linear algebra - If $AB = I$ then $BA = I$: is my proof right?

I want to prove that for matrices $A,B \in M_n (\mathbb K)$ where $\mathbb K \in \{\mathbb R, \mathbb C, \mathbb H\}$ if $AB = I$ then $BA = I$.

My proof is really short so I'm not sure it's right:

If $AB = I$ then $(BA)B = B$ and therefore $BA=I$?

Answer

The implication $(BA)B=B \Rightarrow BA=I$ is a little quick and not always true...

But observe that

$$1=\det(BA)= \det(B)\det(A)$$
thus $B$ is invertible and it follows that
$$BA= BA(BB^{-1}) = B(AB)B^{-1}=BB^{-1}=I.$$