Let α ∈(0, ∞) and A= [1, 2, α; 1, 0, 1; 0, 1, 2]. If det ( adj (2 A-A^T ) · adj (A-2 A^T ))=2^8, then ( det (A))^2 is equal to:

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