In the paper, Counting Domineering Positions, there is this theorem-
The polynomial profile of Domineering on an $m \times n$ board is the (1,1) entry of $G_{0,n}^{m}$ where $G_{1,0}=[0], G_{0,0} = [1]$, and $$G_{0,q+1} = \begin{bmatrix} G_{0,q}+G_{1,q} & xG_{0,q} \\ G_{0,q} & 0 \end{bmatrix}$$ $$G_{1,q+1} = \begin{bmatrix} yG_{0,q} & 0 \\ 0 & 0 \end{bmatrix}$$
Now, say we have matrices $\begin{bmatrix} x+1 & 0 \\ 0 & 0 \end{bmatrix}$ , $\begin{bmatrix} y & 0 \\ 0 & 0 \end{bmatrix}$.
Is it possible to write a MAPLE code which would allow matrix multiplication in a manner such that the polynomial entries to the matrix are multiplied and added in accordance to the rules of polynomial multiplication?
Furthermore, is there a way to extend this to block matrices with polynomial entries? If so, I'd be obliged if someone could illustrate it. Thank you.