Prove or Disprove Invertible Matrix Identitiy

Question

Prove or disprove:

if $A,B\in M_{n\times n}(F)$ are invertible, and also $A^{-1}B^{-1}AB=cI_n$, $c \in F$ then $c^n=1$.

I think it has to do with multiplying both sides by $A^{-1}B^{-1}$ but I don't know what I can do from there.

Answer 1

HINT

Take a determinant from both sides of the equation.

Remember: $\det(AB)=\det(A) \det(B)$, $\det(I)=1$ and $\det(cA)=c^n \det(A)$.