Matrix Entrywise Addition forms Abelian Group/Examples/2x2 Matrices over Rational Numbers

Example of Matrix Entrywise Addition over Group forms Group
Let $\Q^{2 \times 2}$ denote the set of order $2$ square matrices over the set $\Q$ of rational numbers.

Then the algebraic structure $\struct {\Q^{2 \times 2}, +}$, where $+$ denotes matrix entrywise addition, is an abelian group.

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

It follows from Matrix Entrywise Addition over Group forms Group that $\struct {\Q^{2 \times 2}, +}$ is a group.

It follows from Matrix Entrywise Addition is Commutative that $\struct {\Q^{2 \times 2}, +}$ is abelian.