Matrix Entrywise Addition over Ring is Associative/Proof 2

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

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

The result then follows directly from Associativity of Hadamard Product.