Negative Matrix is Inverse for Matrix Entrywise Addition over Ring

Theorem
Let $\struct {R, +, \circ}$ be a ring whose zero is $0_R$.

Let $\map {\MM_R} {m, n}$ be a $m \times n$ matrix space over $\struct {R, +, \circ}$.

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

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 $\mathbf A = \sqbrk a_{m n} \in \map {\MM_R} {m, n}$.

Then:

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

Also see

 * Negative Matrix is Inverse for Hadamard Product