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

Example of Matrix Entrywise Addition forms Abelian 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 Infinite Abelian Group, $\struct {\Q, +}$ is an abelian group.

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

$\blacksquare$

Sources

• 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Definition of Group Structure: $\S 29 \alpha \ (4)$