Matrix Multiplication Distributes over Matrix Addition

Theorem
Matrix multiplication is distributive over matrix 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 $$\left({\mathbf{B} + \mathbf{C}}\right) \mathbf{A} = \left({\mathbf{B} \mathbf{A}}\right) + \left({\mathbf{C} \mathbf{A}}\right)$$.