Diagonalization & Matrix Powers · Medium problem

A sequence satisfies $a_0 = 0$, $a_1 = 2$ and $a_{n+1} = 4a_n - 3a_{n-1}$. Writing $\begin{pmatrix}a_{n+1}\\a_n\end{pmatrix} = \begin{pmatrix}4 & -3\\1 & 0\end{pmatrix}\begin{pmatrix}a_n\\a_{n-1}\end{pmatrix}$, use the eigenvalues of this matrix to find a closed form, and give $a_5$.