Negative Matrix is Inverse for Hadamard Product

Theorem
Let $\struct {G, \cdot}$ be a group whose identity is $e$.

Let $\map {\MM_G} {m, n}$ be a $m \times n$ matrix space over $\struct {G, \cdot}$.

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

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

Then $-\mathbf A$ is the inverse for the operation $\circ$, where $\circ$ is the Hadamard product.

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

Then:

The result follows from Zero Matrix is Identity for Hadamard Product.

Also see

 * Negative Matrix is Inverse for Matrix Entrywise Addition