Elementary Linear Algebra 7th Edition

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

Chapter 3 - Determinants - Review Exercises - Page 138: 26


(a) $|A^T|= -1.$ (b) $|A^3|=| -1.$ (c) $|A^TA|= 1 .$ (d) $|5A|= -125.$

Work Step by Step

Let $A$ be given by $$A=\left[ \begin {array}{cccc} 3&0&1\\ -1&0&0\\ 2&1&2 \end {array} \right] ,$$ (a) $|A^T|=|A|=-1.$ (b) $|A^3|=|A|^3=-1^3=-1.$ (c) $|A^TA|=|A^T||A|=|A|^2=1 .$ (d) $|5A|=5^3|A|=(125)(-1)=-125.$
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