$$A=\begin{bmatrix}2&-1&2\\-1&2&-1\\1&-1&2\end{bmatrix}$$
I am supposed to find $A^4$.
I tried to find the characteristic equation and got
$$-x^3+6x^2-8x+3=0$$
From here I can go the long route and multiply by $x$ the equation and solve the other side for $x^4$ but it felt time consuming. Is there any other way to proceed?