The Factor After Three Ticks

A two-dimensional system is updated once per tick by multiplying its state by the fixed matrix

$A = \begin{bmatrix} 9 & -12 \\ 1 & 1 \end{bmatrix}$

so the state after $t$ ticks is $\mathbf{x}_t = A^t \mathbf{x}_0$.

The system is started from

$\mathbf{x}_0 = \begin{bmatrix} 2 \\ 1 \end{bmatrix}$

Run it for three ticks. It turns out that $\mathbf{x}_3$ points along exactly the same line as $\mathbf{x}_0$ — the three ticks have not rotated the state at all, only rescaled it. So there is a single number $c$ with

$A^3 \mathbf{x}_0 = c\,\mathbf{x}_0$

What is $c$?

The answer is an exact integer — enter it with no rounding.