Zero Matrix is Identity for Matrix Entrywise Addition/Proof 2

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Theorem

Let $\map \MM {m, n}$ be a $m \times n$ matrix space over one of the standard number systems.

Let $\mathbf 0 = \sqbrk 0_{m n}$ be the zero matrix of $\map \MM {m, n}$.


Then $\mathbf 0$ is the identity element for matrix entrywise addition.


Proof

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

Then:

\(\ds \mathbf A + \mathbf 0\) \(=\) \(\ds \sqbrk a_{m n} + \sqbrk 0_{m n}\) Definition of $\mathbf A$ and $\mathbf 0_R$
\(\ds \) \(=\) \(\ds \sqbrk {a + 0}_{m n}\) Definition of Matrix Entrywise Addition
\(\ds \) \(=\) \(\ds \sqbrk a_{m n}\) Identity Element of Addition on Numbers
\(\ds \leadsto \ \ \) \(\ds \mathbf A + \mathbf 0\) \(=\) \(\ds \mathbf A\) Definition of Zero Matrix


Similarly:

\(\ds \mathbf 0 + \mathbf A\) \(=\) \(\ds \sqbrk 0_{m n} + \sqbrk a_{m n}\) Definition of $\mathbf A$ and $\mathbf 0_R$
\(\ds \) \(=\) \(\ds \sqbrk {0 + a}_{m n}\) Definition of Matrix Entrywise Addition
\(\ds \) \(=\) \(\ds \sqbrk a_{m n}\) Identity Element of Addition on Numbers
\(\ds \leadsto \ \ \) \(\ds \mathbf 0 + \mathbf A\) \(=\) \(\ds \mathbf A\) Definition of Zero Matrix

$\blacksquare$


Sources

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