By Cramer's rule, if $A$ is an $n \times n$ matrix $(A-tI)^{-1} = \dfrac{Adj(A-tI)}{\det(A-tI)}$. So if $\lambda$ is a simple real eigenvalue of the real matrix $A$, but not an eigenvalue of any of its first minors, the signs of all entries of $(A-tI)^{-1}$ will flip as $t$ crosses $\lambda$.

By Cramer's rule, if $A$ is an $n \times n$ matrix $(A-tI)^{-1} = \dfrac{Adj(A-tI)}{\det(A-tI)}$. So if $\lambda$ is a simple real eigenvalue of the real matrix $A$, but all the first minors of $A - \lambda I$ are nonzero, then the signs of all entries of $(A-tI)^{-1}$ will flip as $t$ crosses $\lambda$.