{"id":20003,"date":"2024-04-15T05:47:04","date_gmt":"2024-04-15T05:47:04","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20003"},"modified":"2024-04-15T05:47:04","modified_gmt":"2024-04-15T05:47:04","slug":"the-system-of-linear-equations-left-beginarray20c-213-301-125-endarray-rightleft-beginarray20c-texta-textb-textc","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-system-of-linear-equations-left-beginarray20c-213-301-125-endarray-rightleft-beginarray20c-texta-textb-textc\/","title":{"rendered":"The system of linear equations \\[\\left[ {\\begin{array}{*{20}{c}} 2&#038;1&#038;3 \\\\ 3&#038;0&#038;1 \\\\ 1&#038;2&#038;5 \\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}} {\\text{a}} \\\\ {\\text{b}} \\\\ {\\text{c}} \\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}} 5 \\\\ { &#8211; 4} \\\\ {14} \\end{array}} \\right]\\] has A. a unique solution B. infinitely many solutions C. no solution D. exactly two solutions"},"content":{"rendered":"<p>[amp_mcq option1=&#8221;a unique solution&#8221; option2=&#8221;infinitely many solutions&#8221; option3=&#8221;no solution&#8221; option4=&#8221;exactly two solutions&#8221; correct=&#8221;option3&#8243;]<!--more--><\/p>\n<p>The correct answer is $\\boxed{\\text{A}}$.<\/p>\n<p>To solve a system of linear equations, we can use Gaussian elimination. In this case, we can reduce the system to the following form:<\/p>\n<p>$$\\left[ {\\begin{array}{<em>{20}{c}} 1&amp;0&amp;-2 \\ 0&amp;1&amp;-3 \\ 0&amp;0&amp;1 \\end{array}} \\right]\\left[ {\\begin{array}{<\/em>{20}{c}} {\\text{a}} \\ {\\text{b}} \\ {\\text{c}} \\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}} 5 \\ { &#8211; 4} \\ {14} \\end{array}} \\right]$$<\/p>\n<p>This means that the system has a unique solution, which is $\\boxed{\\text{a} = 5, \\text{b} = -4, \\text{c} = 14}$.<\/p>\n<p>Here is a brief explanation of each option:<\/p>\n<ul>\n<li>Option $\\boxed{\\text{A}}$: The system has a unique solution. This is because the reduced matrix is invertible, which means that there is a unique solution to the system of equations.<\/li>\n<li>Option $\\boxed{\\text{B}}$: The system has infinitely many solutions. This is because the reduced matrix is not invertible, which means that there are infinitely many solutions to the system of equations.<\/li>\n<li>Option $\\boxed{\\text{C}}$: The system has no solution. This is because the reduced matrix is singular, which means that there is no solution to the system of equations.<\/li>\n<li>Option $\\boxed{\\text{D}}$: The system has exactly two solutions. This is not possible, because the system either has a unique solution, infinitely many solutions, or no solution.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>[amp_mcq option1=&#8221;a unique solution&#8221; option2=&#8221;infinitely many solutions&#8221; option3=&#8221;no solution&#8221; option4=&#8221;exactly two solutions&#8221; correct=&#8221;option3&#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-20003","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 system of linear equations \\[\\left[ {\\begin{array}{*{20}{c}} 2&amp;1&amp;3 \\\\ 3&amp;0&amp;1 \\\\ 1&amp;2&amp;5 \\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}} {\\text{a}} \\\\ {\\text{b}} \\\\ {\\text{c}} \\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}} 5 \\\\ { - 4} \\\\ {14} \\end{array}} \\right]\\] has A. a unique solution B. infinitely many solutions C. no solution D. exactly two solutions<\/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-system-of-linear-equations-left-beginarray20c-213-301-125-endarray-rightleft-beginarray20c-texta-textb-textc\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The system of linear equations \\[\\left[ {\\begin{array}{*{20}{c}} 2&amp;1&amp;3 \\\\ 3&amp;0&amp;1 \\\\ 1&amp;2&amp;5 \\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}} {\\text{a}} \\\\ {\\text{b}} \\\\ {\\text{c}} \\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}} 5 \\\\ { - 4} \\\\ {14} \\end{array}} \\right]\\] has A. a unique solution B. infinitely many solutions C. no solution D. exactly two solutions\" \/>\n<meta property=\"og:description\" content=\"[amp_mcq option1=&#8221;a unique solution&#8221; option2=&#8221;infinitely many solutions&#8221; option3=&#8221;no solution&#8221; option4=&#8221;exactly two solutions&#8221; correct=&#8221;option3&#8243;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/exam.pscnotes.com\/mcq\/the-system-of-linear-equations-left-beginarray20c-213-301-125-endarray-rightleft-beginarray20c-texta-textb-textc\/\" \/>\n<meta property=\"og:site_name\" content=\"MCQ and Quiz for Exams\" \/>\n<meta property=\"article:published_time\" content=\"2024-04-15T05:47:04+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":"The system of linear equations \\[\\left[ {\\begin{array}{*{20}{c}} 2&1&3 \\\\ 3&0&1 \\\\ 1&2&5 \\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}} {\\text{a}} \\\\ {\\text{b}} \\\\ {\\text{c}} \\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}} 5 \\\\ { - 4} \\\\ {14} \\end{array}} \\right]\\] has A. a unique solution B. infinitely many solutions C. no solution D. exactly two solutions","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\/the-system-of-linear-equations-left-beginarray20c-213-301-125-endarray-rightleft-beginarray20c-texta-textb-textc\/","og_locale":"en_US","og_type":"article","og_title":"The system of linear equations \\[\\left[ {\\begin{array}{*{20}{c}} 2&1&3 \\\\ 3&0&1 \\\\ 1&2&5 \\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}} {\\text{a}} \\\\ {\\text{b}} \\\\ {\\text{c}} \\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}} 5 \\\\ { - 4} \\\\ {14} \\end{array}} \\right]\\] has A. a unique solution B. infinitely many solutions C. no solution D. exactly two solutions","og_description":"[amp_mcq option1=&#8221;a unique solution&#8221; option2=&#8221;infinitely many solutions&#8221; option3=&#8221;no solution&#8221; option4=&#8221;exactly two solutions&#8221; correct=&#8221;option3&#8243;]","og_url":"https:\/\/exam.pscnotes.com\/mcq\/the-system-of-linear-equations-left-beginarray20c-213-301-125-endarray-rightleft-beginarray20c-texta-textb-textc\/","og_site_name":"MCQ and Quiz for Exams","article_published_time":"2024-04-15T05:47:04+00:00","author":"rawan239","twitter_card":"summary_large_image","twitter_misc":{"Written by":"rawan239","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"WebPage","@id":"https:\/\/exam.pscnotes.com\/mcq\/the-system-of-linear-equations-left-beginarray20c-213-301-125-endarray-rightleft-beginarray20c-texta-textb-textc\/","url":"https:\/\/exam.pscnotes.com\/mcq\/the-system-of-linear-equations-left-beginarray20c-213-301-125-endarray-rightleft-beginarray20c-texta-textb-textc\/","name":"The system of linear equations \\[\\left[ {\\begin{array}{*{20}{c}} 2&1&3 \\\\ 3&0&1 \\\\ 1&2&5 \\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}} {\\text{a}} \\\\ {\\text{b}} \\\\ {\\text{c}} \\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}} 5 \\\\ { - 4} \\\\ {14} \\end{array}} \\right]\\] has A. a unique solution B. infinitely many solutions C. no solution D. exactly two solutions","isPartOf":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#website"},"datePublished":"2024-04-15T05:47:04+00:00","dateModified":"2024-04-15T05:47:04+00:00","author":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209"},"breadcrumb":{"@id":"https:\/\/exam.pscnotes.com\/mcq\/the-system-of-linear-equations-left-beginarray20c-213-301-125-endarray-rightleft-beginarray20c-texta-textb-textc\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/exam.pscnotes.com\/mcq\/the-system-of-linear-equations-left-beginarray20c-213-301-125-endarray-rightleft-beginarray20c-texta-textb-textc\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/exam.pscnotes.com\/mcq\/the-system-of-linear-equations-left-beginarray20c-213-301-125-endarray-rightleft-beginarray20c-texta-textb-textc\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/exam.pscnotes.com\/mcq\/"},{"@type":"ListItem","position":2,"name":"mcq","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/"},{"@type":"ListItem","position":3,"name":"Linear Algebra","item":"https:\/\/exam.pscnotes.com\/mcq\/category\/mcq\/linear-algebra\/"},{"@type":"ListItem","position":4,"name":"The system of linear equations \\[\\left[ {\\begin{array}{*{20}{c}} 2&#038;1&#038;3 \\\\ 3&#038;0&#038;1 \\\\ 1&#038;2&#038;5 \\end{array}} \\right]\\left[ {\\begin{array}{*{20}{c}} {\\text{a}} \\\\ {\\text{b}} \\\\ {\\text{c}} \\end{array}} \\right] = \\left[ {\\begin{array}{*{20}{c}} 5 \\\\ { &#8211; 4} \\\\ {14} \\end{array}} \\right]\\] has A. a unique solution B. infinitely many solutions C. no solution D. exactly two solutions"}]},{"@type":"WebSite","@id":"https:\/\/exam.pscnotes.com\/mcq\/#website","url":"https:\/\/exam.pscnotes.com\/mcq\/","name":"MCQ and Quiz for Exams","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/exam.pscnotes.com\/mcq\/?s={search_term_string}"},"query-input":"required name=search_term_string"}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/5807dafeb27d2ec82344d6cbd6c3d209","name":"rawan239","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/exam.pscnotes.com\/mcq\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/761a7274f9cce048fa5b921221e7934820d74514df93ef195a9d22af0c1c9001?s=96&d=mm&r=g","caption":"rawan239"},"sameAs":["https:\/\/exam.pscnotes.com"],"url":"https:\/\/exam.pscnotes.com\/mcq\/author\/rawan239\/"}]}},"amp_enabled":true,"_links":{"self":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/20003","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/comments?post=20003"}],"version-history":[{"count":0,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/posts\/20003\/revisions"}],"wp:attachment":[{"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/media?parent=20003"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/categories?post=20003"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/exam.pscnotes.com\/mcq\/wp-json\/wp\/v2\/tags?post=20003"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}