If $B=\left[\begin{array}{ccc}3 & \alpha & -1 \ 1 & 3 & 1 \ -1 & 1 & 3\end{array} ight]$ is the adjoint of $a$ $3 imes 3$ matrix $\mathrm{A}$ and $|\mathrm{A}|=4$, then $\alpha$ is equal to8

Question

If $B=\left[\begin{array}{ccc}3 & \alpha & -1 \ 1 & 3 & 1 \ -1 & 1 & 3\end{array}ight]$ is the adjoint of $a$ $3 	imes 3$ matrix $\mathrm{A}$ and $|\mathrm{A}|=4$, then $\alpha$ is equal to8

Aicharya · Accepted Answer

$\begin{aligned} & \quad 	ext { Using }|\operatorname{adj} A|=|A|^{\mathrm{n}-1} \ & \quad 	ext { But } \mathrm{B}=\operatorname{Adj}(\mathrm{A}) ...[Given]\ & 	herefore \quad|\mathrm{B}|=|\mathrm{A}|^2 \ & \Rightarrow\left|\begin{array}{ccc} 3 & \alpha & -1 \ 1 & 3 & 1 \ -1 & 1 & 3 \end{array}ight|=|\mathrm{A}|^2 \ & \Rightarrow 24-4 \alpha-4=4^2 \ & \Rightarrow 20-4 \alpha=16 \ & \Rightarrow \alpha=1 \end{aligned}$

If $B=\left[\begin{array}{ccc}3 & \alpha & -1 \\ 1 & 3 & 1 \\ -1 & 1 & 3\end{array}\right]$ is the adjoint…

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