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

Question

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

Aicharya · Accepted Answer

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

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

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