# Distributive Property Of Matrix Multiplication

When solving an exercise I have made the following step where $\alpha,\beta \in \mathbb{F}$ and $A,B,T\in M_{n\times n}$

$$(\alpha A+\beta B)T=\alpha AT+\beta BT$$

Then I recalled the distributivity is not a property of a vector space, I know that left/right distributivity hold for matrices multiplication. So there must be vector space with "Multiplication" that has no distributivity? Or there is just left/right distributivity in vector spaces?

• @Jack sorry my bad it is n by n matrices
– gbox
Jan 3, 2018 at 15:28

Assuming dimensions work out, look at what feeding a vector on both sides does, you will see that it is the same, maybe setting $Tv=w$, another vector, will make things clearer $$(\alpha A+\beta B)Tv=(\alpha A+\beta B)w\stackrel{\text{linearity}}{=} \alpha Aw+\beta Bw\\ =\alpha ATv+\beta BTv\implies (\alpha A+\beta B)T=\alpha AT+\beta BT$$