Question

If $A$ and $B$ are non-singular matrices of order three such that ${adj}(A B)=\begin{bmatrix}1 & 1 & 1 \\ 1 & p & 1 \\ 1 & 1 & p\end{bmatrix}$ and $\left|B^{2} {adj}(A)\right|=p^{2}-3$, then $p$ is

Answer 1

${adj}(A B)=|A B|^{2}=(p-1)^{2}$
$\left|B^{2} {adj} A\right|=\left|B^{2}\right||A|^{2}=|A B|^{2}$
$\Rightarrow(p-1)^{2}=p^{2}-3$
$\Rightarrow p=2$