Let $A,B,C,D$ be four complex $n\times n$ matrices such that $AB^T$ and $CD^T$ are symmetric and $AD^T-BC^T=I$. Show that $A^TD-C^TB=I$.
The point is that I can't see how any theory can be used here. It appears to be a mere exercise of calculus which, either way, I'm not able to solve.
I think that some multiplications must be done, in order to use the fact that the identity is involved (and not just show that $A^TD-C^TB=AD^T-BC^T)$. I tried to get help from particular cases, but found nothing.
