Matrix Powers Magic

Question

Let $A = [\begin{matrix} 1 & 4 \\ 2 & 3 \end{matrix}]$ and let $x = [\begin{matrix} 2 \\ - 1 \end{matrix}]$ . Let $t = A^{1107} x$ . What is the value of $t_{1}$ ?

The answer is $- 2$ .

Note that $A x = [\begin{matrix} - 2 \\ 1 \end{matrix}] = - x$ . Hence, $t = A^{1107} x = (- 1)^{1107} x = - x = [\begin{matrix} - 2 \\ 1 \end{matrix}]$ .