Proceeding as per the given equation,
PAP' = -A'
det(PAP') = det(-A') which equals,
det(P)det(A)det(P') = - det(A') Now since, det(X)=det(X'), I have the following equation as a result.
(det(P)^2) = -1
How does this corresponds to the above equations? Did I proceed correctly?
If $\det(A)\neq0$, then you got that $\det^2(P)=-1$, which is impossible. Therefore, $\det(A)=0$. So $\det(P)$ can be any real number.