Matrix Entrywise Addition over Ring is Commutative/Proof 2

Proof
By definition, matrix entrywise addition is the Hadamard product of $\mathbf A$ and $\mathbf B$ with respect to ring addition.

We have from Ring Axiom $\text A 2$ that ring addition is commutative.

The result then follows directly from Commutativity of Hadamard Product.