\displaystyle\begin{aligned}  \det(R-I)&=\det((R-I)^T)=\det(R^T-I)=\det(R^{-1}-I)\\  &=\det(R^{-1}(I-R))=\det(R^{-1})\det(I-R)\\  &=-(\det R^{-1})\det(R-I)=-\det(R-I),  \end{aligned}