# 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:

 $\ds \mathbf A + \paren {-\mathbf A}$ $=$ $\ds \sqbrk a_{m n} + \paren {-\sqbrk a_{m n} }$ Definition of $\mathbf A$ $\ds$ $=$ $\ds \sqbrk a_{m n} + \sqbrk {-a}_{m n}$ Definition of Negative Matrix $\ds$ $=$ $\ds \sqbrk {a + \paren {-a} }_{m n}$ Definition of Matrix Entrywise Addition $\ds$ $=$ $\ds \sqbrk 0_{m n}$ $\ds \leadsto \ \$ $\ds \mathbf A + \paren {-\mathbf A}$ $=$ $\ds \mathbf 0$ Definition of Zero Matrix

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

$\blacksquare$