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: 3


$$\operatorname{adj}(A)=\left[ \begin {array}{ccc} 0&0&0\\ 0&-12&-6 \\ 0&4&2\end {array} \right]. $$ $A$ is singular matrix and has no inverse.

Work Step by Step

The matrix is given by $A=\left[ \begin {array}{ccc} 1&0&0\\ 0&2&6 \\ 0&-4&-12\end {array} \right] .$ To find $\operatorname{adj}(A)$, we calculate first the cofactor matrix of $A$ as follows $$\left[ \begin {array}{ccc} 0&0&0\\ 0&-12&4 \\ 0&-6&2\end {array} \right] .$$ Now, the adjoint of $A$ is $$\operatorname{adj}(A)=\left[ \begin {array}{ccc} 0&0&0\\ 0&-12&-6 \\ 0&4&2\end {array} \right] $$ To find $A^{-1}$, we have to calculate $\det(A)$ which is given by $$\det(A)=0.$$ Hence, $A$ is singular matrix and has no inverse.
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