Negative Matrix is Inverse for Matrix Entrywise Addition

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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.


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


\(\ds \mathbf A + \paren {-\mathbf A}\) \(=\) \(\ds \sqbrk a_{m n} + \paren {-\sqbrk a_{m n} }\) 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.


Also see