Zero Matrix is Identity for Matrix Entrywise Addition/Proof 1
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.
Proof
From:
the standard number systems $\Z$, $\Q$, $\R$ and $\C$ are rings whose zero is the number $0$ (zero).
Hence we can apply Zero Matrix is Identity for Matrix Entrywise Addition over Ring.
$\Box$
The above cannot be applied to the natural numbers $\N$, as they do not form a ring.
However, from Natural Numbers under Addition form Commutative Monoid, the algebraic structure $\struct {\N, +}$ is a commutative monoid whose identity is $0$ (zero).
By definition, matrix entrywise addition is the Hadamard product with respect to addition of numbers.
The result follows from Zero Matrix is Identity for Hadamard Product.
$\blacksquare$