]” option2=”[left[ {begin{array}{*{20}{c}} 1&2 \\ { – 2}&{ – 4} end{array}} right]]” option3=”[left[ {begin{array}{*{20}{c}} { – 1}&0 \\ 0&{ – 2} end{array}} right]]” option4=”[left[ {begin{array}{*{20}{c}} 0&1 \\ { – 2}&{ – 3} end{array}} right]]” correct=”option3″]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/a-matrix-has-eigen-values-1-and-2-the-corresponding-eigen-vectors-are-left-beginarray20c-1-1-endarray-right-and-left-beginarray20c-1\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T05:47:06+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":"A matrix has eigen values -1 and -2. The corresponding eigen vectors are \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { - 1} \\end{array}} \\right]\\] and \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { - 2} \\end{array}} \\right]\\] respectively. The matrix is A. \\[\\left[ {\\begin{array}{*{20}{c}} 1&1 \\\\ { - 1}&{ - 2} \\end{array}} \\right]\\] B. \\[\\left[ {\\begin{array}{*{20}{c}} 1&2 \\\\ { - 2}&{ - 4} \\end{array}} \\right]\\] C. \\[\\left[ {\\begin{array}{*{20}{c}} { - 1}&0 \\\\ 0&{ - 2} \\end{array}} \\right]\\] D. \\[\\left[ {\\begin{array}{*{20}{c}} 0&1 \\\\ { - 2}&{ - 3} \\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\/a-matrix-has-eigen-values-1-and-2-the-corresponding-eigen-vectors-are-left-beginarray20c-1-1-endarray-right-and-left-beginarray20c-1\/","og_locale":"en_US","og_type":"article","og_title":"A matrix has eigen values -1 and -2. The corresponding eigen vectors are \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { - 1} \\end{array}} \\right]\\] and \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { - 2} \\end{array}} \\right]\\] respectively. The matrix is A. \\[\\left[ {\\begin{array}{*{20}{c}} 1&1 \\\\ { - 1}&{ - 2} \\end{array}} \\right]\\] B. \\[\\left[ {\\begin{array}{*{20}{c}} 1&2 \\\\ { - 2}&{ - 4} \\end{array}} \\right]\\] C. \\[\\left[ {\\begin{array}{*{20}{c}} { - 1}&0 \\\\ 0&{ - 2} \\end{array}} \\right]\\] D. \\[\\left[ {\\begin{array}{*{20}{c}} 0&1 \\\\ { - 2}&{ - 3} \\end{array}} \\right]\\]","og_description":"[amp_mcq option1=”[left[ {begin{array}{*{20}{c}} 1&1 \\ { – 1}&{ – 2} end{array}} right]]” option2=”[left[ {begin{array}{*{20}{c}} 1&2 \\ { – 2}&{ – 4} end{array}} right]]” option3=”[left[ {begin{array}{*{20}{c}} { – 1}&0 \\ 0&{ – 2} end{array}} right]]” option4=”[left[ {begin{array}{*{20}{c}} 0&1 \\ { – 2}&{ – 3} end{array}} right]]” correct=”option3″]","og_url":"https:\/\/exam.pscnotes.com\/mcq\/a-matrix-has-eigen-values-1-and-2-the-corresponding-eigen-vectors-are-left-beginarray20c-1-1-endarray-right-and-left-beginarray20c-1\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T05:47:06+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\/a-matrix-has-eigen-values-1-and-2-the-corresponding-eigen-vectors-are-left-beginarray20c-1-1-endarray-right-and-left-beginarray20c-1\/","url":"https:\/\/exam.pscnotes.com\/mcq\/a-matrix-has-eigen-values-1-and-2-the-corresponding-eigen-vectors-are-left-beginarray20c-1-1-endarray-right-and-left-beginarray20c-1\/","name":"A matrix has eigen values -1 and -2. The corresponding eigen vectors are \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { - 1} \\end{array}} \\right]\\] and \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { - 2} \\end{array}} \\right]\\] respectively. The matrix is A. \\[\\left[ {\\begin{array}{*{20}{c}} 1&1 \\\\ { - 1}&{ - 2} \\end{array}} \\right]\\] B. \\[\\left[ {\\begin{array}{*{20}{c}} 1&2 \\\\ { - 2}&{ - 4} \\end{array}} \\right]\\] C. \\[\\left[ {\\begin{array}{*{20}{c}} { - 1}&0 \\\\ 0&{ - 2} \\end{array}} \\right]\\] D. \\[\\left[ {\\begin{array}{*{20}{c}} 0&1 \\\\ { - 2}&{ - 3} \\end{array}} \\right]\\]","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2024-04-15T05:47:06+00:00","dateModified":"2024-04-15T05:47:06+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/a-matrix-has-eigen-values-1-and-2-the-corresponding-eigen-vectors-are-left-beginarray20c-1-1-endarray-right-and-left-beginarray20c-1\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/a-matrix-has-eigen-values-1-and-2-the-corresponding-eigen-vectors-are-left-beginarray20c-1-1-endarray-right-and-left-beginarray20c-1\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/a-matrix-has-eigen-values-1-and-2-the-corresponding-eigen-vectors-are-left-beginarray20c-1-1-endarray-right-and-left-beginarray20c-1\/#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":"A matrix has eigen values -1 and -2. The corresponding eigen vectors are \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { – 1} \\end{array}} \\right]\\] and \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ { – 2} \\end{array}} \\right]\\] respectively. The matrix is A. \\[\\left[ {\\begin{array}{*{20}{c}} 1&1 \\\\ { – 1}&{ – 2} \\end{array}} \\right]\\] B. \\[\\left[ {\\begin{array}{*{20}{c}} 1&2 \\\\ { – 2}&{ – 4} \\end{array}} \\right]\\] C. \\[\\left[ {\\begin{array}{*{20}{c}} { – 1}&0 \\\\ 0&{ – 2} \\end{array}} \\right]\\] D. \\[\\left[ {\\begin{array}{*{20}{c}} 0&1 \\\\ { – 2}&{ – 3} \\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\/20006","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=20006"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/20006\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=20006"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=20006"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=20006"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}