{"id":232239,"date":"2025-06-11T21:43:16","date_gmt":"2025-06-11T21:43:16","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=232239"},"modified":"2025-06-11T21:43:18","modified_gmt":"2025-06-11T21:43:18","slug":"determine-if-the-system-has-a-nontrivial-solution-you-do-not-need-to-completely-solve-the-system","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/11\/determine-if-the-system-has-a-nontrivial-solution-you-do-not-need-to-completely-solve-the-system\/","title":{"rendered":"Determine if the system has a nontrivial solution (you do not need to completely solve the system)."},"content":{"rendered":"\n

Determine if the system has a nontrivial solution (you do not need to completely solve the system). x1 \u2013 3×2 + 7×3 = 0 -2×1 + x2 \u2013 4×3 = 0 x1 + 2×2 + 9×3 = 0<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To determine if the system has a nontrivial solution<\/strong>, we need to analyze whether the homogeneous system of linear equations has more than just the trivial solution<\/strong> (i.e., all variables equal to zero). A nontrivial solution<\/strong> exists if the system has infinitely many solutions<\/strong>\u2014this happens when the equations are linearly dependent<\/strong>, or equivalently, when the coefficient matrix has a determinant of zero<\/strong> or is not of full rank<\/strong>.<\/p>\n\n\n\n

Given System of Equations:<\/h3>\n\n\n\n

x1\u22123×2+7×3=0\u22122×1+x2\u22124×3=0x1+2×2+9×3=0\\begin{aligned} x_1 – 3x_2 + 7x_3 &= 0 \\\\ -2x_1 + x_2 – 4x_3 &= 0 \\\\ x_1 + 2x_2 + 9x_3 &= 0 \\end{aligned}x1\u200b\u22123×2\u200b+7×3\u200b\u22122×1\u200b+x2\u200b\u22124×3\u200bx1\u200b+2×2\u200b+9×3\u200b\u200b=0=0=0\u200b<\/p>\n\n\n\n

Step 1: Write the coefficient matrix:<\/h3>\n\n\n\n

A=[1\u221237\u221221\u22124129]A = \\begin{bmatrix} 1 & -3 & 7 \\\\ -2 & 1 & -4 \\\\ 1 & 2 & 9 \\end{bmatrix}A=\u200b1\u221221\u200b\u2212312\u200b7\u221249\u200b\u200b<\/p>\n\n\n\n

We want to determine if the homogeneous system<\/strong> Ax\u20d7=0A\\vec{x} = 0Ax=0 has a nontrivial solution<\/strong>. This will be true if and only if<\/strong> the matrix AAA is not of full rank<\/strong>, i.e., the rows are linearly dependent<\/strong>.<\/p>\n\n\n\n

Step 2: Perform row reduction (Gaussian elimination)<\/h3>\n\n\n\n

Start with:[1\u221237\u221221\u22124129]\\begin{bmatrix} 1 & -3 & 7 \\\\ -2 & 1 & -4 \\\\ 1 & 2 & 9 \\end{bmatrix}\u200b1\u221221\u200b\u2212312\u200b7\u221249\u200b\u200b<\/p>\n\n\n\n

    \n
  • Add 2\u00d7Row1 to Row2:<\/li>\n<\/ul>\n\n\n\n

    R2=R2+2R1\u21d2(0,\u22125,10)R_2 = R_2 + 2R_1 \\Rightarrow (0, -5, 10)R2\u200b=R2\u200b+2R1\u200b\u21d2(0,\u22125,10)<\/p>\n\n\n\n

      \n
    • Subtract Row1 from Row3:<\/li>\n<\/ul>\n\n\n\n

      R3=R3\u2212R1\u21d2(0,5,2)R_3 = R_3 – R_1 \\Rightarrow (0, 5, 2)R3\u200b=R3\u200b\u2212R1\u200b\u21d2(0,5,2)<\/p>\n\n\n\n

      New matrix:[1\u2212370\u2212510052]\\begin{bmatrix} 1 & -3 & 7 \\\\ 0 & -5 & 10 \\\\ 0 & 5 & 2 \\end{bmatrix}\u200b100\u200b\u22123\u221255\u200b7102\u200b\u200b<\/p>\n\n\n\n

        \n
      • Add Row2 to Row3:<\/li>\n<\/ul>\n\n\n\n

        R3=R3+R2\u21d2(0,0,12)R_3 = R_3 + R_2 \\Rightarrow (0, 0, 12)R3\u200b=R3\u200b+R2\u200b\u21d2(0,0,12)<\/p>\n\n\n\n

        Final matrix:[1\u2212370\u22125100012]\\begin{bmatrix} 1 & -3 & 7 \\\\ 0 & -5 & 10 \\\\ 0 & 0 & 12 \\end{bmatrix}\u200b100\u200b\u22123\u221250\u200b71012\u200b\u200b<\/p>\n\n\n\n

        This is an upper triangular matrix with 3 leading entries<\/strong>, so the matrix has rank 3<\/strong>.<\/p>\n\n\n\n

        Since there are 3 equations and 3 variables<\/strong> and the matrix has full rank<\/strong>, the only solution<\/strong> is the trivial solution<\/strong>.<\/p>\n\n\n\n

        \n\n\n\n

        \u2705 Final Answer:<\/strong><\/h3>\n\n\n\n

        No, the system does not have a nontrivial solution.<\/strong><\/p>\n\n\n\n

        \n\n\n\n

        \u270d\ufe0f Explanation <\/h3>\n\n\n\n

        The given system of three homogeneous linear equations involves three unknowns: x1,x2,x_1, x_2,x1\u200b,x2\u200b, and x3x_3x3\u200b. Homogeneous systems always have at least one solution\u2014the trivial solution<\/strong>, where all variables are zero. The key question here is whether any nontrivial solution<\/strong> (a solution with at least one variable non-zero) exists.<\/p>\n\n\n\n

        To find this, we analyze the coefficient matrix of the system. A homogeneous system has a nontrivial solution if and only if<\/strong> the coefficient matrix is not of full rank<\/strong>, meaning its rows are linearly dependent. This would imply there are more variables than pivot positions after row reduction.<\/p>\n\n\n\n

        We performed Gaussian elimination on the coefficient matrix:[1\u221237\u221221\u22124129]\\begin{bmatrix} 1 & -3 & 7 \\\\ -2 & 1 & -4 \\\\ 1 & 2 & 9 \\end{bmatrix}\u200b1\u221221\u200b\u2212312\u200b7\u221249\u200b\u200b<\/p>\n\n\n\n

        After row operations, we obtained a row echelon form with three pivot positions\u2014one in each row and column. This indicates that the matrix has full rank (rank = 3). In a 3\u00d73 system, full rank means there is exactly one solution<\/strong>, which for homogeneous systems, is the trivial solution<\/strong>.<\/p>\n\n\n\n

        Therefore, since the rank of the matrix equals the number of unknowns, the system is consistent and independent<\/strong>, and has no free variables<\/strong>. This implies that there is no nontrivial solution<\/strong>.<\/p>\n\n\n\n

        In conclusion, because the system\u2019s coefficient matrix is of full rank, the system does not<\/strong> have a nontrivial solution.<\/p>\n\n\n\n

        \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

        Determine if the system has a nontrivial solution (you do not need to completely solve the system). x1 \u2013 3×2 + 7×3 = 0 -2×1 + x2 \u2013 4×3 = 0 x1 + 2×2 + 9×3 = 0 The Correct Answer and Explanation is: To determine if the system has a nontrivial solution, we need […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-232239","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/232239","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=232239"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/232239\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=232239"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=232239"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=232239"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}