Question

Let $A=\left\{a_{i j}\right\}$ be a $3 \times 3$ matrix, where $a_{i j}=\begin{cases}(-1)^{j-1} & \text { if } i < j, \\ 2 & \text { if } i=j, \\ (-1)^{i+j} & \text { if } i>j, \\ \end{cases}$ then det$\left(3 A d j\left(2 A^{-1}\right)\right)$ is equal to _______.

Answer 1

$A=\begin{bmatrix}2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2\end{bmatrix}$
$|A|=4$
$\left|3 a d j\left(2 A^{-1}\right)\right|=\left|3.2^{2} a d j\left(A^{-1}\right)\right|$
$=12^{3}\left|\text{adj}\left(A^{-1}\right)\right|=12^{3}\left|A^{-1}\right|^{2}$
$=\frac{12^{3}}{|A|^{2}}=\frac{12^{3}}{16}=108$