Negative Matrix is Inverse for Matrix Entrywise Addition

Theorem
Let $\Bbb F$ denote one of the standard number systems.

Let $\map \MM {m, n}$ be a $m \times n$ matrix space over $\Bbb F$.

Let $\mathbf A$ be an element of $\map \MM {m, n}$.

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

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

Proof
Let $\mathbf A = \sqbrk a_{m n} \in \map \MM {m, n}$.

Then:

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

Also see

 * Negative Matrix is Inverse for Matrix Entrywise Addition over Ring
 * Negative Matrix is Inverse for Hadamard Product