JEE Main 2022 · 26 Jun · Shift 2, Mathematics question 24

Let $X=\begin{bmatrix}0&1&0\\0&0&1\\0&0&0\end{bmatrix}$, $Y=\alpha I+\beta X+\gamma X^2$ and $Z=\alpha^2I-\alpha\beta X+\left(\beta^2-\alpha\gamma\right)X^2$, $\alpha,\beta,\gamma\in\mathbb R$. If $Y^{-1}=\begin{bmatrix}\frac15&-\frac25&\frac15\\0&\frac15&-\frac25\\0&0&\frac15\end{bmatrix}$, then $(\alpha-\beta+\gamma)^2$ is equal to ______.