Possible matrix-determinant identity Is $$\det(I+ ABB^*A^*C^{-1})=\det(I+ B^*A^*ABC^{-1})$$
where $I$ is identity matrix, $A,B,C$ are complex valued matrices. And $C$ is $(I+X)$ where $X$ is PSD.
I know that this makes $ABB^*A^*$ and $B^*A^*AB$ positive semi/definite. And I know for square matrices $X,Y,Z$ we have $\det(XYZ)=\det(YXZ)$ so $\det (ABB^*A^*C^{-1})=\det(B^*A^*ABC^{-1})$. And also $\det (I + ABB^*A^*C^{-1})=\det(I + B^*A^*ABC^{-1})$. Please comment if this is correct. I have a feeling I'm missing something since I didn't use properties of $C$. 
Simulations seem to hold. At least when I used $A=\pmatrix{1&1\\ 0&0},\ B=I$ and $C=\pmatrix{2&0\\ 0&2}$ gave 2 in both sides.
Thanks a lot in advance.
PS. Not home work.

Edit: Try $A=\pmatrix{1&1\\ 0&0},\ B=I,\ C=\pmatrix{2&1\\ 1&2}$ and $C^{-1}=\frac13\pmatrix{2&-1\\ -1&2}$. The two determinants are $\frac53$ and $\frac73$, which are not equal. For questions like this, you may first do a few numerical experiments with a computer to see if the hypothesis can be easily falsified.
