{"id":2826,"date":"2026-01-19T15:54:42","date_gmt":"2026-01-19T10:24:42","guid":{"rendered":"https:\/\/www.manabadi.co.in\/boards\/?p=2826"},"modified":"2026-01-19T15:55:36","modified_gmt":"2026-01-19T10:25:36","slug":"ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h","status":"publish","type":"post","link":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/","title":{"rendered":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(h)"},"content":{"rendered":"\n<h3>Question 1.<br> Solve the following systems of equations.<br> (i) By using Cramer\u2019s rule and Matrix inversion method, when the coefficient matrix is non-singular.<br> (ii) By using Gauss Jordan Method. Also deter-mine whether the system has a unique solution or infinite number of solutions or no solution and find the solutions if exist.<br> I) 5x \u2013 6y + 4z = 15<br> 7x + 4y-3z = 17<br> 2x + y + 6z = 46<\/h3>\n\nAnswer:<br> i) Cramer\u2019s rule<br> \u0394 = <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: 6.992em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.014em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.87em, 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: 0.282em; 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-5\"><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, 6.06em, -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, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 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;\">5<\/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: -3.959em; right: 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;\">7<\/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-26\"><span class=\"mrow\" id=\"MathJax-Span-27\"><span class=\"mn\" id=\"MathJax-Span-28\" 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: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.06em, -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, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-9\"><span class=\"mrow\" id=\"MathJax-Span-10\"><span class=\"mo\" id=\"MathJax-Span-11\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-12\" style=\"font-family: MathJax_Main;\">6<\/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: -3.959em; right: 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;\">4<\/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: -2.561em; right: 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;\">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: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1001.26em, 6.153em, -999.998em); top: -4.518em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-13\"><span class=\"mrow\" id=\"MathJax-Span-14\"><span class=\"mn\" id=\"MathJax-Span-15\" style=\"font-family: MathJax_Main;\">4<\/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-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, 1000.47em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-32\"><span class=\"mrow\" id=\"MathJax-Span-33\"><span class=\"mn\" id=\"MathJax-Span-34\" 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: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-35\" 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-1\">\\left|\\begin{array}{rrr} 5 & -6 & 4 \\\\ 7 & 4 & -3 \\\\ 2 & 1 & 6 \\end{array}\\right|<\/script><br> = 5(24 + 3) + 6(42 + 6) + 4(7 \u2013 8)<br> = 135 + 288-4 = 419 \u2260 0<br> Hence Cramer\u2019s rule is applicable.<\/p>\n\n<p>\u0394<sub>1<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-2-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-36\" style=\"width: 7.551em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.479em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.34em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-37\"><span class=\"mrow\" id=\"MathJax-Span-38\"><span class=\"mo\" id=\"MathJax-Span-39\" 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-40\"><span style=\"display: inline-block; position: relative; width: 5.548em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.98em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.93em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-41\"><span class=\"mrow\" id=\"MathJax-Span-42\"><span class=\"mn\" id=\"MathJax-Span-43\" style=\"font-family: MathJax_Main;\">15<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.98em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-51\"><span class=\"mrow\" id=\"MathJax-Span-52\"><span class=\"mn\" id=\"MathJax-Span-53\" style=\"font-family: MathJax_Main;\">19<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.98em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-61\"><span class=\"mrow\" id=\"MathJax-Span-62\"><span class=\"mn\" id=\"MathJax-Span-63\" style=\"font-family: MathJax_Main;\">46<\/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, 6.06em, -999.998em); top: -4.471em; left: 2.006em;\"><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-44\"><span class=\"mrow\" id=\"MathJax-Span-45\"><span class=\"mo\" id=\"MathJax-Span-46\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-47\" style=\"font-family: MathJax_Main;\">6<\/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: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-54\"><span class=\"mrow\" id=\"MathJax-Span-55\"><span class=\"mn\" id=\"MathJax-Span-56\" style=\"font-family: MathJax_Main;\">4<\/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: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-64\"><span class=\"mrow\" id=\"MathJax-Span-65\"><span class=\"mn\" id=\"MathJax-Span-66\" 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: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.379em, 1001.26em, 6.153em, -999.998em); top: -4.518em; left: 4.289em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-48\"><span class=\"mrow\" id=\"MathJax-Span-49\"><span class=\"mn\" id=\"MathJax-Span-50\" style=\"font-family: MathJax_Main;\">4<\/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-57\"><span class=\"mrow\" id=\"MathJax-Span-58\"><span class=\"mo\" id=\"MathJax-Span-59\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-60\" 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-67\"><span class=\"mrow\" id=\"MathJax-Span-68\"><span class=\"mn\" id=\"MathJax-Span-69\" 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: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-70\" 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}{rrr} 15 & -6 & 4 \\\\ 19 & 4 & -3 \\\\ 46 & 1 & 6 \\end{array}\\right|<\/script><br> = 15(24 + 3) + 6(114 + 138) + 4(19 -184)<br> = 405 + 1512 \u2013 660<br> = 1917 \u2013 660<br> = 1257<\/p>\n\n<p>\u0394<sub>2<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-3-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-71\" style=\"width: 6.666em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.734em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.59em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-72\"><span class=\"mrow\" id=\"MathJax-Span-73\"><span class=\"mo\" id=\"MathJax-Span-74\" 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-75\"><span style=\"display: inline-block; position: relative; width: 4.802em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -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, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-76\"><span class=\"mrow\" id=\"MathJax-Span-77\"><span class=\"mn\" id=\"MathJax-Span-78\" style=\"font-family: MathJax_Main;\">5<\/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: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-85\"><span class=\"mrow\" id=\"MathJax-Span-86\"><span class=\"mn\" id=\"MathJax-Span-87\" style=\"font-family: MathJax_Main;\">7<\/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-95\"><span class=\"mrow\" id=\"MathJax-Span-96\"><span class=\"mn\" id=\"MathJax-Span-97\" 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: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.98em, 6.107em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.93em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-79\"><span class=\"mrow\" id=\"MathJax-Span-80\"><span class=\"mn\" id=\"MathJax-Span-81\" style=\"font-family: MathJax_Main;\">15<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.98em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-88\"><span class=\"mrow\" id=\"MathJax-Span-89\"><span class=\"mn\" id=\"MathJax-Span-90\" style=\"font-family: MathJax_Main;\">19<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.98em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-98\"><span class=\"mrow\" id=\"MathJax-Span-99\"><span class=\"mn\" id=\"MathJax-Span-100\" style=\"font-family: MathJax_Main;\">46<\/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(2.379em, 1001.26em, 6.153em, -999.998em); top: -4.518em; left: 3.497em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.171em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-82\"><span class=\"mrow\" id=\"MathJax-Span-83\"><span class=\"mn\" id=\"MathJax-Span-84\" style=\"font-family: MathJax_Main;\">4<\/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-91\"><span class=\"mrow\" id=\"MathJax-Span-92\"><span class=\"mo\" id=\"MathJax-Span-93\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-94\" 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-101\"><span class=\"mrow\" id=\"MathJax-Span-102\"><span class=\"mn\" id=\"MathJax-Span-103\" 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: 4.522em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-104\" 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-3\">\\left|\\begin{array}{rrr} 5 & 15 & 4 \\\\ 7 & 19 & -3 \\\\ 2 & 46 & 6 \\end{array}\\right|<\/script><br> = = 5(114 + 138) -15(42 + 6) + 4(322 \u2013 38)<br> = 1260 \u2013 720 + 1136<br> = 1676<\/p>\n\n<p>\u0394<sub>3<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-4-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-105\" style=\"width: 6.666em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.734em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.59em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-106\"><span class=\"mrow\" id=\"MathJax-Span-107\"><span class=\"mo\" id=\"MathJax-Span-108\" 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-109\"><span style=\"display: inline-block; position: relative; width: 4.802em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -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, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-110\"><span class=\"mrow\" id=\"MathJax-Span-111\"><span class=\"mn\" id=\"MathJax-Span-112\" style=\"font-family: MathJax_Main;\">5<\/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: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-120\"><span class=\"mrow\" id=\"MathJax-Span-121\"><span class=\"mn\" id=\"MathJax-Span-122\" style=\"font-family: MathJax_Main;\">7<\/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-129\"><span class=\"mrow\" id=\"MathJax-Span-130\"><span class=\"mn\" id=\"MathJax-Span-131\" 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: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.06em, -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, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-113\"><span class=\"mrow\" id=\"MathJax-Span-114\"><span class=\"mo\" id=\"MathJax-Span-115\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-116\" style=\"font-family: MathJax_Main;\">6<\/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: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-123\"><span class=\"mrow\" id=\"MathJax-Span-124\"><span class=\"mn\" id=\"MathJax-Span-125\" style=\"font-family: MathJax_Main;\">4<\/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: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-132\"><span class=\"mrow\" id=\"MathJax-Span-133\"><span class=\"mn\" id=\"MathJax-Span-134\" 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: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.98em, 6.107em, -999.998em); top: -4.471em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.93em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><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;\">15<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.98em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-126\"><span class=\"mrow\" id=\"MathJax-Span-127\"><span class=\"mn\" id=\"MathJax-Span-128\" style=\"font-family: MathJax_Main;\">19<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.171em, 1000.98em, 4.15em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-135\"><span class=\"mrow\" id=\"MathJax-Span-136\"><span class=\"mn\" id=\"MathJax-Span-137\" style=\"font-family: MathJax_Main;\">46<\/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><\/span><span class=\"mo\" id=\"MathJax-Span-138\" 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-4\">\\left|\\begin{array}{rrr} 5 & -6 & 15 \\\\ 7 & 4 & 19 \\\\ 2 & 1 & 46 \\end{array}\\right|<\/script><br> = 5(184 \u2013 19) + 6 (322 \u2013 38) + 15 (7 \u2013 8)<br> = 825 + 1704 \u2013 15<br> = 2529 \u2013 15<br> = 2514<\/p>\n\n<p>\u2234 x = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-5-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-139\" style=\"width: 5.221em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.476em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1004.48em, 2.891em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-140\"><span class=\"mfrac\" id=\"MathJax-Span-141\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.89em, 4.243em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msubsup\" id=\"MathJax-Span-142\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-143\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.912em; left: 0.608em;\"><span class=\"mn\" id=\"MathJax-Span-144\" style=\"font-size: 50%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.277em;\"><span class=\"mi\" id=\"MathJax-Span-145\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.03em, 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: 1.027em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-146\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-147\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.4em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.697em;\"><span class=\"mn\" id=\"MathJax-Span-148\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1257<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1001.03em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.51em;\"><span class=\"mn\" id=\"MathJax-Span-149\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">419<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.54em, 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: 1.54em; 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: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.538em; border-left: 0px solid; width: 0px; height: 1.732em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-5\">\\frac{\\Delta_1}{\\Delta}=\\frac{1257}{419}<\/script> = 3<br> y = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-6-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-150\" style=\"width: 5.221em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.476em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1004.48em, 2.891em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-151\"><span class=\"mfrac\" id=\"MathJax-Span-152\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.89em, 4.243em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msubsup\" id=\"MathJax-Span-153\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-154\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.912em; left: 0.608em;\"><span class=\"mn\" id=\"MathJax-Span-155\" style=\"font-size: 50%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.277em;\"><span class=\"mi\" id=\"MathJax-Span-156\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.03em, 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: 1.027em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-157\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-158\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.4em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.697em;\"><span class=\"mn\" id=\"MathJax-Span-159\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">1676<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1001.03em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.51em;\"><span class=\"mn\" id=\"MathJax-Span-160\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">419<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.54em, 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: 1.54em; 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: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.538em; border-left: 0px solid; width: 0px; height: 1.732em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-6\">\\frac{\\Delta_2}{\\Delta}=\\frac{1676}{419}<\/script> = 4<br> z = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-7-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-161\" style=\"width: 5.221em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.476em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1004.48em, 2.891em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-162\"><span class=\"mfrac\" id=\"MathJax-Span-163\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.89em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msubsup\" id=\"MathJax-Span-164\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-165\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.912em; left: 0.608em;\"><span class=\"mn\" id=\"MathJax-Span-166\" style=\"font-size: 50%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.277em;\"><span class=\"mi\" id=\"MathJax-Span-167\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.03em, 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: 1.027em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-168\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-169\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.54em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.4em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.697em;\"><span class=\"mn\" id=\"MathJax-Span-170\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">2514<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1001.03em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.51em;\"><span class=\"mn\" id=\"MathJax-Span-171\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">419<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.54em, 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: 1.54em; 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: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.538em; border-left: 0px solid; width: 0px; height: 1.786em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-7\">\\frac{\\Delta_3}{\\Delta}=\\frac{2514}{419}<\/script> = 6<br>\n\n\u2234 Solution is x = 3, y = 4, and z = 6<\/p>\n\n<p>ii) Matrix Inversion method:<br> Use the formula A<sup>-1<\/sup> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-8-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-172\" style=\"width: 2.519em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.146em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1002.15em, 2.845em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-173\"><span class=\"mfrac\" id=\"MathJax-Span-174\"><span style=\"display: inline-block; position: relative; width: 1.866em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1001.73em, 4.289em, -999.998em); top: -4.564em; left: 50%; margin-left: -0.836em;\"><span class=\"mrow\" id=\"MathJax-Span-175\"><span class=\"texatom\" id=\"MathJax-Span-176\"><span class=\"mrow\" id=\"MathJax-Span-177\"><span class=\"mi\" id=\"MathJax-Span-178\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-179\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">d<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-180\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">j<\/span><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-181\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.49em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.743em;\"><span class=\"mrow\" id=\"MathJax-Span-182\"><span class=\"texatom\" id=\"MathJax-Span-183\"><span class=\"mrow\" id=\"MathJax-Span-184\"><span class=\"mi\" id=\"MathJax-Span-185\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">d<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-186\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">e<\/span><span class=\"mi\" id=\"MathJax-Span-187\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">t<\/span><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-188\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.87em, 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: 1.866em; 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: -0.538em; border-left: 0px solid; width: 0px; height: 1.786em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-8\">\\frac{{Adj} A}{{det} A}<\/script><br>\n\n<img fetchpriority=\"high\" decoding=\"async\" class=\"alignnone size-full wp-image-6331\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-1.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 1\" width=\"314\" height=\"1062\" sizes=\"(max-width: 314px) 100vw, 314px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 1\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\"><\/p>\n\n<p>iii) Gauss Jordan Method:<br> Augmented matrix of the system is<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6330\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-2.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 2\" width=\"318\" height=\"636\" sizes=\"auto, (max-width: 318px) 100vw, 318px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 2\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\"><br> The given system is consistent and has a unique solution given by x = 3, y = 4, z = 6.<\/p>\n\n<h3>Question 2.<br> x + y + z = 1<br> 2x + 2y + 3z = 6<br> x + 4y + 9z = 3<\/h3>\n\n<p>Answer:<br> i) Cramer\u2019s rule :<br> \u0394 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-9-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-189\" style=\"width: 5.128em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.429em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.29em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-190\"><span class=\"mrow\" id=\"MathJax-Span-191\"><span class=\"mo\" id=\"MathJax-Span-192\" 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-193\"><span style=\"display: inline-block; position: relative; width: 3.497em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -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-194\"><span class=\"mrow\" id=\"MathJax-Span-195\"><span class=\"mn\" id=\"MathJax-Span-196\" 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-203\"><span class=\"mrow\" id=\"MathJax-Span-204\"><span class=\"mn\" id=\"MathJax-Span-205\" 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, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-212\"><span class=\"mrow\" id=\"MathJax-Span-213\"><span class=\"mn\" id=\"MathJax-Span-214\" 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: 4.476em;\"><\/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-197\"><span class=\"mrow\" id=\"MathJax-Span-198\"><span class=\"mn\" id=\"MathJax-Span-199\" 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-206\"><span class=\"mrow\" id=\"MathJax-Span-207\"><span class=\"mn\" id=\"MathJax-Span-208\" 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(3.171em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-215\"><span class=\"mrow\" id=\"MathJax-Span-216\"><span class=\"mn\" id=\"MathJax-Span-217\" 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(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 2.985em;\"><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-200\"><span class=\"mrow\" id=\"MathJax-Span-201\"><span class=\"mn\" id=\"MathJax-Span-202\" 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-209\"><span class=\"mrow\" id=\"MathJax-Span-210\"><span class=\"mn\" id=\"MathJax-Span-211\" 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; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-218\"><span class=\"mrow\" id=\"MathJax-Span-219\"><span class=\"mn\" id=\"MathJax-Span-220\" 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: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-221\" 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-9\">\\left|\\begin{array}{lll} 1 & 1 & 1 \\\\ 2 & 2 & 3 \\\\ 1 & 4 & 9 \\end{array}\\right|<\/script><br> = 1(18 \u2013 12) \u2013 1(18 \u2013 3) + 1(8 \u2013 2)<br> = 6- 15 + 6<br> = -3<\/p>\n\n<p>\u0394<sub>1<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-10-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-222\" style=\"width: 5.128em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.429em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.29em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-223\"><span class=\"mrow\" id=\"MathJax-Span-224\"><span class=\"mo\" id=\"MathJax-Span-225\" 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-226\"><span style=\"display: inline-block; position: relative; width: 3.497em; 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-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, 1000.47em, 4.15em, -999.998em); top: -3.959em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-236\"><span class=\"mrow\" id=\"MathJax-Span-237\"><span class=\"mn\" id=\"MathJax-Span-238\" style=\"font-family: MathJax_Main;\">6<\/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-245\"><span class=\"mrow\" id=\"MathJax-Span-246\"><span class=\"mn\" id=\"MathJax-Span-247\" 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, 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-230\"><span class=\"mrow\" id=\"MathJax-Span-231\"><span class=\"mn\" id=\"MathJax-Span-232\" 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-239\"><span class=\"mrow\" id=\"MathJax-Span-240\"><span class=\"mn\" id=\"MathJax-Span-241\" 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(3.171em, 1000.47em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-248\"><span class=\"mrow\" id=\"MathJax-Span-249\"><span class=\"mn\" id=\"MathJax-Span-250\" 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(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 2.985em;\"><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-233\"><span class=\"mrow\" id=\"MathJax-Span-234\"><span class=\"mn\" id=\"MathJax-Span-235\" 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-242\"><span class=\"mrow\" id=\"MathJax-Span-243\"><span class=\"mn\" id=\"MathJax-Span-244\" 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; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-251\"><span class=\"mrow\" id=\"MathJax-Span-252\"><span class=\"mn\" id=\"MathJax-Span-253\" 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: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-254\" style=\"vertical-align: 2.146em;\"><span style=\"display: inline-block; position: relative; width: 0.282em; 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-10\">\\left|\\begin{array}{lll} 1 & 1 & 1 \\\\ 6 & 2 & 3 \\\\ 3 & 4 & 9 \\end{array}\\right|<\/script><br> = 1(18 \u2013 12) \u2013 1(54 \u2013 9) + 1(24 \u2013 6)<br> = 6 \u2013 45 + 18<br> = -21<\/p>\n\n<p>\u0394<sub>2<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-11-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-255\" style=\"width: 5.128em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.429em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.29em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-256\"><span class=\"mrow\" id=\"MathJax-Span-257\"><span class=\"mo\" id=\"MathJax-Span-258\" 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-259\"><span style=\"display: inline-block; position: relative; width: 3.497em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -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-260\"><span class=\"mrow\" id=\"MathJax-Span-261\"><span class=\"mn\" id=\"MathJax-Span-262\" 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-269\"><span class=\"mrow\" id=\"MathJax-Span-270\"><span class=\"mn\" id=\"MathJax-Span-271\" 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, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-278\"><span class=\"mrow\" id=\"MathJax-Span-279\"><span class=\"mn\" id=\"MathJax-Span-280\" 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: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -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-263\"><span class=\"mrow\" id=\"MathJax-Span-264\"><span class=\"mn\" id=\"MathJax-Span-265\" 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-272\"><span class=\"mrow\" id=\"MathJax-Span-273\"><span class=\"mn\" id=\"MathJax-Span-274\" style=\"font-family: MathJax_Main;\">6<\/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-281\"><span class=\"mrow\" id=\"MathJax-Span-282\"><span class=\"mn\" id=\"MathJax-Span-283\" 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, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 2.985em;\"><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-266\"><span class=\"mrow\" id=\"MathJax-Span-267\"><span class=\"mn\" id=\"MathJax-Span-268\" 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-275\"><span class=\"mrow\" id=\"MathJax-Span-276\"><span class=\"mn\" id=\"MathJax-Span-277\" 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; left: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-284\"><span class=\"mrow\" id=\"MathJax-Span-285\"><span class=\"mn\" id=\"MathJax-Span-286\" 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: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-287\" 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-11\">\\left|\\begin{array}{lll} 1 & 1 & 1 \\\\ 2 & 6 & 3 \\\\ 1 & 3 & 9 \\end{array}\\right|<\/script><br> = 1(54 \u2013 9) \u2013 1(18 \u2013 3) + 1(6 \u2013 6)<br> = 45 \u2013 15<br> = 30<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6329\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-3.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 3\" width=\"246\" height=\"299\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 3\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\">\n\n<br> \u2234 Solution is x = 7, y = -10, and z = 4.<\/p>\n\n<p>ii) Matrix Inversion Method:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6328\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-4.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 4\" width=\"310\" height=\"1093\" sizes=\"auto, (max-width: 310px) 100vw, 310px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 4\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\"><br> \u2234 Solution is x = 7, y = -10 and z = 4<\/p>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6327\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-5.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 5\" width=\"187\" height=\"605\" sizes=\"auto, (max-width: 187px) 100vw, 187px\" data-pin-nopin=\"true\"><br> The given system is consistent and has a unique solution given by x = 7,y = -10,z = 4. [\u2235 \u03c1(A) = \u03c1(AB)]<\/p>\n\n<p>Question 3.<br> x \u2013 y + 3z = 5 (March 2015-T.S)<br> 4x + 2y \u2013 z = 0<br> -x + 3y + z = 5<\/h3>\n\n<p>Answer:<br> i) Cramer\u2019s rule<br> \u0394 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-12-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-288\" 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, 1006.62em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-289\"><span class=\"mrow\" id=\"MathJax-Span-290\"><span class=\"mo\" id=\"MathJax-Span-291\" 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-292\"><span style=\"display: inline-block; position: relative; width: 5.827em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><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-293\"><span class=\"mrow\" id=\"MathJax-Span-294\"><span class=\"mn\" id=\"MathJax-Span-295\" 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: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-303\"><span class=\"mrow\" id=\"MathJax-Span-304\"><span class=\"mn\" id=\"MathJax-Span-305\" style=\"font-family: MathJax_Main;\">4<\/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-313\"><span class=\"mrow\" id=\"MathJax-Span-314\"><span class=\"mo\" id=\"MathJax-Span-315\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-316\" 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: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 2.286em;\"><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-296\"><span class=\"mrow\" id=\"MathJax-Span-297\"><span class=\"mo\" id=\"MathJax-Span-298\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-299\" 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-306\"><span class=\"mrow\" id=\"MathJax-Span-307\"><span class=\"mn\" id=\"MathJax-Span-308\" 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-317\"><span class=\"mrow\" id=\"MathJax-Span-318\"><span class=\"mn\" id=\"MathJax-Span-319\" 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, 6.06em, -999.998em); top: -4.471em; left: 4.569em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-300\"><span class=\"mrow\" id=\"MathJax-Span-301\"><span class=\"mn\" id=\"MathJax-Span-302\" 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: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-309\"><span class=\"mrow\" id=\"MathJax-Span-310\"><span class=\"mo\" id=\"MathJax-Span-311\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-312\" 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: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-320\"><span class=\"mrow\" id=\"MathJax-Span-321\"><span class=\"mn\" id=\"MathJax-Span-322\" 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: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-323\" 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-12\">\\left|\\begin{array}{rrr} 1 & -1 & 3 \\\\ 4 & 2 & -1 \\\\ -1 & 3 & 1 \\end{array}\\right|<\/script><br> = 1(2 + 3) + 1(4-1) + 3(12 + 2)<br> = 5 + 3 + 42<br> = 50 \u2260 0<br> Cramer\u2019s rule is applicable.<\/p><p>\u0394<sub>1<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-13-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-324\" style=\"width: 6.992em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.014em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.87em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-325\"><span class=\"mrow\" id=\"MathJax-Span-326\"><span class=\"mo\" id=\"MathJax-Span-327\" 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-328\"><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, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-329\"><span class=\"mrow\" id=\"MathJax-Span-330\"><span class=\"mn\" id=\"MathJax-Span-331\" style=\"font-family: MathJax_Main;\">5<\/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-339\"><span class=\"mrow\" id=\"MathJax-Span-340\"><span class=\"mn\" id=\"MathJax-Span-341\" 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-349\"><span class=\"mrow\" id=\"MathJax-Span-350\"><span class=\"mn\" id=\"MathJax-Span-351\" 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, 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, 1001.21em, 4.243em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-332\"><span class=\"mrow\" id=\"MathJax-Span-333\"><span class=\"mo\" id=\"MathJax-Span-334\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-335\" 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-342\"><span class=\"mrow\" id=\"MathJax-Span-343\"><span class=\"mn\" id=\"MathJax-Span-344\" 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-352\"><span class=\"mrow\" id=\"MathJax-Span-353\"><span class=\"mn\" id=\"MathJax-Span-354\" 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, 6.06em, -999.998em); top: -4.471em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-336\"><span class=\"mrow\" id=\"MathJax-Span-337\"><span class=\"mn\" id=\"MathJax-Span-338\" 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: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-345\"><span class=\"mrow\" id=\"MathJax-Span-346\"><span class=\"mo\" id=\"MathJax-Span-347\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-348\" 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: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-355\"><span class=\"mrow\" id=\"MathJax-Span-356\"><span class=\"mn\" id=\"MathJax-Span-357\" 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: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-358\" 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-13\">\\left|\\begin{array}{rrr} 5 & -1 & 3 \\\\ 0 & 2 & -1 \\\\ 5 & 3 & 1 \\end{array}\\right|<\/script><br> = 5(2 + 3) + 1(0+ 5) + 3(0 -10)<br> = 25 + 5 \u2013 30<br> = 0<\/p><p>\u0394<sub>2<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-14-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-359\" style=\"width: 6.992em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.014em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.87em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-360\"><span class=\"mrow\" id=\"MathJax-Span-361\"><span class=\"mo\" id=\"MathJax-Span-362\" 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-363\"><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(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><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-364\"><span class=\"mrow\" id=\"MathJax-Span-365\"><span class=\"mn\" id=\"MathJax-Span-366\" 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: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-373\"><span class=\"mrow\" id=\"MathJax-Span-374\"><span class=\"mn\" id=\"MathJax-Span-375\" style=\"font-family: MathJax_Main;\">4<\/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-383\"><span class=\"mrow\" id=\"MathJax-Span-384\"><span class=\"mo\" id=\"MathJax-Span-385\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-386\" 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: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-367\"><span class=\"mrow\" id=\"MathJax-Span-368\"><span class=\"mn\" id=\"MathJax-Span-369\" style=\"font-family: MathJax_Main;\">5<\/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-376\"><span class=\"mrow\" id=\"MathJax-Span-377\"><span class=\"mn\" id=\"MathJax-Span-378\" 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-387\"><span class=\"mrow\" id=\"MathJax-Span-388\"><span class=\"mn\" id=\"MathJax-Span-389\" 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, 1001.21em, 6.06em, -999.998em); top: -4.471em; left: 3.777em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-370\"><span class=\"mrow\" id=\"MathJax-Span-371\"><span class=\"mn\" id=\"MathJax-Span-372\" 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: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-379\"><span class=\"mrow\" id=\"MathJax-Span-380\"><span class=\"mo\" id=\"MathJax-Span-381\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-382\" 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: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-390\"><span class=\"mrow\" id=\"MathJax-Span-391\"><span class=\"mn\" id=\"MathJax-Span-392\" 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: 4.476em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-393\" 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-14\">\\left|\\begin{array}{rrr} 1 & 5 & 3 \\\\ 4 & 0 & -1 \\\\ -1 & 5 & 1 \\end{array}\\right|<\/script><br> = 1(0+ 5)-5(4-1)+ 3(20)<br> = 5 \u2013 15 + 60 = 0<\/p>\n\n<p>\u0394<sub>3<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-15-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-394\" style=\"width: 6.992em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.014em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.87em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-395\"><span class=\"mrow\" id=\"MathJax-Span-396\"><span class=\"mo\" id=\"MathJax-Span-397\" 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-398\"><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(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 0em;\"><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-399\"><span class=\"mrow\" id=\"MathJax-Span-400\"><span class=\"mn\" id=\"MathJax-Span-401\" 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: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-409\"><span class=\"mrow\" id=\"MathJax-Span-410\"><span class=\"mn\" id=\"MathJax-Span-411\" style=\"font-family: MathJax_Main;\">4<\/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-418\"><span class=\"mrow\" id=\"MathJax-Span-419\"><span class=\"mo\" id=\"MathJax-Span-420\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-421\" 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: 4.569em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 2.286em;\"><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-402\"><span class=\"mrow\" id=\"MathJax-Span-403\"><span class=\"mo\" id=\"MathJax-Span-404\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-405\" 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-412\"><span class=\"mrow\" id=\"MathJax-Span-413\"><span class=\"mn\" id=\"MathJax-Span-414\" 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-422\"><span class=\"mrow\" id=\"MathJax-Span-423\"><span class=\"mn\" id=\"MathJax-Span-424\" 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, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 4.569em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-406\"><span class=\"mrow\" id=\"MathJax-Span-407\"><span class=\"mn\" id=\"MathJax-Span-408\" style=\"font-family: MathJax_Main;\">5<\/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-415\"><span class=\"mrow\" id=\"MathJax-Span-416\"><span class=\"mn\" id=\"MathJax-Span-417\" 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-425\"><span class=\"mrow\" id=\"MathJax-Span-426\"><span class=\"mn\" id=\"MathJax-Span-427\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-428\" 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-15\">\\left|\\begin{array}{rrr} 1 & -1 & 5 \\\\ 4 & 2 & 0 \\\\ -1 & 3 & 5 \\end{array}\\right|<\/script><br> = 1(10 \u2013 0) + 1(20 \u2013 0)+ 5(12 + 2)<br> = 10 + 20 + 70<br> = 100<\/p><p>\u2234 x = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-16-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-429\" style=\"width: 1.54em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 1.307em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1001.31em, 2.845em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-430\"><span class=\"mfrac\" id=\"MathJax-Span-431\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.89em, 4.243em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msubsup\" id=\"MathJax-Span-432\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-433\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.912em; left: 0.608em;\"><span class=\"mn\" id=\"MathJax-Span-434\" style=\"font-size: 50%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.277em;\"><span class=\"mi\" id=\"MathJax-Span-435\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.03em, 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: 1.027em; 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: -0.538em; border-left: 0px solid; width: 0px; height: 1.732em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-16\">\\frac{\\Delta_1}{\\Delta}<\/script> = 0<br> y = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-17-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-436\" style=\"width: 4.383em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.777em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1003.78em, 2.891em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-437\"><span class=\"mfrac\" id=\"MathJax-Span-438\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.89em, 4.243em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msubsup\" id=\"MathJax-Span-439\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-440\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.912em; left: 0.608em;\"><span class=\"mn\" id=\"MathJax-Span-441\" style=\"font-size: 50%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.277em;\"><span class=\"mi\" id=\"MathJax-Span-442\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.03em, 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: 1.027em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-443\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-444\" style=\"padding-left: 0.282em;\"><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(3.404em, 1000.7em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-445\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">50<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.7em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-446\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">50<\/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: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.538em; border-left: 0px solid; width: 0px; height: 1.732em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-17\">\\frac{\\Delta_2}{\\Delta}=\\frac{50}{50}<\/script> = 1<br> z = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-18-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-447\" style=\"width: 4.755em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.103em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1004.1em, 2.891em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-448\"><span class=\"mfrac\" id=\"MathJax-Span-449\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.89em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msubsup\" id=\"MathJax-Span-450\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-451\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.912em; left: 0.608em;\"><span class=\"mn\" id=\"MathJax-Span-452\" style=\"font-size: 50%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.277em;\"><span class=\"mi\" id=\"MathJax-Span-453\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.03em, 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: 1.027em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-454\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-455\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.167em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.03em, 4.15em, -999.998em); top: -4.425em; left: 50%; margin-left: -0.51em;\"><span class=\"mn\" id=\"MathJax-Span-456\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">100<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.7em, 4.15em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.37em;\"><span class=\"mn\" id=\"MathJax-Span-457\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">50<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.17em, 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: 1.167em; 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: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.538em; border-left: 0px solid; width: 0px; height: 1.786em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-18\">\\frac{\\Delta_3}{\\Delta}=\\frac{100}{50}<\/script> = 2<br> \u2234 x = 0, y = 1 and z = 2.<\/p>\n\n<p>ii) Matrix Inversion Method:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6326\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-6.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 6\" width=\"315\" height=\"979\" sizes=\"auto, (max-width: 315px) 100vw, 315px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 6\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\"><\/p>\n\n<p>iii) Gauss Jordan Method:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6325\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-7.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 7\" width=\"314\" height=\"695\" sizes=\"auto, (max-width: 314px) 100vw, 314px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 7\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\"><br> The given system of equations is consistent since \u03c1(A) = \u03c1(AB) = 3 and the system has a unique solution. x = 0, y = 1, z = 2.<\/p>\n\n<h3>Question 4.<br> 2x + 6y + 11 = 0<br> 6x + 20y \u2013 6z + 3 = 0<br> 6y \u2013 18z + 1 = 0<\/h3>\n\n<p>Answer:<br> \u0394 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-19-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-458\" style=\"width: 7.225em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.2em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1006.06em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-459\"><span class=\"mrow\" id=\"MathJax-Span-460\"><span class=\"mo\" id=\"MathJax-Span-461\" 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-462\"><span style=\"display: inline-block; position: relative; width: 5.268em; 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, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-463\"><span class=\"mrow\" id=\"MathJax-Span-464\"><span class=\"mn\" id=\"MathJax-Span-465\" 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: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-472\"><span class=\"mrow\" id=\"MathJax-Span-473\"><span class=\"mn\" id=\"MathJax-Span-474\" style=\"font-family: MathJax_Main;\">6<\/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-482\"><span class=\"mrow\" id=\"MathJax-Span-483\"><span class=\"mn\" id=\"MathJax-Span-484\" 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, 1000.98em, 6.107em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-466\"><span class=\"mrow\" id=\"MathJax-Span-467\"><span class=\"mn\" id=\"MathJax-Span-468\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.98em, 4.15em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-475\"><span class=\"mrow\" id=\"MathJax-Span-476\"><span class=\"mn\" id=\"MathJax-Span-477\" style=\"font-family: MathJax_Main;\">20<\/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-485\"><span class=\"mrow\" id=\"MathJax-Span-486\"><span class=\"mn\" id=\"MathJax-Span-487\" 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, 1001.73em, 6.247em, -999.998em); top: -4.564em; left: 3.497em;\"><span style=\"display: inline-block; position: relative; width: 1.773em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-469\"><span class=\"mrow\" id=\"MathJax-Span-470\"><span class=\"mn\" id=\"MathJax-Span-471\" 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, 1001.21em, 4.243em, -999.998em); top: -3.959em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-478\"><span class=\"mrow\" id=\"MathJax-Span-479\"><span class=\"mo\" id=\"MathJax-Span-480\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-481\" style=\"font-family: MathJax_Main;\">6<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1001.73em, 4.243em, -999.998em); top: -2.561em; right: 0em;\"><span class=\"mtd\" id=\"MathJax-Span-488\"><span class=\"mrow\" id=\"MathJax-Span-489\"><span class=\"mo\" id=\"MathJax-Span-490\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-491\" style=\"font-family: MathJax_Main;\">18<\/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-492\" 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-19\">\\left|\\begin{array}{rrr} 2 & 6 & 0 \\\\ 6 & 20 & -6 \\\\ 0 & 6 & -18 \\end{array}\\right|<\/script><br> = 2(- 360 + 36) \u2013 6(-108-0) + 0(36)<br> = -648 + 648 = 0<br> Cramer\u2019s and matrix inversion methods are not applicable.<\/p>\n\n<p>Gauss Jordan Process:<br> The augmented matrix of the given system is<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6324\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-8.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 8\" width=\"187\" height=\"315\" sizes=\"auto, (max-width: 187px) 100vw, 187px\" data-pin-nopin=\"true\"><br> Since \u03c1(A) = 2 and \u03c1(AB) = 3, the given system is not consistent and has no solution.<\/p>\n\n<h3>Question 5.<br> 2x \u2013 y + 3z = 9<br> x + y + z = 6<br> x \u2013 y + z = 2 (May 2014, Mar. \u201914, \u201905, \u201902)<br> Answer:<br> i) Cramer\u2019s rule :<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6323\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-9.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 9\" width=\"256\" height=\"752\" sizes=\"auto, (max-width: 256px) 100vw, 256px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 9\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\"><\/p>\n\n<p>ii) Martix Inversion Method:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6322\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-10.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 10\" width=\"300\" height=\"585\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 10\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\"><br>\n\ndet A = 2(2) + 1(0) + 3(- 2)<br> = 4 -6 = -2 \u2260 0<br> \u2234 A<sup>-1<\/sup> exists and A<sup>-1<\/sup> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-20-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-493\" style=\"width: 2.519em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 2.146em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.121em, 1002.15em, 2.845em, -999.998em); top: -2.281em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-494\"><span class=\"mfrac\" id=\"MathJax-Span-495\"><span style=\"display: inline-block; position: relative; width: 1.866em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1001.68em, 4.289em, -999.998em); top: -4.564em; left: 50%; margin-left: -0.836em;\"><span class=\"mrow\" id=\"MathJax-Span-496\"><span class=\"texatom\" id=\"MathJax-Span-497\"><span class=\"mrow\" id=\"MathJax-Span-498\"><span class=\"mi\" id=\"MathJax-Span-499\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">A<\/span><span class=\"mi\" id=\"MathJax-Span-500\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">d<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-501\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">j<\/span><\/span><\/span><span class=\"texatom\" id=\"MathJax-Span-502\"><span class=\"mrow\" id=\"MathJax-Span-503\"><span class=\"mi\" id=\"MathJax-Span-504\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">A<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1001.45em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.743em;\"><span class=\"mrow\" id=\"MathJax-Span-505\"><span class=\"texatom\" id=\"MathJax-Span-506\"><span class=\"mrow\" id=\"MathJax-Span-507\"><span class=\"mi\" id=\"MathJax-Span-508\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">d<span style=\"display: inline-block; overflow: hidden; height: 1px; width: 0.002em;\"><\/span><\/span><span class=\"mi\" id=\"MathJax-Span-509\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">e<\/span><span class=\"mi\" id=\"MathJax-Span-510\" style=\"font-size: 70.7%; font-family: MathJax_Math-italic;\">t<\/span><\/span><\/span><span class=\"texatom\" id=\"MathJax-Span-511\"><span class=\"mrow\" id=\"MathJax-Span-512\"><span class=\"mi\" id=\"MathJax-Span-513\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">A<\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.87em, 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: 1.866em; 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: -0.538em; border-left: 0px solid; width: 0px; height: 1.786em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-20\">\\frac{{Adj} \\mathrm{A}}{{det} \\mathrm{A}}<\/script>\n\n<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6321\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-11.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 11\" width=\"264\" height=\"377\" sizes=\"auto, (max-width: 264px) 100vw, 264px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 11\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\"><br> \u2234 x = 1, y = 2, z = 3 is the solution.<\/p>\n\n<p>iii) Gauss Jordan Method:<br> Augmented matrix of the system<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6320\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-12.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 12\" width=\"238\" height=\"693\" sizes=\"auto, (max-width: 238px) 100vw, 238px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 12\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\"><br> Since \u03c1(A) = \u03c1(AB) = 3; the system is consistent and has a unique solution given by x = 1, y = 2, z = 3.<\/p>\n\n<h3>Question 6.<br> 2x- y + 8z = 13<br> 3x + 4y + 5z = 18<br> 5x \u2013 2y + 7z = 20 (March 2004, 03, \u201901) (Board New Model Paper)<\/h3>\n\n<p> Answer:\n\n<br> i) Cramer\u2019s rule :<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6319\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-13.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 13\" width=\"322\" height=\"624\" sizes=\"auto, (max-width: 322px) 100vw, 322px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 13\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\"><\/p><p>ii) Matrix Inversion Method:<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6318\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-14.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 14\" width=\"312\" height=\"1251\" sizes=\"auto, (max-width: 312px) 100vw, 312px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 14\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\"><br>\n\n\u2234 Solution is x = 3, y = 1 and z = 1<\/p>\n\n<p>iii) Gauss Jordan Method:<br> Augmented matrix of the system<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6317\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-15.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 15\" width=\"230\" height=\"725\" sizes=\"auto, (max-width: 230px) 100vw, 230px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 15\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\"><br> Since \u03c1(A) = \u03c1(AB) = 3, the system is consistent and has a unique solution given by x = 3, y = 1 and z = 1.<br> Then the cofactors of elements in the matrix A are<\/p>\n\n<h3>Question 7.<br> 2x \u2013 y + 3z = 8<br> -x + 2y + z = 4<br> 3x + y \u2013 4z = 0<\/h3>\n\n<p>Answer:<br> i) Cramer\u2019s rule :<br> \u0394 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-21-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-514\" 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, 1006.62em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-515\"><span class=\"mrow\" id=\"MathJax-Span-516\"><span class=\"mo\" id=\"MathJax-Span-517\" 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-518\"><span style=\"display: inline-block; position: relative; width: 5.827em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-519\"><span class=\"mrow\" id=\"MathJax-Span-520\"><span class=\"mn\" id=\"MathJax-Span-521\" 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: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-529\"><span class=\"mrow\" id=\"MathJax-Span-530\"><span class=\"mo\" id=\"MathJax-Span-531\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-532\" 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-539\"><span class=\"mrow\" id=\"MathJax-Span-540\"><span class=\"mn\" id=\"MathJax-Span-541\" 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, 6.06em, -999.998em); top: -4.471em; left: 2.286em;\"><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-522\"><span class=\"mrow\" id=\"MathJax-Span-523\"><span class=\"mo\" id=\"MathJax-Span-524\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-525\" 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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-533\"><span class=\"mrow\" id=\"MathJax-Span-534\"><span class=\"mn\" id=\"MathJax-Span-535\" 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, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-542\"><span class=\"mrow\" id=\"MathJax-Span-543\"><span class=\"mn\" id=\"MathJax-Span-544\" 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: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 6.247em, -999.998em); top: -4.564em; left: 4.569em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-526\"><span class=\"mrow\" id=\"MathJax-Span-527\"><span class=\"mn\" id=\"MathJax-Span-528\" 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, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-536\"><span class=\"mrow\" id=\"MathJax-Span-537\"><span class=\"mn\" id=\"MathJax-Span-538\" 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, 1001.26em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-545\"><span class=\"mrow\" id=\"MathJax-Span-546\"><span class=\"mo\" id=\"MathJax-Span-547\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-548\" 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: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-549\" 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-21\">\\left|\\begin{array}{ccc} 2 & -1 & 3 \\\\ -1 & 2 & 1 \\\\ 3 & 1 & -4 \\end{array}\\right|<\/script><br> = 2(- 8 \u2013 1) + 1(4 \u2013 3) + 3(- 1-6)<br> = -18 + 1 \u2013 21 = -38 \u2260 0<br> Cramer\u2019s rule is applicable.<\/p>\n\n<p>\u0394<sub>1<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-22-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-550\" style=\"width: 6.992em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.014em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.87em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-551\"><span class=\"mrow\" id=\"MathJax-Span-552\"><span class=\"mo\" id=\"MathJax-Span-553\" 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-554\"><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, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-555\"><span class=\"mrow\" id=\"MathJax-Span-556\"><span class=\"mn\" id=\"MathJax-Span-557\" style=\"font-family: MathJax_Main;\">8<\/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: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-565\"><span class=\"mrow\" id=\"MathJax-Span-566\"><span class=\"mn\" id=\"MathJax-Span-567\" style=\"font-family: MathJax_Main;\">4<\/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-574\"><span class=\"mrow\" id=\"MathJax-Span-575\"><span class=\"mn\" id=\"MathJax-Span-576\" 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, 1001.21em, 6.06em, -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, 1001.21em, 4.243em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-558\"><span class=\"mrow\" id=\"MathJax-Span-559\"><span class=\"mo\" id=\"MathJax-Span-560\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-561\" 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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-568\"><span class=\"mrow\" id=\"MathJax-Span-569\"><span class=\"mn\" id=\"MathJax-Span-570\" 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, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-577\"><span class=\"mrow\" id=\"MathJax-Span-578\"><span class=\"mn\" id=\"MathJax-Span-579\" 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: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.26em, 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, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-562\"><span class=\"mrow\" id=\"MathJax-Span-563\"><span class=\"mn\" id=\"MathJax-Span-564\" 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, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-571\"><span class=\"mrow\" id=\"MathJax-Span-572\"><span class=\"mn\" id=\"MathJax-Span-573\" 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, 1001.26em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-580\"><span class=\"mrow\" id=\"MathJax-Span-581\"><span class=\"mo\" id=\"MathJax-Span-582\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-583\" 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: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-584\" 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-22\">\\left|\\begin{array}{ccc} 8 & -1 & 3 \\\\ 4 & 2 & 1 \\\\ 0 & 1 & -4 \\end{array}\\right|<\/script><br> = 8(- 8 \u2013 1) + 1(- 16 \u2013 0) + 3(4 \u2013 0)<br> = -72 \u2013 16 + 12<br> = -76<\/p>\n\n<p>\u0394<sub>2<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-23-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-585\" style=\"width: 6.992em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.014em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.87em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-586\"><span class=\"mrow\" id=\"MathJax-Span-587\"><span class=\"mo\" id=\"MathJax-Span-588\" 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-589\"><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, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-590\"><span class=\"mrow\" id=\"MathJax-Span-591\"><span class=\"mn\" id=\"MathJax-Span-592\" 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: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-599\"><span class=\"mrow\" id=\"MathJax-Span-600\"><span class=\"mo\" id=\"MathJax-Span-601\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-602\" 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-609\"><span class=\"mrow\" id=\"MathJax-Span-610\"><span class=\"mn\" id=\"MathJax-Span-611\" 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, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 2.286em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-593\"><span class=\"mrow\" id=\"MathJax-Span-594\"><span class=\"mn\" id=\"MathJax-Span-595\" style=\"font-family: MathJax_Main;\">8<\/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: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-603\"><span class=\"mrow\" id=\"MathJax-Span-604\"><span class=\"mn\" id=\"MathJax-Span-605\" style=\"font-family: MathJax_Main;\">4<\/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-612\"><span class=\"mrow\" id=\"MathJax-Span-613\"><span class=\"mn\" id=\"MathJax-Span-614\" 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(2.425em, 1001.26em, 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, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-596\"><span class=\"mrow\" id=\"MathJax-Span-597\"><span class=\"mn\" id=\"MathJax-Span-598\" 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, 1000.42em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-606\"><span class=\"mrow\" id=\"MathJax-Span-607\"><span class=\"mn\" id=\"MathJax-Span-608\" 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, 1001.26em, 4.243em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-615\"><span class=\"mrow\" id=\"MathJax-Span-616\"><span class=\"mo\" id=\"MathJax-Span-617\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-618\" 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: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-619\" 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-23\">\\left|\\begin{array}{ccc} 2 & 8 & 3 \\\\ -1 & 4 & 1 \\\\ 3 & 0 & -4 \\end{array}\\right|<\/script><br> = 2(- 16 \u2013 0) \u2013 8(4 \u2013 3) + 3(0 \u2013 12)<br> = \u2013 32 \u2013 8 \u2013 36 = \u2013 76<\/p>\n\n<p>\u0394<sub>3<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-24-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-620\" style=\"width: 6.992em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 6.014em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.87em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-621\"><span class=\"mrow\" id=\"MathJax-Span-622\"><span class=\"mo\" id=\"MathJax-Span-623\" 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-624\"><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, 1001.21em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 1.26em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-625\"><span class=\"mrow\" id=\"MathJax-Span-626\"><span class=\"mn\" id=\"MathJax-Span-627\" 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: -3.959em; left: 50%; margin-left: -0.65em;\"><span class=\"mtd\" id=\"MathJax-Span-635\"><span class=\"mrow\" id=\"MathJax-Span-636\"><span class=\"mo\" id=\"MathJax-Span-637\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-638\" 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-645\"><span class=\"mrow\" id=\"MathJax-Span-646\"><span class=\"mn\" id=\"MathJax-Span-647\" 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, 6.06em, -999.998em); top: -4.471em; left: 2.286em;\"><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-628\"><span class=\"mrow\" id=\"MathJax-Span-629\"><span class=\"mo\" id=\"MathJax-Span-630\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-631\" 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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-639\"><span class=\"mrow\" id=\"MathJax-Span-640\"><span class=\"mn\" id=\"MathJax-Span-641\" 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, 1000.42em, 4.15em, -999.998em); top: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-648\"><span class=\"mrow\" id=\"MathJax-Span-649\"><span class=\"mn\" id=\"MathJax-Span-650\" 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: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.107em, -999.998em); top: -4.471em; left: 4.569em;\"><span style=\"display: inline-block; position: relative; width: 0.515em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-632\"><span class=\"mrow\" id=\"MathJax-Span-633\"><span class=\"mn\" id=\"MathJax-Span-634\" style=\"font-family: MathJax_Main;\">8<\/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: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-642\"><span class=\"mrow\" id=\"MathJax-Span-643\"><span class=\"mn\" id=\"MathJax-Span-644\" style=\"font-family: MathJax_Main;\">4<\/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-651\"><span class=\"mrow\" id=\"MathJax-Span-652\"><span class=\"mn\" id=\"MathJax-Span-653\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-654\" 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-24\">\\left|\\begin{array}{ccc} 2 & -1 & 8 \\\\ -1 & 2 & 4 \\\\ 3 & 1 & 0 \\end{array}\\right|<\/script><br> = 2(0 \u2013 4) + 1(0 \u2013 12) + 8(-1 \u2013 6)<br> = -8 \u2013 12 \u2013 56<br> = -76<br> \u2234 x = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-25-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-655\" style=\"width: 5.035em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.336em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1004.34em, 2.891em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-656\"><span class=\"mfrac\" id=\"MathJax-Span-657\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.89em, 4.243em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msubsup\" id=\"MathJax-Span-658\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-659\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.912em; left: 0.608em;\"><span class=\"mn\" id=\"MathJax-Span-660\" style=\"font-size: 50%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.277em;\"><span class=\"mi\" id=\"MathJax-Span-661\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.03em, 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: 1.027em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-662\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-663\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.4em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.21em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.603em;\"><span class=\"mrow\" id=\"MathJax-Span-664\"><span class=\"mo\" id=\"MathJax-Span-665\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-666\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">76<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1001.21em, 4.196em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.603em;\"><span class=\"mrow\" id=\"MathJax-Span-667\"><span class=\"mo\" id=\"MathJax-Span-668\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-669\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">38<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.4em, 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: 1.4em; 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: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.538em; border-left: 0px solid; width: 0px; height: 1.786em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-25\">\\frac{\\Delta_1}{\\Delta}=\\frac{-76}{-38}<\/script> = 2<br> y = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-26-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-670\" style=\"width: 5.035em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.336em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1004.34em, 2.891em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-671\"><span class=\"mfrac\" id=\"MathJax-Span-672\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.89em, 4.243em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msubsup\" id=\"MathJax-Span-673\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-674\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.912em; left: 0.608em;\"><span class=\"mn\" id=\"MathJax-Span-675\" style=\"font-size: 50%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.277em;\"><span class=\"mi\" id=\"MathJax-Span-676\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.03em, 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: 1.027em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-677\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-678\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.4em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.21em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.603em;\"><span class=\"mrow\" id=\"MathJax-Span-679\"><span class=\"mo\" id=\"MathJax-Span-680\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-681\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">76<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1001.21em, 4.196em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.603em;\"><span class=\"mrow\" id=\"MathJax-Span-682\"><span class=\"mo\" id=\"MathJax-Span-683\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-684\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">38<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.4em, 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: 1.4em; 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: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.538em; border-left: 0px solid; width: 0px; height: 1.786em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-26\">\\frac{\\Delta_2}{\\Delta}=\\frac{-76}{-38}<\/script> = 2<br> z = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-27-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-685\" style=\"width: 5.035em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.336em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1004.34em, 2.891em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-686\"><span class=\"mfrac\" id=\"MathJax-Span-687\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.89em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msubsup\" id=\"MathJax-Span-688\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-689\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.912em; left: 0.608em;\"><span class=\"mn\" id=\"MathJax-Span-690\" style=\"font-size: 50%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.277em;\"><span class=\"mi\" id=\"MathJax-Span-691\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.03em, 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: 1.027em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-692\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-693\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.4em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.21em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.603em;\"><span class=\"mrow\" id=\"MathJax-Span-694\"><span class=\"mo\" id=\"MathJax-Span-695\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-696\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">76<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1001.21em, 4.196em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.603em;\"><span class=\"mrow\" id=\"MathJax-Span-697\"><span class=\"mo\" id=\"MathJax-Span-698\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-699\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">38<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.4em, 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: 1.4em; 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: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.538em; border-left: 0px solid; width: 0px; height: 1.786em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-27\">\\frac{\\Delta_3}{\\Delta}=\\frac{-76}{-38}<\/script> = 2<br> \u2234 Solution is x = 2, y =2, z = 2.<\/p>\n\n<p>Matrix Inversion Method:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6316\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-16.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 16\" width=\"328\" height=\"1116\" sizes=\"auto, (max-width: 328px) 100vw, 328px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 16\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\"><br> \u2234 Solution is x = 2, y = 2 and z = 2<\/p>\n\n<p>iii) Gauss Jordan Method:<br> The augmented matrix of the system is<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6315\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-17.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 17\" width=\"227\" height=\"770\" sizes=\"auto, (max-width: 227px) 100vw, 227px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 17\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\"><br> \u2234 \u03c1(A) = \u03c1(AB) = 3 ; the system is consistent and has a unique solution, x = 2, y = 2 and z = 2.<\/p>\n\n<h3>Question 8.<br> x + y + z = 9<br> 2x + 5y + 7z = 52<br> 2x + y-z = 0 (May 2011)<\/h3>\n\n<p>Answer:<br> i) Cramer\u2019s rule :<br> \u0394 = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-28-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-700\" 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-701\"><span class=\"mrow\" id=\"MathJax-Span-702\"><span class=\"mo\" id=\"MathJax-Span-703\" 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-704\"><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, 6.06em, -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-705\"><span class=\"mrow\" id=\"MathJax-Span-706\"><span class=\"mn\" id=\"MathJax-Span-707\" 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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-714\"><span class=\"mrow\" id=\"MathJax-Span-715\"><span class=\"mn\" id=\"MathJax-Span-716\" 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-723\"><span class=\"mrow\" id=\"MathJax-Span-724\"><span class=\"mn\" id=\"MathJax-Span-725\" 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: 4.476em;\"><\/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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-708\"><span class=\"mrow\" id=\"MathJax-Span-709\"><span class=\"mn\" id=\"MathJax-Span-710\" 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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-717\"><span class=\"mrow\" id=\"MathJax-Span-718\"><span class=\"mn\" id=\"MathJax-Span-719\" style=\"font-family: MathJax_Main;\">5<\/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: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-726\"><span class=\"mrow\" id=\"MathJax-Span-727\"><span class=\"mn\" id=\"MathJax-Span-728\" 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: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 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, 1000.42em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-711\"><span class=\"mrow\" id=\"MathJax-Span-712\"><span class=\"mn\" id=\"MathJax-Span-713\" 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: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-720\"><span class=\"mrow\" id=\"MathJax-Span-721\"><span class=\"mn\" id=\"MathJax-Span-722\" style=\"font-family: MathJax_Main;\">7<\/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-729\"><span class=\"mrow\" id=\"MathJax-Span-730\"><span class=\"mo\" id=\"MathJax-Span-731\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-732\" 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: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-733\" 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-28\">\\left|\\begin{array}{ccc} 1 & 1 & 1 \\\\ 2 & 5 & 7 \\\\ 2 & 1 & -1 \\end{array}\\right|<\/script><br> = 1(-5 \u2013 7) \u2013 1(-2 \u2013 14) + 1(2 \u2013 10)<br> = \u2013 12 + 16-8 = -4 \u2260 0<br> \u2234 The Cramer\u2019s method is applicable.<\/p>\n\n<p>\u0394<sub>1<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-29-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-734\" style=\"width: 6.666em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.734em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.59em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-735\"><span class=\"mrow\" id=\"MathJax-Span-736\"><span class=\"mo\" id=\"MathJax-Span-737\" 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-738\"><span style=\"display: inline-block; position: relative; width: 4.802em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.93em, 6.107em, -999.998em); top: -4.471em; left: 0em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-739\"><span class=\"mrow\" id=\"MathJax-Span-740\"><span class=\"mn\" id=\"MathJax-Span-741\" style=\"font-family: MathJax_Main;\">9<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.93em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-748\"><span class=\"mrow\" id=\"MathJax-Span-749\"><span class=\"mn\" id=\"MathJax-Span-750\" style=\"font-family: MathJax_Main;\">52<\/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-757\"><span class=\"mrow\" id=\"MathJax-Span-758\"><span class=\"mn\" id=\"MathJax-Span-759\" 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, 1000.47em, 6.06em, -999.998em); top: -4.471em; left: 2.006em;\"><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-742\"><span class=\"mrow\" id=\"MathJax-Span-743\"><span class=\"mn\" id=\"MathJax-Span-744\" 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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-751\"><span class=\"mrow\" id=\"MathJax-Span-752\"><span class=\"mn\" id=\"MathJax-Span-753\" style=\"font-family: MathJax_Main;\">5<\/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: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-760\"><span class=\"mrow\" id=\"MathJax-Span-761\"><span class=\"mn\" id=\"MathJax-Span-762\" 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: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 3.497em;\"><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-745\"><span class=\"mrow\" id=\"MathJax-Span-746\"><span class=\"mn\" id=\"MathJax-Span-747\" 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: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-754\"><span class=\"mrow\" id=\"MathJax-Span-755\"><span class=\"mn\" id=\"MathJax-Span-756\" style=\"font-family: MathJax_Main;\">7<\/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-763\"><span class=\"mrow\" id=\"MathJax-Span-764\"><span class=\"mo\" id=\"MathJax-Span-765\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-766\" 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: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-767\" 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-29\">\\left|\\begin{array}{ccc} 9 & 1 & 1 \\\\ 52 & 5 & 7 \\\\ 0 & 1 & -1 \\end{array}\\right|<\/script><br> = 9(-5 \u2013 7) -1(-52) + 1(52)<br> = -108 + 52 + 52 =-4<\/p>\n\n<p>\u0394<sub>2<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-30-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-768\" style=\"width: 6.666em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 5.734em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1005.59em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-769\"><span class=\"mrow\" id=\"MathJax-Span-770\"><span class=\"mo\" id=\"MathJax-Span-771\" 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-772\"><span style=\"display: inline-block; position: relative; width: 4.802em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -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-773\"><span class=\"mrow\" id=\"MathJax-Span-774\"><span class=\"mn\" id=\"MathJax-Span-775\" 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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-782\"><span class=\"mrow\" id=\"MathJax-Span-783\"><span class=\"mn\" id=\"MathJax-Span-784\" 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-791\"><span class=\"mrow\" id=\"MathJax-Span-792\"><span class=\"mn\" id=\"MathJax-Span-793\" 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: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.93em, 6.107em, -999.998em); top: -4.471em; left: 1.493em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-776\"><span class=\"mrow\" id=\"MathJax-Span-777\"><span class=\"mn\" id=\"MathJax-Span-778\" style=\"font-family: MathJax_Main;\">9<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.93em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-785\"><span class=\"mrow\" id=\"MathJax-Span-786\"><span class=\"mn\" id=\"MathJax-Span-787\" style=\"font-family: MathJax_Main;\">52<\/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-794\"><span class=\"mrow\" id=\"MathJax-Span-795\"><span class=\"mn\" id=\"MathJax-Span-796\" 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(2.425em, 1001.21em, 6.247em, -999.998em); top: -4.564em; left: 3.497em;\"><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-779\"><span class=\"mrow\" id=\"MathJax-Span-780\"><span class=\"mn\" id=\"MathJax-Span-781\" 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: -3.959em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-788\"><span class=\"mrow\" id=\"MathJax-Span-789\"><span class=\"mn\" id=\"MathJax-Span-790\" style=\"font-family: MathJax_Main;\">7<\/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-797\"><span class=\"mrow\" id=\"MathJax-Span-798\"><span class=\"mo\" id=\"MathJax-Span-799\" style=\"font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-800\" 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: 4.569em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-801\" 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-30\">\\left|\\begin{array}{ccc} 1 & 9 & 1 \\\\ 2 & 52 & 7 \\\\ 2 & 0 & -1 \\end{array}\\right|<\/script><br> = 1(- 52 \u2013 0) \u2013 9(- 2 \u2013 14) + 1(0 \u2013 104)<br> = -52 + 144 \u2013 104<br> = -12<\/p>\n\n<p>\u0394<sub>3<\/sub> = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-31-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-802\" style=\"width: 5.734em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.942em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(2.146em, 1004.8em, 6.2em, -999.998em); top: -4.425em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-803\"><span class=\"mrow\" id=\"MathJax-Span-804\"><span class=\"mo\" id=\"MathJax-Span-805\" 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-806\"><span style=\"display: inline-block; position: relative; width: 4.01em; height: 0px; margin-right: 0.189em; margin-left: 0.189em;\"><span style=\"position: absolute; clip: rect(2.332em, 1000.47em, 6.06em, -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-807\"><span class=\"mrow\" id=\"MathJax-Span-808\"><span class=\"mn\" id=\"MathJax-Span-809\" 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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-816\"><span class=\"mrow\" id=\"MathJax-Span-817\"><span class=\"mn\" id=\"MathJax-Span-818\" 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-825\"><span class=\"mrow\" id=\"MathJax-Span-826\"><span class=\"mn\" id=\"MathJax-Span-827\" 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: 4.476em;\"><\/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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-810\"><span class=\"mrow\" id=\"MathJax-Span-811\"><span class=\"mn\" id=\"MathJax-Span-812\" 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: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-819\"><span class=\"mrow\" id=\"MathJax-Span-820\"><span class=\"mn\" id=\"MathJax-Span-821\" style=\"font-family: MathJax_Main;\">5<\/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: -2.561em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-828\"><span class=\"mrow\" id=\"MathJax-Span-829\"><span class=\"mn\" id=\"MathJax-Span-830\" 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: 4.476em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(2.332em, 1000.93em, 6.107em, -999.998em); top: -4.471em; left: 2.985em;\"><span style=\"display: inline-block; position: relative; width: 0.981em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.218em, 1000.47em, 4.15em, -999.998em); top: -5.356em; left: 50%; margin-left: -0.231em;\"><span class=\"mtd\" id=\"MathJax-Span-813\"><span class=\"mrow\" id=\"MathJax-Span-814\"><span class=\"mn\" id=\"MathJax-Span-815\" style=\"font-family: MathJax_Main;\">9<\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.218em, 1000.93em, 4.15em, -999.998em); top: -3.959em; left: 50%; margin-left: -0.51em;\"><span class=\"mtd\" id=\"MathJax-Span-822\"><span class=\"mrow\" id=\"MathJax-Span-823\"><span class=\"mn\" id=\"MathJax-Span-824\" style=\"font-family: MathJax_Main;\">52<\/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-831\"><span class=\"mrow\" id=\"MathJax-Span-832\"><span class=\"mn\" id=\"MathJax-Span-833\" 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><\/span><span class=\"mo\" id=\"MathJax-Span-834\" 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-31\">\\left|\\begin{array}{ccc} 1 & 1 & 9 \\\\ 2 & 5 & 52 \\\\ 2 & 1 & 0 \\end{array}\\right|<\/script><br> = 1(0 \u2013 52) -1(0 \u2013 104) + 9(2 \u2013 10)<br> = -52 + 104 \u2013 72<br> = -20<br> \u2234 x = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-32-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-835\" style=\"width: 4.616em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 3.963em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1003.96em, 2.891em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-836\"><span class=\"mfrac\" id=\"MathJax-Span-837\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.89em, 4.243em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msubsup\" id=\"MathJax-Span-838\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-839\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.912em; left: 0.608em;\"><span class=\"mn\" id=\"MathJax-Span-840\" style=\"font-size: 50%; font-family: MathJax_Main;\">1<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.277em;\"><span class=\"mi\" id=\"MathJax-Span-841\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.03em, 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: 1.027em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-842\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-843\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1000.89em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.464em;\"><span class=\"mrow\" id=\"MathJax-Span-844\"><span class=\"mo\" id=\"MathJax-Span-845\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-846\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.89em, 4.196em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.464em;\"><span class=\"mrow\" id=\"MathJax-Span-847\"><span class=\"mo\" id=\"MathJax-Span-848\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-849\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.03em, 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: 1.027em; 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: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.538em; border-left: 0px solid; width: 0px; height: 1.786em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-32\">\\frac{\\Delta_1}{\\Delta}=\\frac{-4}{-4}<\/script> = 1<br> y = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-33-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-850\" style=\"width: 5.035em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.336em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1004.34em, 2.891em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-851\"><span class=\"mfrac\" id=\"MathJax-Span-852\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.89em, 4.243em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msubsup\" id=\"MathJax-Span-853\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-854\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.912em; left: 0.608em;\"><span class=\"mn\" id=\"MathJax-Span-855\" style=\"font-size: 50%; font-family: MathJax_Main;\">2<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.277em;\"><span class=\"mi\" id=\"MathJax-Span-856\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.03em, 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: 1.027em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-857\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-858\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.4em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.21em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.603em;\"><span class=\"mrow\" id=\"MathJax-Span-859\"><span class=\"mo\" id=\"MathJax-Span-860\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-861\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">12<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.89em, 4.196em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.464em;\"><span class=\"mrow\" id=\"MathJax-Span-862\"><span class=\"mo\" id=\"MathJax-Span-863\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-864\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.4em, 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: 1.4em; 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: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.538em; border-left: 0px solid; width: 0px; height: 1.786em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-33\">\\frac{\\Delta_2}{\\Delta}=\\frac{-12}{-4}<\/script> = 3<br> z = <span class=\"MathJax_Preview\" style=\"\"><\/span><span class=\"MathJax\" id=\"MathJax-Element-34-Frame\" tabindex=\"0\" style=\"\"><nobr><span class=\"math\" id=\"MathJax-Span-865\" style=\"width: 5.035em; display: inline-block;\"><span style=\"display: inline-block; position: relative; width: 4.336em; height: 0px; font-size: 116%;\"><span style=\"position: absolute; clip: rect(1.167em, 1004.34em, 2.891em, -999.998em); top: -2.328em; left: 0em;\"><span class=\"mrow\" id=\"MathJax-Span-866\"><span class=\"mfrac\" id=\"MathJax-Span-867\"><span style=\"display: inline-block; position: relative; width: 1.027em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.89em, 4.289em, -999.998em); top: -4.518em; left: 50%; margin-left: -0.464em;\"><span class=\"msubsup\" id=\"MathJax-Span-868\"><span style=\"display: inline-block; position: relative; width: 0.888em; height: 0px;\"><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -4.005em; left: 0em;\"><span class=\"mi\" id=\"MathJax-Span-869\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; top: -3.912em; left: 0.608em;\"><span class=\"mn\" id=\"MathJax-Span-870\" style=\"font-size: 50%; font-family: MathJax_Main;\">3<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.357em, 1000.56em, 4.15em, -999.998em); top: -3.586em; left: 50%; margin-left: -0.277em;\"><span class=\"mi\" id=\"MathJax-Span-871\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u0394<\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.03em, 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: 1.027em; height: 0px;\"><\/span><span style=\"display: inline-block; width: 0px; height: 1.074em;\"><\/span><\/span><\/span><\/span><span class=\"mo\" id=\"MathJax-Span-872\" style=\"font-family: MathJax_Main; padding-left: 0.282em;\">=<\/span><span class=\"mfrac\" id=\"MathJax-Span-873\" style=\"padding-left: 0.282em;\"><span style=\"display: inline-block; position: relative; width: 1.4em; height: 0px; margin-right: 0.142em; margin-left: 0.142em;\"><span style=\"position: absolute; clip: rect(3.404em, 1001.21em, 4.196em, -999.998em); top: -4.471em; left: 50%; margin-left: -0.603em;\"><span class=\"mrow\" id=\"MathJax-Span-874\"><span class=\"mo\" id=\"MathJax-Span-875\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-876\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">20<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(3.404em, 1000.89em, 4.196em, -999.998em); top: -3.632em; left: 50%; margin-left: -0.464em;\"><span class=\"mrow\" id=\"MathJax-Span-877\"><span class=\"mo\" id=\"MathJax-Span-878\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">\u2212<\/span><span class=\"mn\" id=\"MathJax-Span-879\" style=\"font-size: 70.7%; font-family: MathJax_Main;\">4<\/span><\/span><span style=\"display: inline-block; width: 0px; height: 4.01em;\"><\/span><\/span><span style=\"position: absolute; clip: rect(0.888em, 1001.4em, 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: 1.4em; 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: 2.332em;\"><\/span><\/span><\/span><span style=\"display: inline-block; overflow: hidden; vertical-align: -0.538em; border-left: 0px solid; width: 0px; height: 1.786em;\"><\/span><\/span><\/nobr><\/span><script type=\"math\/tex\" id=\"MathJax-Element-34\">\\frac{\\Delta_3}{\\Delta}=\\frac{-20}{-4}<\/script> = 5<br> x = 1, y = 3, z = 5 is a solution.<\/p>\n\n<p>ii) Matrix Inversion Method:<br> <img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6314\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-18.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 18\" width=\"312\" height=\"945\" sizes=\"auto, (max-width: 312px) 100vw, 312px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 18\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\"><br> \u2234 x = 1, y = 3, z = 5 is the solution.<\/p>\n\n<p>iii) Gauss Jordan Method:<br> Augmented matrix<br>\n\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-6313\" src=\"https:\/\/cdn.manabadi.co.in\/2026-img\/Intet-Math\/TS-Inter-1st-Year-Maths-1A-Solutions-Chapter-3-Matrices-Ex-3h-19.png\" alt=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 19\" width=\"235\" height=\"607\" sizes=\"auto, (max-width: 235px) 100vw, 235px\" data-pin-description=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h) 19\" data-pin-title=\"TS Inter 1st Year Maths 1A Solutions Chapter 3 Matrices Ex 3(h)\"><br> \u03c1(A) = \u03c1(AB) = 3 and the system is consistent.<br> The system has a unique solution given by<br> x = 1, y = 3 and z = 5.<\/p><\/div>\n","protected":false},"excerpt":{"rendered":"<p>Question 1. Solve the following systems of equations. (i) By using Cramer\u2019s rule and Matrix inversion method, when the coefficient matrix is non-singular. (ii) By using Gauss Jordan Method. Also deter-mine whether the system has a unique solution or infinite number of solutions or no solution and find the solutions if exist. I) 5x \u2013 [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":2930,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1],"tags":[],"class_list":{"0":"post-2826","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-other"},"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\r\n<title>TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(h)<\/title>\r\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\r\n<link rel=\"canonical\" href=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/\" \/>\r\n<meta property=\"og:locale\" content=\"en_US\" \/>\r\n<meta property=\"og:type\" content=\"article\" \/>\r\n<meta property=\"og:title\" content=\"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(h)\" \/>\r\n<meta property=\"og:description\" content=\"Question 1. Solve the following systems of equations. (i) By using Cramer\u2019s rule and Matrix inversion method, when the coefficient matrix is non-singular. (ii) By using Gauss Jordan Method. Also deter-mine whether the system has a unique solution or infinite number of solutions or no solution and find the solutions if exist. I) 5x \u2013 [&hellip;]\" \/>\r\n<meta property=\"og:url\" content=\"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/\" \/>\r\n<meta property=\"og:site_name\" content=\"Manabadi Boards\" \/>\r\n<meta property=\"article:published_time\" content=\"2026-01-19T10:24:42+00:00\" \/>\r\n<meta property=\"article:modified_time\" content=\"2026-01-19T10:25:36+00:00\" \/>\r\n<meta property=\"og:image\" content=\"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3h.jpg\" \/>\r\n\t<meta property=\"og:image:width\" content=\"900\" \/>\r\n\t<meta property=\"og:image:height\" content=\"600\" \/>\r\n\t<meta property=\"og:image:type\" content=\"image\/jpeg\" \/>\r\n<meta name=\"author\" content=\"Junaid\" \/>\r\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\r\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"Junaid\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"11 minutes\" \/>\r\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\\\/\"},\"author\":{\"name\":\"Junaid\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#\\\/schema\\\/person\\\/a55695e135e9dc26536b290e31b4179d\"},\"headline\":\"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(h)\",\"datePublished\":\"2026-01-19T10:24:42+00:00\",\"dateModified\":\"2026-01-19T10:25:36+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\\\/\"},\"wordCount\":593,\"commentCount\":0,\"image\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/boardscdn.manabadi.co.in\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/07114619\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3h.jpg\",\"articleSection\":[\"Other\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\\\/\",\"url\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\\\/\",\"name\":\"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(h)\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\\\/#primaryimage\"},\"image\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/boardscdn.manabadi.co.in\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/07114619\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3h.jpg\",\"datePublished\":\"2026-01-19T10:24:42+00:00\",\"dateModified\":\"2026-01-19T10:25:36+00:00\",\"author\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#\\\/schema\\\/person\\\/a55695e135e9dc26536b290e31b4179d\"},\"breadcrumb\":{\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\\\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\\\/#primaryimage\",\"url\":\"https:\\\/\\\/boardscdn.manabadi.co.in\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/07114619\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3h.jpg\",\"contentUrl\":\"https:\\\/\\\/boardscdn.manabadi.co.in\\\/wp-content\\\/uploads\\\/2026\\\/01\\\/07114619\\\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3h.jpg\",\"width\":900,\"height\":600,\"caption\":\"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(h)\"},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(h)\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#website\",\"url\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/\",\"name\":\"Manabadi Boards\",\"description\":\"\",\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Person\",\"@id\":\"https:\\\/\\\/www.manabadi.co.in\\\/boards\\\/#\\\/schema\\\/person\\\/a55695e135e9dc26536b290e31b4179d\",\"name\":\"Junaid\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g\",\"url\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g\",\"contentUrl\":\"https:\\\/\\\/secure.gravatar.com\\\/avatar\\\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g\",\"caption\":\"Junaid\"}}]}<\/script>\r\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(h)","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:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/","og_locale":"en_US","og_type":"article","og_title":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(h)","og_description":"Question 1. Solve the following systems of equations. (i) By using Cramer\u2019s rule and Matrix inversion method, when the coefficient matrix is non-singular. (ii) By using Gauss Jordan Method. Also deter-mine whether the system has a unique solution or infinite number of solutions or no solution and find the solutions if exist. I) 5x \u2013 [&hellip;]","og_url":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/","og_site_name":"Manabadi Boards","article_published_time":"2026-01-19T10:24:42+00:00","article_modified_time":"2026-01-19T10:25:36+00:00","og_image":[{"width":900,"height":600,"url":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3h.jpg","type":"image\/jpeg"}],"author":"Junaid","twitter_card":"summary_large_image","twitter_misc":{"Written by":"Junaid","Est. reading time":"11 minutes"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/#article","isPartOf":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/"},"author":{"name":"Junaid","@id":"https:\/\/www.manabadi.co.in\/boards\/#\/schema\/person\/a55695e135e9dc26536b290e31b4179d"},"headline":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(h)","datePublished":"2026-01-19T10:24:42+00:00","dateModified":"2026-01-19T10:25:36+00:00","mainEntityOfPage":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/"},"wordCount":593,"commentCount":0,"image":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/#primaryimage"},"thumbnailUrl":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3h.jpg","articleSection":["Other"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/","url":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/","name":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(h)","isPartOf":{"@id":"https:\/\/www.manabadi.co.in\/boards\/#website"},"primaryImageOfPage":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/#primaryimage"},"image":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/#primaryimage"},"thumbnailUrl":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3h.jpg","datePublished":"2026-01-19T10:24:42+00:00","dateModified":"2026-01-19T10:25:36+00:00","author":{"@id":"https:\/\/www.manabadi.co.in\/boards\/#\/schema\/person\/a55695e135e9dc26536b290e31b4179d"},"breadcrumb":{"@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/"]}]},{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/#primaryimage","url":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3h.jpg","contentUrl":"https:\/\/boardscdn.manabadi.co.in\/wp-content\/uploads\/2026\/01\/07114619\/ts-inter-1st-year-maths-1a-solutions-cp-3-matrices-ex-3h.jpg","width":900,"height":600,"caption":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(h)"},{"@type":"BreadcrumbList","@id":"https:\/\/www.manabadi.co.in\/boards\/ts-inter-1st-year-maths-1a-matrices-solutions-exercise-3h\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/www.manabadi.co.in\/boards\/"},{"@type":"ListItem","position":2,"name":"TS Inter 1st Year Maths 1A Matrices Solutions Exercise 3(h)"}]},{"@type":"WebSite","@id":"https:\/\/www.manabadi.co.in\/boards\/#website","url":"https:\/\/www.manabadi.co.in\/boards\/","name":"Manabadi Boards","description":"","potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/www.manabadi.co.in\/boards\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Person","@id":"https:\/\/www.manabadi.co.in\/boards\/#\/schema\/person\/a55695e135e9dc26536b290e31b4179d","name":"Junaid","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/secure.gravatar.com\/avatar\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g","url":"https:\/\/secure.gravatar.com\/avatar\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/cf93d7547c6d8b1489ad346979c0e0d181d53fe095610d9d059830de5f332e0e?s=96&d=mm&r=g","caption":"Junaid"}}]}},"_links":{"self":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2826","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=2826"}],"version-history":[{"count":1,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2826\/revisions"}],"predecessor-version":[{"id":2827,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/posts\/2826\/revisions\/2827"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/media\/2930"}],"wp:attachment":[{"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/media?parent=2826"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/categories?post=2826"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.manabadi.co.in\/boards\/wp-json\/wp\/v2\/tags?post=2826"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}