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.

$\blacksquare$


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