Hadamard Product over Group forms Group

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

Let $\map {\mathcal M_G} {m, n}$ be a $m \times n$ matrix space over $\struct {G, \circ}$.

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

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

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

The result follows.