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