Elementary Linear Algebra 7th Edition

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

Chapter 1 - Systems of Linear Equations - 1.2 Gaussian Elimination and Gauss-Jordan Elimination - 1.2 Exercises - Page 24: 61


$$a\neq 0, \quad bc-ad\neq 0.$$

Work Step by Step

We have the matrix $$ \left[ \begin {array}{ccc} a&b\\ c&d \end {array} \right]. $$ Multiply the first row by $c$ and adding it to $-a$ times the second row, we get $$\left[ \begin {array}{ccc} a&b\\ 0&bc-ad\end {array} \right]. $$ Dividing the second row on $bc-ad$, we get $$\left[ \begin {array}{ccc} a&b\\ 0&1\end {array} \right]. $$ Multiply the second row by $-b$ and adding it to the first row, we get $$\left[ \begin {array}{ccc} a&0\\ 0&1\end {array} \right].$$ Dividing the first row on $a$, we get $$ \left[ \begin {array}{ccc} 1&0\\ 0&1 \end {array} \right].$$ Now, the matrix $ \left[ \begin {array}{ccc} a&b\\ c&d \end {array} \right]$ is row-equivalent to $ \left[ \begin {array}{ccc} 1&0\\ 0&1 \end {array} \right]$ provided that $$a\neq 0, \quad bc-ad\neq 0.$$
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