{"id":20010,"date":"2024-04-15T05:47:10","date_gmt":"2024-04-15T05:47:10","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20010"},"modified":"2024-04-15T05:47:10","modified_gmt":"2024-04-15T05:47:10","slug":"the-rank-of-the-matrix-left-beginarray20c-41-1-1-1-1-7-31-endarray-right-is-a-1-b-2-c-3-d-4","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-rank-of-the-matrix-left-beginarray20c-41-1-1-1-1-7-31-endarray-right-is-a-1-b-2-c-3-d-4\/","title":{"rendered":"The rank of the matrix \\[\\left[ {\\begin{array}{*{20}{c}} { &#8211; 4}&#038;1&#038;{ &#8211; 1} \\\\ { &#8211; 1}&#038;{ &#8211; 1}&#038;{ &#8211; 1} \\\\ 7&#038;{ &#8211; 3}&#038;1 \\end{array}} \\right]\\] is A. 1 B. 2 C. 3 D. 4"},"content":{"rendered":"<p>[amp_mcq option1=&#8221;1&#8243; option2=&#8221;2&#8243; option3=&#8221;3&#8243; option4=&#8221;4&#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 perform the following row operations:<\/p>\n<ul>\n<li>Add $\\frac{1}{4}$ of row 1 to row 2:<\/li>\n<\/ul>\n<p>$$\\left[ {\\begin{array}{*{20}{c}} { &#8211; 4}&amp;1&amp;{ &#8211; 1} \\ 0&amp;-\\frac{3}{4}&amp;-\\frac{3}{4} \\ 7&amp;{ &#8211; 3}&amp;1 \\end{array}} \\right]$$<\/p>\n<ul>\n<li>Add $\\frac{7}{4}$ of row 1 to row 3:<\/li>\n<\/ul>\n<p>$$\\left[ {\\begin{array}{*{20}{c}} { &#8211; 4}&amp;1&amp;{ &#8211; 1} \\ 0&amp;-\\frac{3}{4}&amp;-\\frac{3}{4} \\ 0&amp;-\\frac{11}{4}&amp;-\\frac{3}{4} \\end{array}} \\right]$$<\/p>\n<ul>\n<li>Swap row 2 with row 3:<\/li>\n<\/ul>\n<p>$$\\left[ {\\begin{array}{*{20}{c}} { &#8211; 4}&amp;1&amp;{ &#8211; 1} \\ 0&amp;-\\frac{11}{4}&amp;-\\frac{3}{4} \\ 0&amp;-\\frac{3}{4}&amp;-\\frac{3}{4} \\end{array}} \\right]$$<\/p>\n<p>*Subtract $\\frac{3}{11}$ of row 2 from row 3:<\/p>\n<p>$$\\left[ {\\begin{array}{*{20}{c}} { &#8211; 4}&amp;1&amp;{ &#8211; 1} \\ 0&amp;-\\frac{11}{4}&amp;-\\frac{3}{4} \\ 0&amp;0&amp;-\\frac{3}{11} \\end{array}} \\right]$$<\/p>\n<p>Therefore, the rank of the matrix $M$ is $\\boxed{2}$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>[amp_mcq option1=&#8221;1&#8243; option2=&#8221;2&#8243; option3=&#8221;3&#8243; option4=&#8221;4&#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-20010","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 \\[\\left[ {\\begin{array}{*{20}{c}} { - 4}&amp;1&amp;{ - 1} \\\\ { - 1}&amp;{ - 1}&amp;{ - 1} \\\\ 7&amp;{ - 3}&amp;1 \\end{array}} \\right]\\] is A. 1 B. 2 C. 3 D. 4<\/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-left-beginarray20c-41-1-1-1-1-7-31-endarray-right-is-a-1-b-2-c-3-d-4\/\" \/>\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 \\[\\left[ {\\begin{array}{*{20}{c}} { - 4}&amp;1&amp;{ - 1} \\\\ { - 1}&amp;{ - 1}&amp;{ - 1} \\\\ 7&amp;{ - 3}&amp;1 \\end{array}} \\right]\\] is A. 1 B. 2 C. 3 D. 4\" \/>\n<meta property=\"og:description\" content=\"[amp_mcq option1=&#8221;1&#8243; 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