Zero Matrix is Identity for Matrix Entrywise Addition

Theorem
Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems.

Let $\mathbf 0 = \sqbrk 0_{m n}$ be the zero matrix of $\map \MM {m, n}$.

Then $\mathbf 0$ is the identity element for matrix entrywise addition.

Also see

 * Zero Matrix is Identity for Hadamard Product