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 - 3.4 Applications of Determinants - 3.4 Exercises - Page 136: 5


$$\operatorname{adj}(A)=\left[ \begin {array}{ccc} -7&-12&13\\ 2&3&-5 \\ 2&3&-2\end {array} \right] .$$ $$A^{-1}= -\frac{1}{3}\left[ \begin {array}{ccc} -7&-12&13\\ 2&3&-5 \\ 2&3&-2\end {array} \right].$$

Work Step by Step

The matrix is given by $A=\left[ \begin {array}{ccc} -3&-5&-7\\ 2&4&3 \\ 0&1&-1\end {array} \right] .$ To find $\operatorname{adj}(A)$, we calculate first the cofactor matrix of $A$ as follows $$\left[ \begin {array}{ccc} -7&2&2\\ -12&3&3 \\ 13&-5&-2\end {array} \right] .$$ Now, the adjoint of $A$ is $$\operatorname{adj}(A)=\left[ \begin {array}{ccc} -7&-12&13\\ 2&3&-5 \\ 2&3&-2\end {array} \right] .$$ To find $A^{-1}$, we have to calculate $\det(A)$ which is given by $$\det(A)=-3.$$ Finally, we have $$A^{-1}=\frac{1}{\det(A)}\operatorname{adj}(A)=-\frac{1}{3}\left[ \begin {array}{ccc} -7&-12&13\\ 2&3&-5 \\ 2&3&-2\end {array} \right].$$
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