Matrix Multiplication Distributes over Matrix Addition

Theorem
Matrix multiplication (conventional) is distributive over matrix entrywise addition.

Proof
Let $$\mathbf A = \left[{a}\right]_{m n}, \mathbf B = \left[{b}\right]_{n p}, \mathbf C = \left[{c}\right]_{n p}$$ be matrices over a ring $$\left({R, +, \circ}\right)$$.

Consider $$\mathbf A \left({\mathbf B + \mathbf C}\right)$$.

Let $$\mathbf R = \left[{r}\right]_{n p} = \mathbf B + \mathbf C, \mathbf S = \left[{s}\right]_{m p} = \mathbf A \left({\mathbf B + \mathbf C}\right)$$.

Let $$\mathbf G = \left[{g}\right]_{m p} = \mathbf A \mathbf B, \mathbf H = \left[{h}\right]_{m p} = \mathbf A \mathbf C$$.

Then:

$$ $$ $$ $$ $$

Thus $$\mathbf A \left({\mathbf B + \mathbf C}\right) = \left({\mathbf A \mathbf B}\right) + \left({\mathbf A \mathbf C}\right)$$.

A similar construction shows that $$\left({\mathbf B + \mathbf C}\right) \mathbf A = \left({\mathbf B \mathbf A}\right) + \left({\mathbf C \mathbf A}\right)$$.