transpose and inverse multiplication

Question

Given:

$$A_{(n,n)} , B_{(n,n)}$$ A and B are invertible, is it possible that :

$$(A^t B^t)^{-1} A^{-1} B^{-1} = I$$

I guess no, should this be true only if the AB=BA= orthogonal matrix ?

Answer 1

Certainly it is possible.
The condition is equivalent to $I = B A (B A)^t$, i.e. that $BA$ is an orthogonal matrix. No need for $BA = AB$, or for $AB$ to be orthogonal.