Negative Matrix is Inverse for Matrix Entrywise Addition over Ring
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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:
\(\ds \mathbf A + \paren {-\mathbf A}\) | \(=\) | \(\ds \sqbrk a_{m n} + \paren {-\sqbrk a_{m n} }\) | Definition of $\mathbf A$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqbrk a_{m n} + \sqbrk {-a}_{m n}\) | Definition of Negative Matrix over Ring | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqbrk {a + \paren {-a} }_{m n}\) | Definition of Matrix Entrywise Addition over Ring | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqbrk {0_R}_{m n}\) | Definition of Ring Negative | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \mathbf A + \paren {-\mathbf A}\) | \(=\) | \(\ds \mathbf 0_R\) | Definition of Zero Matrix over Ring |
The result follows from Zero Matrix is Identity for Matrix Entrywise Addition over Ring.
$\blacksquare$