TITLE: Matrix multiplication by scalar is commutative
QUESTION [5 upvotes]: Is matrix multiplication by scalar commutative, i.e. $(\alpha M)N=M(\alpha N)$? If so, can we prove it without induction?

REPLY [2 votes]: Yes it's true and also easy to prove. Let $f: X \rightarrow Y$ be the linear function with matrix $N$ and $g: Y \rightarrow Z$ be the linear function with matrix $M$. You have for each vector $x \in X$: $$\begin{align}((\alpha M)N)x & = ((\alpha g) \circ f)(x) \\
& = (\alpha g)(f(x)) \\
& = \alpha \cdot g(f(x)) \\
& \left\downarrow \ g\text{ is linear}\right. \\
& = g(\alpha \cdot f(x)) \\
& = (g \circ \alpha f)(x) \\
& = (M(\alpha N))(x) \end{align}$$