Matrix Entrywise Addition is Associative/Proof 1

Proof
From:
 * Integers form Ring
 * Rational Numbers form Ring
 * Real Numbers form Ring
 * Complex Numbers form Ring

the standard number systems $\Z$, $\Q$, $\R$ and $\C$ are rings.

Hence we can apply Matrix Entrywise Addition over Ring is Associative.

The above cannot be applied to the natural numbers $\N$, as they do not form a ring.

However, from Natural Numbers under Addition form Commutative Monoid, the algebraic structure $\struct {\N, +}$ is a commutative monoid.

By definition, matrix entrywise addition is the Hadamard product with respect to addition of numbers.

The result follows from Associativity of Hadamard Product.