Hadamard Product over Group forms Group

Theorem
Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Let $\mathcal M_G \left({m, n}\right)$ be a $m \times n$ matrix space over $\left({G, \circ}\right)$.

Then $\left({\mathcal M_G \left({m, n}\right), +}\right)$, where where $+$ is matrix entrywise addition, is also a group.

Proof
As $\left({G, \circ}\right)$, being a group, is also a monoid, it follows from Matrix Space Semigroup under Addition‎ that $\mathcal M_G \left({m, n}\right)$ is also a monoid.

As $\left({G, \circ}\right)$ is a group, it follows from Negative Matrix that all elements of $\mathcal M_G \left({m, n}\right)$ have an inverse.

The result follows.