Definition:Negative Matrix

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 $\left({G, \circ}\right)$.

Then every element $\mathbf A = \left[{a}\right]_{m n}$ of $\left({\mathcal M_G \left({m, n}\right), +}\right)$, where $+$ is matrix entrywise addition, has an inverse for the operation $+$.

This inverse is called the negative (matrix) of $\left[{a}\right]_{m n}$, and is written $-\left[{a}\right]_{m n}$ or $- \mathbf A$.

So:
 * $- \mathbf A = - \left[{a}\right]_{m n} = \left[{a^{-1}}\right]_{m n}$

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

Then:

Thus $- \left[{a}\right]_{m n}$, the negative of $\left[{a}\right]_{m n}$, is defined as follows.

Let $\left[{b}\right]_{m n} = - \left[{a}\right]_{m n}$.

Then:
 * $\forall \left({i, j}\right) \in \left[{1 \, . \, . \, m}\right] \times \left[{1 \, . \, . \, n}\right]: b_{i j} = a_{i j}^{-1}$