{"id":20014,"date":"2024-04-15T05:47:13","date_gmt":"2024-04-15T05:47:13","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20014"},"modified":"2024-04-15T05:47:13","modified_gmt":"2024-04-15T05:47:13","slug":"given-matrix-left-texta-right-left-beginarray20c-4213-6347-2101-endarray-right-the-rank-of-the-matrix-is-a-4-b-3-c-2-d-1","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/given-matrix-left-texta-right-left-beginarray20c-4213-6347-2101-endarray-right-the-rank-of-the-matrix-is-a-4-b-3-c-2-d-1\/","title":{"rendered":"Given Matrix \\[\\left[ {\\text{A}} \\right] = \\left[ {\\begin{array}{*{20}{c}} 4&#038;2&#038;1&#038;3 \\\\ 6&#038;3&#038;4&#038;7 \\\\ 2&#038;1&#038;0&#038;1 \\end{array}} \\right],\\] the rank of the matrix is A. 4 B. 3 C. 2 D. 1"},"content":{"rendered":"<p>[amp_mcq option1=&#8221;4&#8243; option2=&#8221;3&#8243; option3=&#8221;2&#8243; option4=&#8221;1&#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 coefficients of the rows above the main diagonal are all 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 we have reduced the matrix to row echelon form, the rank of the matrix is the number of non-zero rows in the row echelon form.<\/p>\n<p>For the matrix $A$, we can reduce it to row echelon form as follows:<\/p>\n<p>[\\left[ {\\text{A}} \\right] = \\left[ {\\begin{array}{<em>{20}{c}} 4&amp;2&amp;1&amp;3 \\ 6&amp;3&amp;4&amp;7 \\ 2&amp;1&amp;0&amp;1 \\end{array}} \\right]\\xrightarrow{R_2-\\frac{3}{2}R_1\\to R_2} \\left[ {\\begin{array}{<\/em>{20}{c}} 4&amp;2&amp;1&amp;3 \\ 0&amp;1&amp;\\frac{5}{2}&amp;\\frac{5}{2} \\ 2&amp;1&amp;0&amp;1 \\end{array}} \\right]\\xrightarrow{\\frac{1}{2}R_3\\to R_3} \\left[ {\\begin{array}{*{20}{c}} 4&amp;2&amp;1&amp;3 \\ 0&amp;1&amp;\\frac{5}{2}&amp;\\frac{5}{2} \\ 0&amp;0&amp;-\\frac{1}{2}&amp;\\frac{3}{2} \\end{array}} \\right]]<\/p>\n<p>The row echelon form of the matrix $A$ has 3 non-zero rows, so the rank of the matrix $A$ is 3.<\/p>\n<p>Therefore, the correct answer is $\\boxed{\\text{C}}$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>[amp_mcq option1=&#8221;4&#8243; option2=&#8221;3&#8243; option3=&#8221;2&#8243; option4=&#8221;1&#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-20014","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>Given Matrix \\[\\left[ {\\text{A}} \\right] = \\left[ {\\begin{array}{*{20}{c}} 4&amp;2&amp;1&amp;3 \\\\ 6&amp;3&amp;4&amp;7 \\\\ 2&amp;1&amp;0&amp;1 \\end{array}} \\right],\\] the rank of the matrix is A. 4 B. 3 C. 2 D. 1<\/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\/given-matrix-left-texta-right-left-beginarray20c-4213-6347-2101-endarray-right-the-rank-of-the-matrix-is-a-4-b-3-c-2-d-1\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Given Matrix \\[\\left[ {\\text{A}} \\right] = \\left[ {\\begin{array}{*{20}{c}} 4&amp;2&amp;1&amp;3 \\\\ 6&amp;3&amp;4&amp;7 \\\\ 2&amp;1&amp;0&amp;1 \\end{array}} \\right],\\] the rank of the matrix is A. 4 B. 3 C. 2 D. 1\" \/>\n<meta property=\"og:description\" content=\"[amp_mcq option1=&#8221;4&#8243; 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