# Negative Matrix is Inverse for Matrix Entrywise Addition

## Theorem

Let $\left({G, \circ}\right)$ be a group whose identity is $e$.

Let $\mathcal M_G \left({m, n}\right)$ be a $m \times n$ matrix space over $\left({G, \circ}\right)$.

Let $\mathbf A$ be an element of $\mathcal M_G \left({m, n}\right)$.

Let $-\mathbf A$ be the negative of $\mathbf A$.

Then $-\mathbf A$ is the inverse for the operation $+$, where $+$ is matrix entrywise addition, has an inverse for the operation $+$.

## Proof

Let $\mathbf A \in$

Let $\mathbf A = \left[{a}\right]_{m n} \in \mathcal M_G \left({m, n}\right)$.

Then:

 $\displaystyle \mathbf A + \left({-\mathbf A}\right)$ $=$ $\displaystyle \left[{a}\right]_{m n} + \left({-\left[{a}\right]_{m n} }\right)$ $\displaystyle$ $=$ $\displaystyle \left[{a}\right]_{m n} - \left[{a}\right]_{m n}$ $\displaystyle$ $=$ $\displaystyle 0$ $\displaystyle \implies \ \$ $\displaystyle \mathbf A + \left({-\mathbf A}\right)$ $=$ $\displaystyle \mathbf 0$

The result follows from Zero Matrix is Identity for Matrix Entrywise Addition.

$\blacksquare$