Let \(\alpha \in (0,\infty )\) and\(A=\left[\begin{matrix}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{matrix}\right]\).…

Let \(\alpha \in (0,\infty )\) and\(A=\left[\begin{matrix}1 & 2 & \alpha \\ 1 & 0 & 1 \\ 0 & 1 & 2\end{matrix}\right]\). If \(\det \left(adj\left(2A-{A}^{T}\right)\cdot adj\left(A-2{A}^{T}\right)\right)={2}^{8}\), then \((\det (A){)}^{2}\) is equal to :

[JEE Main 2024, 4 Apr (Shift 1)]