Matrix Multiplication is Homogeneous of Degree 1

Theorem
Let $\mathbf A$ be an $m \times n$ matrix, $\mathbf B$ 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 $\lambda \in \mathbb F \in \left \{\R, \C \right\}$ be a scalar.

Then:

Proof
Let $\mathbf A = \left[{a}\right]_{m n}, \mathbf B = \left[{b}\right]_{n p}$