If invertible matrices $A$, $B$ satisfy $AB=cBA$, then $c^n=1$

Question

Suppose $A$ and $B$ are invertible $n\times n$ matrix such that $AB=cBA$ where $c\in\mathbb{C}$. Show that $c^n=1$.

I have tried this question from different angles. $n=1$ is obvious. But even for $n=2$, I am not able to show this, even after multiplying, squaring the equations etc.

Answer 1

$$AB=cBA\implies \det(AB)=c^n\det(BA)\implies c^n=1.$$