Zero Matrix is Identity for Matrix Entrywise Addition/Proof 2

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

$\blacksquare$