Elementary Linear Algebra 7th Edition

Published by Cengage Learning
ISBN 10: 1-13311-087-8
ISBN 13: 978-1-13311-087-3

Chapter 2 - Matrices - Review Exercises - Page 98: 9


$$x_1=\frac{2}{3} , \quad x_2=-{\frac {17}{3}}, \quad x_3=-\frac{11}{3}$$.

Work Step by Step

The system can written on the form $Ax=b$ as follows $$x_1\left[ \begin {array}{cc} {-3} \\{2}\\{1} \end {array} \right]+x_2\left[ \begin {array}{cc} {-1}\\ {4} \\{-2}\end {array} \right]+x_3\left[ \begin {array}{cc} {1}\\ {-5} \\{3}\end {array} \right]=\left[ \begin {array}{cc} {0}\\ {-3}\\{1} \end {array} \right].$$ To solve system using Gaussian elimination, we form the augmented matrix as follows $$\left[ \begin {array}{cccc} -3&-1&1&0\\ 2&4&-5&-3 \\ 1&-2&3&1\end {array} \right] . $$ The reduced row echelon form is given by $$\left[ \begin {array}{cccc} 1&0&0&\frac{2}{3}\\ 0&1&0&-{ \frac {17}{3}}\\ 0&0&1&-\frac{11}{3}\end {array} \right] . $$ From which, we have $$x_1=\frac{2}{3} , \quad x_2=-{ \frac {17}{3}}, \quad x_3=-\frac{11}{3}$$.
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