{"id":2828,"date":"2026-01-19T15:52:25","date_gmt":"2026-01-19T10:22:25","guid":{"rendered":"https:\/\/www.manabadi.co.in\/boards\/?p=2828"},"modified":"2026-01-19T15:52:28","modified_gmt":"2026-01-19T10:22:28","slug":"ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3i","status":"publish","type":"post","link":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3i\/","title":{"rendered":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(i)"},"content":{"rendered":"\n<h2>TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(i)<\/h2>\n\n<h3>Question 1.<br> 2x + 3y \u2013 z = 0,<br> x \u2013 y \u2013 2z = 0,<br> 3x + y + 3z = 0<\/h3>\n\n<p>Answer:<br> The coefficient matrix obtained from the given equations is<br>\n\n<img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-6481\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3i-1s.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(i) 1\" width=\"323\" height=\"170\" sizes=\"(max-width: 323px) 100vw, 323px\" data-pin-nopin=\"true\"><br> Since the determinant of the coefficient matrix \u2260 0 the system has a trivial solution, x = y = z = 0 and \u03c1(A) = 3.<\/p>\n\n<h3>Question 3.<br> x + y \u2013 2z = 0,<br> 2x + y \u2013 3z = 0,<br> 5x + 4y \u2013 9z = 0<\/h3>\n\n<p>Answer:<br> The coefficient matrix is<br> A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-1-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-1\" style=\"width: 6.945em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.967em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.64em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-2\"><span class=\"mrow\" id=\"MathJax-Span-3\"><span class=\"mo\" id=\"MathJax-Span-4\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-5\"><span style=\"display: inline-block; position: relative; width: 4.289em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-6\"><span class=\"mrow\" id=\"MathJax-Span-7\"><span class=\"mn\" id=\"MathJax-Span-8\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-16\"><span class=\"mrow\" id=\"MathJax-Span-17\"><span class=\"mn\" id=\"MathJax-Span-18\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-26\"><span class=\"mrow\" id=\"MathJax-Span-27\"><span class=\"mn\" id=\"MathJax-Span-28\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-9\"><span class=\"mrow\" id=\"MathJax-Span-10\"><span class=\"mn\" id=\"MathJax-Span-11\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-19\"><span class=\"mrow\" id=\"MathJax-Span-20\"><span class=\"mn\" id=\"MathJax-Span-21\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-29\"><span class=\"mrow\" id=\"MathJax-Span-30\"><span class=\"mn\" id=\"MathJax-Span-31\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 2.985em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-12\"><span class=\"mrow\" id=\"MathJax-Span-13\"><span class=\"mo\" id=\"MathJax-Span-14\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-15\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-22\"><span class=\"mrow\" id=\"MathJax-Span-23\"><span class=\"mo\" id=\"MathJax-Span-24\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-25\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-32\"><span class=\"mrow\" id=\"MathJax-Span-33\"><span class=\"mo\" id=\"MathJax-Span-34\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-35\" style=\"font-family: MathJax_Main;\">9<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-36\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-1\">\\left[\\begin{array}{lll} 1 & 1 & -2 \\\\ 2 & 1 & -3 \\\\ 5 & 4 & -9 \\end{array}\\right]<\/script><br> and <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-2-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-37\" style=\"width: 6.06em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.221em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.08em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-38\"><span class=\"mrow\" id=\"MathJax-Span-39\"><span class=\"mo\" id=\"MathJax-Span-40\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-41\"><span style=\"display: inline-block; position: relative; width: 4.289em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-42\"><span class=\"mrow\" id=\"MathJax-Span-43\"><span class=\"mn\" id=\"MathJax-Span-44\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-52\"><span class=\"mrow\" id=\"MathJax-Span-53\"><span class=\"mn\" id=\"MathJax-Span-54\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-62\"><span class=\"mrow\" id=\"MathJax-Span-63\"><span class=\"mn\" id=\"MathJax-Span-64\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-45\"><span class=\"mrow\" id=\"MathJax-Span-46\"><span class=\"mn\" id=\"MathJax-Span-47\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-55\"><span class=\"mrow\" id=\"MathJax-Span-56\"><span class=\"mn\" id=\"MathJax-Span-57\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-65\"><span class=\"mrow\" id=\"MathJax-Span-66\"><span class=\"mn\" id=\"MathJax-Span-67\" style=\"font-family: MathJax_Main;\">4<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 2.985em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-48\"><span class=\"mrow\" id=\"MathJax-Span-49\"><span class=\"mo\" id=\"MathJax-Span-50\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-51\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-58\"><span class=\"mrow\" id=\"MathJax-Span-59\"><span class=\"mo\" id=\"MathJax-Span-60\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-61\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-68\"><span class=\"mrow\" id=\"MathJax-Span-69\"><span class=\"mo\" id=\"MathJax-Span-70\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-71\" style=\"font-family: MathJax_Main;\">9<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-72\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -0.464em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.328em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -1.396em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-2\">\\left|\\begin{array}{lll} 1 & 1 & -2 \\\\ 2 & 1 & -3 \\\\ 5 & 4 & -9 \\end{array}\\right|<\/script><br> = 1(-9 + 12) \u2013 1(-18 + 15) \u2013 2<br> = 3 + 3 \u2013 6 = 0<br> If <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-3-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-73\" style=\"width: 4.01em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.451em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1003.22em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-74\"><span class=\"mrow\" id=\"MathJax-Span-75\"><span class=\"mo\" id=\"MathJax-Span-76\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-77\"><span style=\"display: inline-block; position: relative; width: 2.006em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-78\"><span class=\"mrow\" id=\"MathJax-Span-79\"><span class=\"mn\" id=\"MathJax-Span-80\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-84\"><span class=\"mrow\" id=\"MathJax-Span-85\"><span class=\"mn\" id=\"MathJax-Span-86\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.42em, 4.895em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-81\"><span class=\"mrow\" id=\"MathJax-Span-82\"><span class=\"mn\" id=\"MathJax-Span-83\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-87\"><span class=\"mrow\" id=\"MathJax-Span-88\"><span class=\"mn\" id=\"MathJax-Span-89\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-90\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-3\">\\left[\\begin{array}{ll} 1 & 1 \\\\ 2 & 1 \\end{array}\\right]<\/script> is any submatrix of order 2 x 2 and<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-4-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-91\" style=\"width: 3.404em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.938em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1002.8em, 4.802em, -999.998em); top: -3.726em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-92\"><span class=\"mrow\" id=\"MathJax-Span-93\"><span class=\"mo\" id=\"MathJax-Span-94\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-95\"><span style=\"display: inline-block; position: relative; width: 2.006em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.47em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-96\"><span class=\"mrow\" id=\"MathJax-Span-97\"><span class=\"mn\" id=\"MathJax-Span-98\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-102\"><span class=\"mrow\" id=\"MathJax-Span-103\"><span class=\"mn\" id=\"MathJax-Span-104\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1000.42em, 4.895em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-99\"><span class=\"mrow\" id=\"MathJax-Span-100\"><span class=\"mn\" id=\"MathJax-Span-101\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-105\"><span class=\"mrow\" id=\"MathJax-Span-106\"><span class=\"mn\" id=\"MathJax-Span-107\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-108\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.73em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-4\">\\left|\\begin{array}{ll} 1 & 1 \\\\ 2 & 1 \\end{array}\\right|<\/script> = 1 \u2013 2 = -1 \u2260 0, \u03c1(A) &lt; 3. Hence the system has a nontrival solution.<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6479\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3i-3.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(i) 3\" width=\"134\" height=\"189\" data-pin-nopin=\"true\"><br> \u2234 System of equations is equivalent to<br> x + y \u2013 2z = 0 and y \u2013 z = 0<br> Let z = k then y = k and x = k<br> \u2234 x = y = z = k for a real number k.<\/p>\n\n<h3>Question 4.<br> x + y \u2013 z= 0<br> x \u2013 2y + z = 0<br> 3x + 6y \u2013 5z = 0<\/h3>\n\n<p>Answer:<br> Coefficient matrix<br> A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-5-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-109\" style=\"width: 7.831em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.759em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.43em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-110\"><span class=\"mrow\" id=\"MathJax-Span-111\"><span class=\"mo\" id=\"MathJax-Span-112\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-113\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-114\"><span class=\"mrow\" id=\"MathJax-Span-115\"><span class=\"mn\" id=\"MathJax-Span-116\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-124\"><span class=\"mrow\" id=\"MathJax-Span-125\"><span class=\"mn\" id=\"MathJax-Span-126\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-134\"><span class=\"mrow\" id=\"MathJax-Span-135\"><span class=\"mn\" id=\"MathJax-Span-136\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-117\"><span class=\"mrow\" id=\"MathJax-Span-118\"><span class=\"mn\" id=\"MathJax-Span-119\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-127\"><span class=\"mrow\" id=\"MathJax-Span-128\"><span class=\"mo\" id=\"MathJax-Span-129\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-130\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-137\"><span class=\"mrow\" id=\"MathJax-Span-138\"><span class=\"mn\" id=\"MathJax-Span-139\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-120\"><span class=\"mrow\" id=\"MathJax-Span-121\"><span class=\"mo\" id=\"MathJax-Span-122\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-123\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-131\"><span class=\"mrow\" id=\"MathJax-Span-132\"><span class=\"mn\" id=\"MathJax-Span-133\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-140\"><span class=\"mrow\" id=\"MathJax-Span-141\"><span class=\"mo\" id=\"MathJax-Span-142\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-143\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-144\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-5\">\\left[\\begin{array}{ccc} 1 & 1 & -1 \\\\ 1 & -2 & 1 \\\\ 3 & 6 & -5 \\end{array}\\right]<\/script><br> |A| = 1(10 \u2013 6) \u2013 1(-5 \u2013 3) \u2013 1(6 + 6)<br> = 4 + 8 \u2013 12 = 0<\/p>\n\n<p>\u2234 If <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-6-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-145\" style=\"width: 4.942em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.243em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.773em, 1004.01em, 4.429em, -999.998em); top: -3.353em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-146\"><span class=\"mrow\" id=\"MathJax-Span-147\"><span class=\"mo\" id=\"MathJax-Span-148\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">[<\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-149\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.42em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-150\"><span class=\"mrow\" id=\"MathJax-Span-151\"><span class=\"mn\" id=\"MathJax-Span-152\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-156\"><span class=\"mrow\" id=\"MathJax-Span-157\"><span class=\"mn\" id=\"MathJax-Span-158\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-153\"><span class=\"mrow\" id=\"MathJax-Span-154\"><span class=\"mn\" id=\"MathJax-Span-155\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-159\"><span class=\"mrow\" id=\"MathJax-Span-160\"><span class=\"mo\" id=\"MathJax-Span-161\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-162\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-163\" style=\"vertical-align: 0em;\"><span style=\"font-family: MathJax_Size3;\">]<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.357em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-6\">\\left[\\begin{array}{cc} 1 & 1 \\\\ 1 & -2 \\end{array}\\right]<\/script> is a submatrix of order 2 and<br> <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-7-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-164\" style=\"width: 4.336em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.73em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1003.59em, 4.802em, -999.998em); top: -3.726em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-165\"><span class=\"mrow\" id=\"MathJax-Span-166\"><span class=\"mo\" id=\"MathJax-Span-167\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-168\"><span style=\"display: inline-block; position: relative; width: 2.798em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.565em, 1000.42em, 4.895em, -999.998em); top: -4.005em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-169\"><span class=\"mrow\" id=\"MathJax-Span-170\"><span class=\"mn\" id=\"MathJax-Span-171\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-175\"><span class=\"mrow\" id=\"MathJax-Span-176\"><span class=\"mn\" id=\"MathJax-Span-177\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.565em, 1001.21em, 4.988em, -999.998em); top: -4.005em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -4.658em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-172\"><span class=\"mrow\" id=\"MathJax-Span-173\"><span class=\"mn\" id=\"MathJax-Span-174\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.26em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-178\"><span class=\"mrow\" id=\"MathJax-Span-179\"><span class=\"mo\" id=\"MathJax-Span-180\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-181\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-182\" style=\"vertical-align: 1.447em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Main; top: -3.26em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Main; top: -1.862em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Main; position: absolute; top: -2.561em; left: 0em;\">\u2223<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 3.73em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.132em; border-left: 0px solid; width: 0px; height: 2.922em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-7\">\\left|\\begin{array}{cc} 1 & 1 \\\\ 1 & -2 \\end{array}\\right|<\/script> = -3 \u2260 0, \u03c1(A) = 2. System has a non-trivial solution \u03c1(A) &lt; 3.<br> A = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-8-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-183\" style=\"width: 7.831em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.759em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.43em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-184\"><span class=\"mrow\" id=\"MathJax-Span-185\"><span class=\"mo\" id=\"MathJax-Span-186\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-187\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-188\"><span class=\"mrow\" id=\"MathJax-Span-189\"><span class=\"mn\" id=\"MathJax-Span-190\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-198\"><span class=\"mrow\" id=\"MathJax-Span-199\"><span class=\"mn\" id=\"MathJax-Span-200\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-208\"><span class=\"mrow\" id=\"MathJax-Span-209\"><span class=\"mn\" id=\"MathJax-Span-210\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-191\"><span class=\"mrow\" id=\"MathJax-Span-192\"><span class=\"mn\" id=\"MathJax-Span-193\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-201\"><span class=\"mrow\" id=\"MathJax-Span-202\"><span class=\"mo\" id=\"MathJax-Span-203\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-204\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-211\"><span class=\"mrow\" id=\"MathJax-Span-212\"><span class=\"mn\" id=\"MathJax-Span-213\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-194\"><span class=\"mrow\" id=\"MathJax-Span-195\"><span class=\"mo\" id=\"MathJax-Span-196\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-197\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-205\"><span class=\"mrow\" id=\"MathJax-Span-206\"><span class=\"mn\" id=\"MathJax-Span-207\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-214\"><span class=\"mrow\" id=\"MathJax-Span-215\"><span class=\"mo\" id=\"MathJax-Span-216\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-217\" style=\"font-family: MathJax_Main;\">5<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-218\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-8\">\\left[\\begin{array}{rrr} 1 & 1 & -1 \\\\ 1 & -2 & 1 \\\\ 3 & 6 & -5 \\end{array}\\right]<\/script><br> Use R<sub>2<\/sub> \u2013 R<sub>1<\/sub> and R<sub>3<\/sub> \u2013 3R<sub>1<\/sub><br> A \u2013 <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-9-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-219\" style=\"width: 7.831em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.759em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.43em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-220\"><span class=\"mrow\" id=\"MathJax-Span-221\"><span class=\"mo\" id=\"MathJax-Span-222\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a1<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a3<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a2<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span class=\"mtable\" id=\"MathJax-Span-223\"><span style=\"display: inline-block; position: relative; width: 5.082em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-224\"><span class=\"mrow\" id=\"MathJax-Span-225\"><span class=\"mn\" id=\"MathJax-Span-226\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-234\"><span class=\"mrow\" id=\"MathJax-Span-235\"><span class=\"mn\" id=\"MathJax-Span-236\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-244\"><span class=\"mrow\" id=\"MathJax-Span-245\"><span class=\"mn\" id=\"MathJax-Span-246\" style=\"font-family: MathJax_Main;\">0<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.26em, 6.107em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.42em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-227\"><span class=\"mrow\" id=\"MathJax-Span-228\"><span class=\"mn\" id=\"MathJax-Span-229\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.26em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-237\"><span class=\"mrow\" id=\"MathJax-Span-238\"><span class=\"mo\" id=\"MathJax-Span-239\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-240\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-247\"><span class=\"mrow\" id=\"MathJax-Span-248\"><span class=\"mn\" id=\"MathJax-Span-249\" style=\"font-family: MathJax_Main;\">3<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-230\"><span class=\"mrow\" id=\"MathJax-Span-231\"><span class=\"mo\" id=\"MathJax-Span-232\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-233\" style=\"font-family: MathJax_Main;\">1<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-241\"><span class=\"mrow\" id=\"MathJax-Span-242\"><span class=\"mn\" id=\"MathJax-Span-243\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.21em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-250\"><span class=\"mrow\" id=\"MathJax-Span-251\"><span class=\"mo\" id=\"MathJax-Span-252\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-253\" style=\"font-family: MathJax_Main;\">2<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-254\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.655em; height: 0px;\"><span style=\"position: absolute; font-family: MathJax_Size4; top: -2.84em; left: 0em;\">\u23a4<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; font-family: MathJax_Size4; top: -0.836em; left: 0em;\">\u23a6<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"font-family: MathJax_Size4; position: absolute; top: -1.815em; left: 0em;\">\u23a5<span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.429em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -1.943em; border-left: 0px solid; width: 0px; height: 4.543em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-9\">\\left[\\begin{array}{rrr} 1 & 1 & -1 \\\\ 0 & -3 & 2 \\\\ 0 & 3 & -2 \\end{array}\\right]<\/script><br> System of equations is equivalent to x + y \u2013 z = 0<br> 3y \u2013 2z = 0<br> Let z = k, then 3y = 2k<br> \u21d2 y = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-10-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-255\" style=\"width: 1.307em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.121em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.12em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-256\"><span class=\"mfrac\" id=\"MathJax-Span-257\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.7em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-258\"><span class=\"mn\" id=\"MathJax-Span-259\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"texatom\" id=\"MathJax-Span-260\"><span class=\"mrow\" id=\"MathJax-Span-261\"><span class=\"mi\" id=\"MathJax-Span-262\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">k<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-263\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.84em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.841em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-10\">\\frac{2 \\mathrm{k}}{3}<\/script><br> x = -y + z = \u2013<span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-11-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-264\" style=\"width: 1.307em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.121em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.12em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-265\"><span class=\"mfrac\" id=\"MathJax-Span-266\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.7em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-267\"><span class=\"mn\" id=\"MathJax-Span-268\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"texatom\" id=\"MathJax-Span-269\"><span class=\"mrow\" id=\"MathJax-Span-270\"><span class=\"mi\" id=\"MathJax-Span-271\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">k<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-272\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.84em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.841em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-11\">\\frac{2 \\mathrm{k}}{3}<\/script> + k = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-12-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-273\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-274\"><span class=\"mfrac\" id=\"MathJax-Span-275\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.38em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-276\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">k<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-277\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-12\">\\frac{k}{3}<\/script><br> x = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-13-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-278\" style=\"width: 0.888em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 0.748em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1000.75em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-279\"><span class=\"mfrac\" id=\"MathJax-Span-280\"><span style=\"display: inline-block; position: relative; width: 0.468em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.38em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.184em;\"><span class=\"mi\" id=\"MathJax-Span-281\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">k<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-282\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.47em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.468em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-13\">\\frac{k}{3}<\/script>, y = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-14-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-283\" style=\"width: 1.307em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.121em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.26em, 1001.12em, 2.798em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-284\"><span class=\"mfrac\" id=\"MathJax-Span-285\"><span style=\"display: inline-block; position: relative; width: 0.841em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.7em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mrow\" id=\"MathJax-Span-286\"><span class=\"mn\" id=\"MathJax-Span-287\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2<\/span><span class=\"texatom\" id=\"MathJax-Span-288\"><span class=\"mrow\" id=\"MathJax-Span-289\"><span class=\"mi\" id=\"MathJax-Span-290\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">k<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.33em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.184em;\"><span class=\"mn\" id=\"MathJax-Span-291\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1000.84em, 1.214em, -999.998em); top: -1.302em; left: 0em;\"><span style=\"display: inline-block; overflow: hidden; vertical-align: 0em; border-top: 1.3px solid; width: 0.841em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 2.286em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.484em; border-left: 0px solid; width: 0px; height: 1.57em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-14\">\\frac{2 \\mathrm{k}}{3}<\/script>, z = k<br> for any real number of k.<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"<p>TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(i) Question 1. 2x + 3y \u2013 z = 0, x \u2013 y \u2013 2z = 0, 3x + y + 3z = 0 Answer: The coefficient matrix obtained from the given equations is Since the determinant of the coefficient matrix \u2260 0 the system has [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":2928,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-2828","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-other"],"_links":{"self":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2828","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/comments?post=2828"}],"version-history":[{"count":1,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2828\/revisions"}],"predecessor-version":[{"id":2830,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2828\/revisions\/2830"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/media\/2928"}],"wp:attachment":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/media?parent=2828"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/categories?post=2828"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/tags?post=2828"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}