Matrix Entrywise Addition forms Abelian Group/Examples/nxn Matrices over Real Numbers

Example of Matrix Entrywise Addition over Group forms Group
Let $\R^{n \times n}$ denote the set of order $n$ square matrices over the set $\R$ of real numbers.

Then the algebraic structure $\struct {\R^{n \times n}, +}$, where $+$ denotes matrix entrywise addition, is an abelian group.

Proof
From Real Numbers under Addition form Abelian Group, $\struct {\R, +}$ is an abelian group.

It follows from Matrix Entrywise Addition over Group forms Group that $\struct {\R^{n \times n}, +}$ is a group.

It follows from Matrix Entrywise Addition is Commutative that $\struct {\R^{n \times n}, +}$ is abelian.