Matrix Entrywise Addition over Group forms Group

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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.

$\blacksquare$


Examples

$2 \times 2$ Matrices over Rational Numbers

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.


$n \times n$ Matrices over Real Numbers

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.