Five Steps Through a Two-Unit Layer

A miniature recurrent layer holds a two-number state $\mathbf{h}_t$ and updates it by the same matrix at every step:

$\mathbf{h}_{t+1} = A\,\mathbf{h}_t, \qquad A = \begin{bmatrix} 5 & -2 \\ 3 & -2 \end{bmatrix}$

After five steps the state is $\mathbf{h}_5 = A^5\mathbf{h}_0$, so the whole five-step behaviour is packed into the single matrix $A^5$. You want the cross term: the entry in row 1, column 2 of $A^5$, which says how strongly unit 2's starting activation feeds into unit 1 five steps later.

$A$ is diagonalizable and both of its eigenvalues are integers, so $A = PDP^{-1}$ for some invertible $P$ and diagonal $D$.

What is the entry of $A^5$ in row 1, column 2? Your answer is an exact integer.