Let $A$, $B$ be $n\times n$ complex matrices and $I$ be the $n\times n$ identity matrix.
Is $\left(\begin{array}{cc}A&I\\I&B\end{array}\right)$ being invertible the same as $\det(AB-I)\ne 0$?
If not, what is the right condition for this $2n\times 2n$ matrix to be invertible? Thank you.