Zero Matrix is Identity for Hadamard Product

Theorem
Let $\struct {S, \circ}$ be a monoid whose identity is $e$.

Let $\map {\MM_S} {m, n}$ be an $m \times n$ matrix space over $S$.

Let $\sqbrk e_{m n}$ be the zero matrix of $\map {\MM_S} {m, n}$

Then $\sqbrk e_{m n}$ is the identity element for matrix entrywise addition.

Proof
Let $\sqbrk a_{m n} \in \map {\MM_S} {m, n}$.

Then:
 * $\forall \tuple {i, j} \in \closedint 1 m \times \closedint 1 n: a_{i j} \circ e = a_{i j} = e \circ a_{i j}$