The recurrence $a_n = 3a_{n-1} - 2a_{n-2}$ with $a_0 = 1, a_1 = 3$ can be written in matrix form:

$\begin{pmatrix} a_n \\ a_{n-1} \end{pmatrix} = M^{n-1} \begin{pmatrix} a_1 \\ a_0 \end{pmatrix}, \quad M = \begin{pmatrix} 3 & -2 \\ 1 & 0 \end{pmatrix}$

Compute $a_3$ using matrix powers.