Matrix Entrywise Addition is Commutative/Proof 2

Theorem

For $\mathbf A, \mathbf B \in \map \MM {m, n}$, let $\mathbf A + \mathbf B$ be defined as the matrix entrywise sum of $\mathbf A$ and $\mathbf B$.

The operation $+$ is commutative on $\map \MM {m, n}$.

That is:

$\mathbf A + \mathbf B = \mathbf B + \mathbf A$

for all $\mathbf A$ and $\mathbf B$ in $\map \MM {m, n}$.

Let $\mathbf A = \sqbrk a_{m n}$ and $\mathbf B = \sqbrk b_{m n}$ be matrices whose order is $m \times n$.

Then:

$\ds \mathbf A + \mathbf B$

$=$

$\ds \sqbrk a_{m n} + \sqbrk b_{m n}$

Definition of $\mathbf A$ and $\mathbf B$

$\ds $

$=$

$\ds \sqbrk {a + b}_{m n}$

$\ds $

$=$

$\ds \sqbrk {b + a}_{m n}$

$\ds $

$=$

$\ds \sqbrk b_{m n} + \sqbrk a_{m n}$

$\ds $

$=$

$\ds \mathbf B + \mathbf A$

Definition of $\mathbf A$ and $\mathbf B$

$\blacksquare$