Let \(A = \begin{bmatrix} 1 & 4\\ 2 & 3 \end{bmatrix}\)
and let \(x = \begin{bmatrix} 2\\ -1 \end{bmatrix}\).
Let \(t = A^{1107}x\).
What is the value of \(t_1\)?
The answer is \(-2\).
Note that \(Ax = \begin{bmatrix} -2 \\ 1\end{bmatrix} = -x\).
Hence,
\(t=A^{1107}x = (-1)^{1107} x = -x = \begin{bmatrix} -2\\1\end{bmatrix}\).