How to prove the statement "If = ⁻¹, and is nonsingular, then det() = det()" if it is true or false?

Question

Statement: If = ⁻¹, and is nonsingular, then det() = det()

Similar Example: Suppose A and B are similar matrices. Then there exists an invertible matrix P such that B = PAP^-1

solution: det B = det(PAP^-1) = (det P)(det A)(det P^-1) = (det P)(det A)(det P)^-1 = det A

Now, how can I prove the given statement above if it is true or false by using the same logic on this similar example?