matrix product equals $n\cdot I$

Question

I'm studying linear algebra on my own time. Came across the following question:

For square $A$ and $B$, if $AB=10I$ then $BA=10I$.

It is quite easy to explain if when $AB=I$ then $BA=I$, but I can't figure out how to explain it if there is a multiple. How do I approach it?

Any help is appreciated.

Answer 1

If $AB=\gamma I$ for some $\gamma\neq0$, then

$$I=\left(\frac{1}{\gamma}A\right)B=B\left(\frac{1}{\gamma}A\right),$$

and therefore $BA=\gamma I$.