Matrix Multiplication is Homogeneous of Degree 1

Theorem
Let $\mathbf A$ be an $m \times n$ matrix and $\mathbf B$ be an $n \times p$ matrix such that the columns of $\mathbf A$ and $\mathbf B$ are members of $\R^m$ and $\R^n$, respectively.

Let $\lambda \in \mathbb F \in \set {\R, \C}$ be a scalar.

Then:
 * $\mathbf A \paren {\lambda \mathbf B} = \lambda \paren {\mathbf A \mathbf B}$

Proof
Let $\mathbf A = \sqbrk a_{m n}, \mathbf B = \sqbrk b_{n p}$.