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 - 2.1 Operations with Matrices - 2.1 Exercises - Page 49: 55


$$\left[\begin{array}{cc}{a} &{b} \\{c}& {d} \end{array}\right]=\left[\begin{array}{cc}{7} &{-4} \\{-\frac{1}{2}}& {\frac{7}{2}} \end{array}\right].$$

Work Step by Step

We have the system $$ \begin{aligned} a+2c&=6 \\ b +2d &=3\\ 3a+4c &=19\\ 3b +4d &=2 \end{aligned}. $$ The augmented matrix is given by $$ \left[ \begin {array}{ccccc} 1&0&2&0&6\\ 0&1&0&2&3 \\ 3&0&4&0&19\\ 0&3&0&4&2 \end {array} \right] . $$ Using Gauss-Jordan elimination, we get the row-reduced echelon form as follows $$\left[ \begin {array}{ccccc} 1&0&0&0&7\\ 0&1&0&0&-4 \\ 0&0&1&0&-\frac{1}{2}\\ 0&0&0&1&\frac{7}{2} \end {array} \right] .$$ From which the solution is $$a=7, \quad b=-4, \quad c=-\frac{1}{2}, \quad d=\frac{7}{2} .$$ Hence, we have $$\left[\begin{array}{cc}{a} &{b} \\{c}& {d} \end{array}\right]=\left[\begin{array}{cc}{7} &{-4} \\{-\frac{1}{2}}& {\frac{7}{2}} \end{array}\right].$$
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