Negative Matrix is Inverse for Matrix Entrywise Addition

From ProofWiki
Jump to navigation Jump to search

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 $\left({G, \circ}\right)$.

Let $\mathbf A$ be an element of $\mathcal M_G \left({m, n}\right)$.

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

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


Proof

Let $\mathbf A \in $

Let $\mathbf A = \left[{a}\right]_{m n} \in \mathcal M_G \left({m, n}\right)$.

Then:

\(\displaystyle \mathbf A + \left({-\mathbf A}\right)\) \(=\) \(\displaystyle \left[{a}\right]_{m n} + \left({-\left[{a}\right]_{m n} }\right)\)
\(\displaystyle \) \(=\) \(\displaystyle \left[{a}\right]_{m n} - \left[{a}\right]_{m n}\)
\(\displaystyle \) \(=\) \(\displaystyle 0\)
\(\displaystyle \implies \ \ \) \(\displaystyle \mathbf A + \left({-\mathbf A}\right)\) \(=\) \(\displaystyle \mathbf 0\)

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

$\blacksquare$