If \(A=\left[\begin{array}{cc}2 & -3 \\ -4 & 1\end{array}\right]\), then adj \(\left(3 A^2+12 A\right)\) is equal to

\(\begin{aligned} & \operatorname{adj}\left(3 \mathrm{~A}^2+12 \mathrm{~A}\right)=\operatorname{adj}\{3 \mathrm{~A}(\mathrm{~A}+4 \mathrm{I})\} \\ & =\operatorname{adj}(\mathrm{A}+4 \mathrm{I}) \cdot \operatorname{adj}(3 \mathrm{~A}) \\ & =\operatorname{adj}\left(\left[\begin{array}{cc}6 & -3 \\ -4 & 5\end{array}\right]\right) \cdot \operatorname{adj}\left(\left[\begin{array}{cc}6 & -9 \\ -12 & 3\end{array}\right]\right) \\ & =\left[\begin{array}{ll}5 & 3 \\ 4 & 6\end{array}\right]\left[\begin{array}{cc}3 & 9 \\ 12 & 6\end{array}\right] \\ & =\left[\begin{array}{ll}51 & 63 \\ 84 & 72\end{array}\right]\end{aligned}\)