Suppose $A$ is a $n \times m$ matrix and $B$ is a $m \times n$ matrix. Then it is known that $det(I_{n}+AB)=det(I_{m}+BA)$.

Is there an analogous identity of the form $det(P_{1}+AB)=det(P_{2}+BA)$, where $P_{1},P_{2}$ are positive definite? Or something like it?

Matrices, Springer-Verlag GTM**216*. More generally, one has $$X^m\det(XI_n-AB)=X^n\det(XI_m-BA).$$ – Denis Serre Jul 5 '12 at 13:30