I'm currently obsessed with the identity det $\det (I \mathbf{I} - At)^{-1} \mathbf{A}t)^{-1} = \exp tr \text{tr } \log (I \mathbf{I} - At)^{-1}. \mathbf{A}t)^{-1}$. It's straightforward to prove algebraically, but its combinatorial meaning is very interesting.