Four Steps Without Multiplying Four Times

A discrete system evolves by repeated application of

$A = \begin{pmatrix}4 & 1\\ 2 & 3\end{pmatrix}$

starting from the state

$x_0 = \begin{pmatrix}3\\ 0\end{pmatrix}$

Compute the state after four steps, $x_4 = A^4 x_0$, by diagonalising $A$ rather than by multiplying matrices repeatedly.