Negative Matrix is Inverse for Matrix Entrywise Addition
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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.
$\blacksquare$
Also see
- Negative Matrix is Inverse for Matrix Entrywise Addition over Ring
- Negative Matrix is Inverse for Hadamard Product
Sources
- 1998: Richard Kaye and Robert Wilson: Linear Algebra ... (previous) ... (next): Part $\text I$: Matrices and vector spaces: $1$ Matrices: $1.2$ Addition and multiplication of matrices: $7$