For invertible matrices $A$ and $C$, prove or disprove that $(C^{-1}BAB^T + I)$ is invertible.

Question

For invertible matrices $A$ and $C$, ($B$ may or may not be invertible) prove or disprove that $(C^{-1}BAB^T + I)$ is invertible.

This problem came up while I was proving an equivalence. I couldn't find a way to prove that it is correct. Can you think of a way to prove it to be correct?

Answer 1

Try, for example, $B=C=I$, $A=-I$.