Question

If $A$ and $B$ are non-singular matrices of order three such that $adj\left(A B\right)=\begin{bmatrix} 1 & 1 & 1 \\ 1 & \alpha & 1 \\ 1 & 1 & \alpha \end{bmatrix}$ and $\left|B^{2} a d j A\right|=\alpha ^{2}+3\alpha -8$ , then the value of $\alpha $ is equal to

• A. $\frac{9}{5}$
• B. $\frac{8}{5}$
• C. $3$
• D. $2$

Answer 1

Answer: (A)

$\left|\left(a d j A B\right)\right|=\begin{vmatrix} 1 & 1 & 1 \\ 1 & \alpha & 1 \\ 1 & 1 & \alpha \end{vmatrix}$
$=1\left(\left(\alpha \right)^{2} - 1\right)-1\left(\alpha - 1\right)+1\left(1 - \alpha \right)$
$=\left(\alpha \right)^{2}-2\alpha +1=\left(\alpha - 1\right)^{2}$
$\Rightarrow \left(\left|A B\right|\right)^{2}=\left(\alpha - 1\right)^{2}$ and $\left|B^{2} a d j A\right|=\left|B\right|^{2}\left|A\right|^{2}$
$=\left(\left|A B\right|\right)^{2}=\left(\alpha - 1\right)^{2}=\left(\alpha \right)^{2}+3\alpha -8$
$\Rightarrow \alpha ^{2}-2\alpha +1=\alpha ^{2}+3\alpha -8$
$\Rightarrow 5\alpha =9$
$\Rightarrow \alpha =\frac{9}{5}$