{"id":19992,"date":"2024-04-15T05:46:56","date_gmt":"2024-04-15T05:46:56","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=19992"},"modified":"2024-04-15T05:46:56","modified_gmt":"2024-04-15T05:46:56","slug":"the-rank-of-the-matrix-textm-left-beginarray20c-51010-102-366-endarray-right-is-a-0-b-1-c-2-d-3","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-rank-of-the-matrix-textm-left-beginarray20c-51010-102-366-endarray-right-is-a-0-b-1-c-2-d-3\/","title":{"rendered":"The rank of the matrix \\[{\\text{M}} = \\left[ {\\begin{array}{*{20}{c}} 5&#038;{10}&#038;{10} \\\\ 1&#038;0&#038;2 \\\\ 3&#038;6&#038;6 \\end{array}} \\right]\\] is A. 0 B. 1 C. 2 D. 3"},"content":{"rendered":"<p>[amp_mcq option1=&#8221;0&#8243; option2=&#8221;1&#8243; option3=&#8221;2&#8243; option4=&#8221;3&#8243; correct=&#8221;option1&#8243;]<!--more--><\/p>\n<p>The rank of a matrix is the number of linearly independent rows or columns in the matrix. To find the rank of a matrix, we can use Gaussian elimination.<\/p>\n<p>In Gaussian elimination, we reduce the matrix to row echelon form. A row echelon form is a matrix in which all the rows below the main diagonal are zero, and the leading coefficient of each non-zero row is 1.<\/p>\n<p>To reduce the matrix to row echelon form, we can use the following operations:<\/p>\n<ul>\n<li>Add or subtract a multiple of one row to another row.<\/li>\n<li>Multiply a row by a non-zero constant.<\/li>\n<li>Swap two rows.<\/li>\n<\/ul>\n<p>Once the matrix is in row echelon form, the rank is the number of non-zero rows in the matrix.<\/p>\n<p>For the matrix $M$, we can reduce it to row echelon form as follows:<\/p>\n<p>[{\\text{M}} = \\left[ {\\begin{array}{*{20}{c}} 5&amp;{10}&amp;{10} \\ 1&amp;0&amp;2 \\ 3&amp;6&amp;6 \\end{array}} \\right]]<\/p>\n<p>Subtract 3 rows of the first row from the third row, we get<\/p>\n<p>[{\\text{M}} = \\left[ {\\begin{array}{*{20}{c}} 5&amp;{10}&amp;{10} \\ 1&amp;0&amp;2 \\ 0&amp;6&amp;-6 \\end{array}} \\right]]<\/p>\n<p>Swap the second row with the first row, we get<\/p>\n<p>[{\\text{M}} = \\left[ {\\begin{array}{*{20}{c}} 1&amp;0&amp;2 \\ 0&amp;6&amp;-6 \\end{array}} \\right]]<\/p>\n<p>Divide the second row by 6, we get<\/p>\n<p>[{\\text{M}} = \\left[ {\\begin{array}{*{20}{c}} 1&amp;0&amp;2 \\ 0&amp;1&amp;-1 \\end{array}} \\right]]<\/p>\n<p>Since there are two non-zero rows in the row echelon form of the matrix, the rank of the matrix is 2.<\/p>\n<p>Therefore, the correct answer is $\\boxed{\\text{C}}$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>[amp_mcq option1=&#8221;0&#8243; option2=&#8221;1&#8243; option3=&#8221;2&#8243; option4=&#8221;3&#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-19992","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>The rank of the matrix \\[{\\text{M}} = \\left[ {\\begin{array}{*{20}{c}} 5&amp;{10}&amp;{10} \\\\ 1&amp;0&amp;2 \\\\ 3&amp;6&amp;6 \\end{array}} \\right]\\] is A. 0 B. 1 C. 2 D. 3<\/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\/the-rank-of-the-matrix-textm-left-beginarray20c-51010-102-366-endarray-right-is-a-0-b-1-c-2-d-3\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The rank of the matrix \\[{\\text{M}} = \\left[ {\\begin{array}{*{20}{c}} 5&amp;{10}&amp;{10} \\\\ 1&amp;0&amp;2 \\\\ 3&amp;6&amp;6 \\end{array}} \\right]\\] is A. 0 B. 1 C. 2 D. 3\" \/>\n<meta property=\"og:description\" content=\"[amp_mcq option1=&#8221;0&#8243; 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