Given that $A$ is a complex square matrix of order $n$, $\lambda$ is an eigenvalue of $A$ with geometric and algebraic multiplicity $1$, and $x,y$ are entrywise nonzero vectors such that $Ax=\lambda x$ and $y^*A=\lambda y^*$. Show that every proper principal submatrix of $\lambda I-A$ has nonzero determinant.
I know that $\operatorname{adj}(\lambda I -A) = \gamma xy^*$ where $\gamma$ is nonzero, hence has only nonzero entries. So I know every principal submatrix of $\lambda I-A$ of size $n-1$ has nonzero determinant. I don't know how to prove this is true for smaller principal submatrices.
Any help is appreciated.