Let X= [0, 1, 0; 0, 0, 1; 0, 0, 0], Y=α I+β X+γ X^2 and Z=α^2 I-α β X+(β^2-α γ) X^2, α, β, γ ∈ R. If Y^-1= [1 / 5, -2 / 5, 1 / 5; 0, 1 / 5, -2 / 5;…

Let $\displaystyle X=\left[\begin{array}{lll}0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{array}\right], Y=\alpha I+\beta X+\gamma X^2$ and $\displaystyle Z=\alpha^2 I-\alpha \beta X+\left(\beta^2-\alpha \gamma\right) X^2, \alpha, \beta, \gamma \in \mathbb{R}$. If $\displaystyle Y^{-1}=\left[\begin{array}{ccc}1 / 5 & -2 / 5 & 1 / 5 \\ 0 & 1 / 5 & -2 / 5 \\ 0 & 0 & 1 / 5\end{array}\right]$, then $\displaystyle (\alpha-\beta+\gamma)^2$ is equal to \_\_\_\_.