Given the Boolean matrices $A_{m \times k}$, $B_{m \times k}$ and $C_{k \times n}$, is it true that $(A \oplus B) * C = (A * C) \oplus (B * C)$, where $\oplus$ denotes elementwise XOR and $*$ denotes Boolean matrix multiplication?
I'm trying to prove the above for a paper, but I'm no mathematician. I've found a related result on Wikipedia, but it does not mention if it holds for Boolean matrices.
I'm not sure if this make any difference, but the matrices $A$ and $B$ in my case are exponentiations of a graph adjacency matrix (no weights and 1 in the diagonal), so it will never happen that $B_{i,j}=1$ and $A_{i,j}=0$.
The equality is false. A counterexample is found by setting $A:=\begin{bmatrix}1 & 1\\0 & 1\end{bmatrix}$, $B:=\begin{bmatrix}1 & 0\\0 & 0\end{bmatrix}$ and $C:=\begin{bmatrix}1 & 1\\1 & 1\end{bmatrix}$. Then:
$(A\oplus B) * C = \begin{bmatrix}1 & 1\\1 & 1\end{bmatrix}$
$(A * C) \oplus (B * C) = \begin{bmatrix}0 & 0\\1 & 1\end{bmatrix}$