Matrix Entrywise Addition forms Abelian Group/Examples/2x2 Matrices over Rational Numbers
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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: $\S 29 \alpha \ (4)$