Powers of a matrix

The characteristic polynomial is $(X-2)^2-1=(X-1)(X-3)$. The vector $(1,-1)$ is an eigenvector for $1$ and $(1,1)$ is one for $3$.

With $P=\begin{pmatrix}1&1\\-1&1\end{pmatrix}$, $D=\textrm{diag}(1,3)$ and $P^{-1}=\dfrac12\begin{pmatrix}1&-1\\1&1\end{pmatrix}$ we have $A=PDP^{-1}$, so
$$A^n=PD^nP^{-1}=\frac12\begin{pmatrix}3^n+1&3^n-1\\3^n-1&3^n+1\end{pmatrix}.$$