Let $A, B, C$ be invertible $n \times n$ matrices. Prove that $(A^T B)^{-1} = B^{-1}(A^{-1})^T$

Question

Let $A, B, C$ be invertible $n \times n$ matrices. Prove that $$(A^T B)^{-1} = B^{-1}(A^{-1})^T$$

I'm confused on how to start the problem and was wondering if the answer ends up being $I$?

Answer 1

Hint:

$$(CD)^{-1}=D^{-1}C^{-1}$$ $$(A^T)^{-1}=(A^{-1})^T$$

Answer 2

Just multiply: $$ (A^TB)(B^{-1}(A^{-1})^T)= A^T(BB^{-1})(A^{-1})^T= A^T(A^{-1})^T= (A^{-1}A)^T=I^T=I $$