{"id":19993,"date":"2024-04-15T05:46:57","date_gmt":"2024-04-15T05:46:57","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=19993"},"modified":"2024-04-15T05:46:57","modified_gmt":"2024-04-15T05:46:57","slug":"one-of-the-eigen-vectors-of-the-matrix-texta-left-beginarray20c-22-13-endarray-right-is-a-left-beginarray20c-2-1-endarray","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/one-of-the-eigen-vectors-of-the-matrix-texta-left-beginarray20c-22-13-endarray-right-is-a-left-beginarray20c-2-1-endarray\/","title":{"rendered":"One of the eigen vectors of the matrix \\[{\\text{A}} = \\left[ {\\begin{array}{*{20}{c}} 2&2 \\\\ 1&3 \\end{array}} \\right]\\] is A. \\[\\left\\{ {\\begin{array}{*{20}{c}} 2 \\\\ { – 1} \\end{array}} \\right\\}\\] B. \\[\\left\\{ {\\begin{array}{*{20}{c}} 2 \\\\ 1 \\end{array}} \\right\\}\\] C. \\[\\left\\{ {\\begin{array}{*{20}{c}} 4 \\\\ 1 \\end{array}} \\right\\}\\] D. \\[\\left\\{ {\\begin{array}{*{20}{c}} 1 \\\\ { – 1} \\end{array}} \\right\\}\\]"},"content":{"rendered":"

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

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

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

$$\\lambda v = A v$$<\/p>\n

where $\\lambda$ is the eigenvalue, $v$ is the eigenvector, and $A$ is the matrix.<\/p>\n

In this case, we have:<\/p>\n

$$\\begin{align}
\n\\lambda v &= A v \\
\n\\lambda \\left( \\begin{array}{c} 2 \\ -1 \\end{array} \\right) &= \\left( \\begin{array}{cc} 2 & 2 \\ 1 & 3 \\end{array} \\right) \\left( \\begin{array}{c} 2 \\ -1 \\end{array} \\right) \\
\n\\lambda \\left( \\begin{array}{c} 2 \\ -1 \\end{array} \\right) &= \\left( \\begin{array}{c} 4 \\ 3 \\end{array} \\right) \\
\n\\lambda &= 4 \\
\n\\end{align<\/em>}$$<\/p>\n

Therefore, one of the eigenvalues of the matrix is $\\lambda = 4$.<\/p>\n

To find the corresponding eigenvector, we can solve the following equation:<\/p>\n

$$\\begin{align}
\nAv &= \\lambda v \\
\n\\left( \\begin{array}{cc} 2 & 2 \\ 1 & 3 \\end{array} \\right) \\left( \\begin{array}{c} x \\ y \\end{array} \\right) &= \\left( \\begin{array}{c} 4 \\ 3 \\end{array} \\right) \\
\n2x + 2y &= 4 \\
\nx + 3y &= 3 \\
\n\\end{align<\/em>}$$<\/p>\n

Solving this system of equations, we get:<\/p>\n

$$x = 2, \\quad y = -1$$<\/p>\n

Therefore, one of the eigenvectors of the matrix is $\\boxed{\\left\\{ {\\begin{array}{*{20}{c}} 2 \\ { – 1} \\end{array}} \\right\\}}$.<\/p>\n

The other eigenvalue and eigenvector of the matrix can be found similarly.<\/p>\n","protected":false},"excerpt":{"rendered":"

[amp_mcq option1=”\\[\\left\\{ {\\begin{array}{*{20}{c}} 2 \\\\ { – 1} \\end{array}} \\right\\}\\]” option2=”\\[\\left\\{ {\\begin{array}{*{20}{c}} 2 \\\\ 1 \\end{array}} \\right\\}\\]” option3=”\\[\\left\\{ {\\begin{array}{*{20}{c}} 4 \\\\ 1 \\end{array}} \\right\\}\\]” option4=”\\[\\left\\{ {\\begin{array}{*{20}{c}} 1 \\\\ { – 1} \\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-19993","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","no-featured-image-padding"],"yoast_head":"\nOne of the eigen vectors of the matrix \\[{\\text{A}} = \\left[ {\\begin{array}{*{20}{c}} 2&2 \\\\ 1&3 \\end{array}} \\right]\\] is A. \\[\\left\\{ {\\begin{array}{*{20}{c}} 2 \\\\ { - 1} \\end{array}} \\right\\}\\] B. \\[\\left\\{ {\\begin{array}{*{20}{c}} 2 \\\\ 1 \\end{array}} \\right\\}\\] C. \\[\\left\\{ {\\begin{array}{*{20}{c}} 4 \\\\ 1 \\end{array}} \\right\\}\\] D. \\[\\left\\{ {\\begin{array}{*{20}{c}} 1 \\\\ { - 1} \\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\/one-of-the-eigen-vectors-of-the-matrix-texta-left-beginarray20c-22-13-endarray-right-is-a-left-beginarray20c-2-1-endarray\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"One of the eigen vectors of the matrix \\[{\\text{A}} = \\left[ {\\begin{array}{*{20}{c}} 2&2 \\\\ 1&3 \\end{array}} \\right]\\] is A. \\[\\left\\{ {\\begin{array}{*{20}{c}} 2 \\\\ { - 1} \\end{array}} \\right\\}\\] B. \\[\\left\\{ {\\begin{array}{*{20}{c}} 2 \\\\ 1 \\end{array}} \\right\\}\\] C. \\[\\left\\{ {\\begin{array}{*{20}{c}} 4 \\\\ 1 \\end{array}} \\right\\}\\] D. \\[\\left\\{ {\\begin{array}{*{20}{c}} 1 \\\\ { - 1} \\end{array}} \\right\\}\\]\" \/>\n<meta property=\"og:description\" content=\"[amp_mcq option1=”[left{ {begin{array}{*{20}{c}} 2 \\ { – 1} end{array}} right}]” option2=”[left{ {begin{array}{*{20}{c}} 2 \\ 1 end{array}} right}]” option3=”[left{ {begin{array}{*{20}{c}} 4 \\ 1 end{array}} right}]” option4=”[left{ {begin{array}{*{20}{c}} 1 \\ { – 1} end{array}} right}]” correct=”option3″]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/one-of-the-eigen-vectors-of-the-matrix-texta-left-beginarray20c-22-13-endarray-right-is-a-left-beginarray20c-2-1-endarray\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T05:46:57+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":"One of the eigen vectors of the matrix \\[{\\text{A}} = \\left[ {\\begin{array}{*{20}{c}} 2&2 \\\\ 1&3 \\end{array}} \\right]\\] is A. \\[\\left\\{ {\\begin{array}{*{20}{c}} 2 \\\\ { - 1} \\end{array}} \\right\\}\\] B. \\[\\left\\{ {\\begin{array}{*{20}{c}} 2 \\\\ 1 \\end{array}} \\right\\}\\] C. \\[\\left\\{ {\\begin{array}{*{20}{c}} 4 \\\\ 1 \\end{array}} \\right\\}\\] D. \\[\\left\\{ {\\begin{array}{*{20}{c}} 1 \\\\ { - 1} \\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\/one-of-the-eigen-vectors-of-the-matrix-texta-left-beginarray20c-22-13-endarray-right-is-a-left-beginarray20c-2-1-endarray\/","og_locale":"en_US","og_type":"article","og_title":"One of the eigen vectors of the matrix \\[{\\text{A}} = \\left[ {\\begin{array}{*{20}{c}} 2&2 \\\\ 1&3 \\end{array}} \\right]\\] is A. \\[\\left\\{ {\\begin{array}{*{20}{c}} 2 \\\\ { - 1} \\end{array}} \\right\\}\\] B. \\[\\left\\{ {\\begin{array}{*{20}{c}} 2 \\\\ 1 \\end{array}} \\right\\}\\] C. \\[\\left\\{ {\\begin{array}{*{20}{c}} 4 \\\\ 1 \\end{array}} \\right\\}\\] D. \\[\\left\\{ {\\begin{array}{*{20}{c}} 1 \\\\ { - 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