{"id":20026,"date":"2024-04-15T05:47:22","date_gmt":"2024-04-15T05:47:22","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20026"},"modified":"2024-04-15T05:47:22","modified_gmt":"2024-04-15T05:47:22","slug":"let-the-eigen-vector-of-the-matrix-left-beginarray20c-12-02-endarray-right-be-written-in-the-form-left-beginarray20c-1-texta-endarra","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/let-the-eigen-vector-of-the-matrix-left-beginarray20c-12-02-endarray-right-be-written-in-the-form-left-beginarray20c-1-texta-endarra\/","title":{"rendered":"Let the Eigen vector of the matrix \\[\\left[ {\\begin{array}{*{20}{c}} 1&#038;2 \\\\ 0&#038;2 \\end{array}} \\right]\\] be written in the form \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ {\\text{a}} \\end{array}} \\right]\\] and \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ {\\text{b}} \\end{array}} \\right]\\]. What is the value of (a + b) = ? A. 0 B. \\[\\frac{1}{2}\\] C. 1 D. 2"},"content":{"rendered":"<p>[amp_mcq option1=&#8221;0&#8243; option2=&#8221;\\[\\frac{1}{2}\\]&#8221; option3=&#8221;1&#8243; option4=&#8221;2&#8243; correct=&#8221;option1&#8243;]<!--more--><\/p>\n<p>The correct answer is $\\boxed{1}$.<\/p>\n<p>The eigenvalues of the matrix $\\left[ {\\begin{array}{<em>{20}{c}} 1&amp;2 \\ 0&amp;2 \\end{array}} \\right]$ are $1$ and $2$. The eigenvectors corresponding to the eigenvalues $1$ and $2$ are $\\left[ {\\begin{array}{<\/em>{20}{c}} 1 \\ 0 \\end{array}} \\right]$ and $\\left[ {\\begin{array}{*{20}{c}} 0 \\ 1 \\end{array}} \\right]$, respectively. Therefore, the value of $(a+b)$ is $1+1=\\boxed{2}$.<\/p>\n<p>Here is a more detailed explanation of each option:<\/p>\n<ul>\n<li>Option A: $0$. This is not possible because the eigenvectors of a matrix are always nonzero vectors.<\/li>\n<li>Option B: $\\frac{1}{2}$. This is also not possible because the eigenvectors of a matrix are always orthogonal to each other, and the vectors $\\left[ {\\begin{array}{<em>{20}{c}} 1 \\ 0 \\end{array}} \\right]$ and $\\left[ {\\begin{array}{<\/em>{20}{c}} 0 \\ 1 \\end{array}} \\right]$ are not orthogonal.<\/li>\n<li>Option C: $1$. This is the correct answer. As explained above, the eigenvectors of the matrix $\\left[ {\\begin{array}{<em>{20}{c}} 1&amp;2 \\ 0&amp;2 \\end{array}} \\right]$ are $\\left[ {\\begin{array}{<\/em>{20}{c}} 1 \\ 0 \\end{array}} \\right]$ and $\\left[ {\\begin{array}{*{20}{c}} 0 \\ 1 \\end{array}} \\right]$. Therefore, the value of $(a+b)$ is $1+1=\\boxed{2}$.<\/li>\n<li>Option D: $2$. This is also not possible because the eigenvectors of a matrix are always linearly independent, and the vectors $\\left[ {\\begin{array}{<em>{20}{c}} 1 \\ 0 \\end{array}} \\right]$ and $\\left[ {\\begin{array}{<\/em>{20}{c}} 0 \\ 1 \\end{array}} \\right]$ are linearly dependent.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>[amp_mcq option1=&#8221;0&#8243; option2=&#8221;\\[\\frac{1}{2}\\]&#8221; option3=&#8221;1&#8243; option4=&#8221;2&#8243; correct=&#8221;option1&#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-20026","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 the Eigen vector of the matrix \\[\\left[ {\\begin{array}{*{20}{c}} 1&amp;2 \\\\ 0&amp;2 \\end{array}} \\right]\\] be written in the form \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ {\\text{a}} \\end{array}} \\right]\\] and \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ {\\text{b}} \\end{array}} \\right]\\]. What is the value of (a + b) = ? A. 0 B. \\[\\frac{1}{2}\\] C. 1 D. 2<\/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-the-eigen-vector-of-the-matrix-left-beginarray20c-12-02-endarray-right-be-written-in-the-form-left-beginarray20c-1-texta-endarra\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Let the Eigen vector of the matrix \\[\\left[ {\\begin{array}{*{20}{c}} 1&amp;2 \\\\ 0&amp;2 \\end{array}} \\right]\\] be written in the form \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ {\\text{a}} \\end{array}} \\right]\\] and \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ {\\text{b}} \\end{array}} \\right]\\]. What is the value of (a + b) = ? 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What is the value of (a + b) = ? A. 0 B. \\[\\frac{1}{2}\\] C. 1 D. 2","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-the-eigen-vector-of-the-matrix-left-beginarray20c-12-02-endarray-right-be-written-in-the-form-left-beginarray20c-1-texta-endarra\/","og_locale":"en_US","og_type":"article","og_title":"Let the Eigen vector of the matrix \\[\\left[ {\\begin{array}{*{20}{c}} 1&2 \\\\ 0&2 \\end{array}} \\right]\\] be written in the form \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ {\\text{a}} \\end{array}} \\right]\\] and \\[\\left[ {\\begin{array}{*{20}{c}} 1 \\\\ {\\text{b}} \\end{array}} \\right]\\]. What is the value of (a + b) = ? 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