Question

If $A=\left[\begin{array}{ccc}a& 2& -1\\ -1& 1& 2\\ 2& -1& 1\end{array}\right]$, $\text{where}(a\ne \frac{11}{3})$ and $det.\left(\text{adj(adj}A\right))=(23{)}^4,$ then the value of $a$ is

Answer 1

$|\text{adj}.\left(\text{adj}A\right)|=|A{|}^{(n-1{)}^2}$
n = 3
$|\text{adj}\left(\text{adj}A\right)|=|A{|}^4={23}^4$
$|A|=23\Rightarrow 3a+11=23\Rightarrow a=4$