Let A= [-1, 1, -1; 1, 0, 1; 0, 0, 1] satisfy A^2+α( adj ( adj ( A )))+β( adj ( A )( adj ( adj ( A ))))= [2, -2, 2; -2, 0, -1; 0, 0, -1] for some α, β…

Let $\displaystyle A=\left[\begin{array}{ccc}-1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 0 & 1\end{array}\right]$ satisfy $\displaystyle \mathrm{A}^2+\alpha(\operatorname{adj}(\operatorname{adj}(\mathrm{A})))+\beta(\operatorname{adj}(\mathrm{A})(\operatorname{adj}(\operatorname{adj}(\mathrm{A}))))=\left[\begin{array}{ccc}2 & -2 & 2 \\ -2 & 0 & -1 \\ 0 & 0 & -1\end{array}\right]$ for some $\displaystyle \alpha, \beta \in \mathbb{R}$. Then $\displaystyle (\alpha-\beta)^2$ is equal to $\displaystyle \_\_\_\_$