Let $A=\left\{a_{i j}\right\}$ be a $3 \times 3$ matrix, where
$\mathrm{a}_{\mathrm{ij}}=\left\{\begin{array}{c}(-1)^{\mathrm{j}-\mathrm{i}} \text { if } \mathrm{i}<\mathrm{j} \\ 2 \quad \text { if } \mathrm{i}=\mathrm{j} \\ (-1)^{\mathrm{i}+\mathrm{j}} \text { if } \mathrm{i}>\mathrm{j}\end{array}\right.$
then $\operatorname{det}\left(3 \operatorname{Adj}\left(2 \mathrm{~A}^{-1}\right)\right)$ is equal to
$A=\left[\begin{array}{rrr}2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2\end{array}\right]$
$|\mathrm{A}|=4$
$\left|3 \operatorname{adj}\left(2 \mathrm{~A}^{-1}\right)\right|=\left|3.2^{2} \operatorname{adj}\left(\mathrm{A}^{-1}\right)\right|$
$=12^{3}\left|\operatorname{adj}\left(\mathrm{A}^{-1}\right)\right|=12^{3}\left|\mathrm{~A}^{-1}\right|^{2}=\frac{12^{3}}{|\mathrm{~A}|^{2}}=\frac{12^{3}}{16}=108$
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