Let $$A=\begin{bmatrix} -1 & 1 & -1\\ 1 & 0 & 1\\ 0 & 0 & 1\end{bmatrix} $$ satisfy $$ A^2+\alpha\bigl(\operatorname{adj}(\operatorname{adj}(A))\bigr) + \beta\bigl(\operatorname{adj}(A)\operatorname{adj}(\operatorname{adj}(A))\bigr) = \begin{bmatrix} 2 & -2 & 2\\ -2 & 0 & -1\\ 0 & 0 & -1 \end{bmatrix}$$ for some $$\alpha,\beta\in\mathbb{R}$$. Then $$(\alpha-\beta)^2$$ is equal to _______.

$$A = \begin{bmatrix} -1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix}$$

$$|A| = 1 \cdot \begin{vmatrix} -1 & 1 \\ 1 & 0 \end{vmatrix} = 1 \cdot (0 - 1) = -1$$

$$\operatorname{adj}(\operatorname{adj}(A)) = |A|^{3-2}A = |A|A$$

$$\operatorname{adj}(A)\operatorname{adj}(\operatorname{adj}(A)) = \operatorname{adj}(A) \cdot (|A|A) = |A| \cdot \big(\operatorname{adj}(A)A\big) = |A| \cdot (|A|I) = |A|^2 I$$

$$\operatorname{adj}(\operatorname{adj}(A)) = -A$$

$$\operatorname{adj}(A)\operatorname{adj}(\operatorname{adj}(A)) = (-1)^2 I = I$$

$$A^2 + \alpha(-A) + \beta(I) = \begin{bmatrix} 2 & -2 & 2 \\ -2 & 0 & -1 \\ 0 & 0 & -1 \end{bmatrix}$$

$$A^2 - \alpha A + \beta I = \begin{bmatrix} 2 & -2 & 2 \\ -2 & 0 & -1 \\ 0 & 0 & -1 \end{bmatrix} \quad \text{--- (Equation 1)}$$

$$A^2 = \begin{bmatrix} -1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} -1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

$$(A^2)_{12} - \alpha(A)_{12} + \beta(I)_{12} = -2$$

$$-1 - \alpha(1) + 0 = -2 \implies -\alpha = -1 \implies \alpha = 1$$

$$(A^2)_{33} - \alpha(A)_{33} + \beta(I)_{33} = -1$$

$$1 - (1)(1) + \beta(1) = -1 \implies \beta = -1$$

$$(\alpha - \beta)^2 = (1 - (-1))^2 = (2)^2 = 4$$