Varieties determined by a characteristic-type of polynomial with the structure of an underlying graph

Question

While writing my master thesis, following problem came up: Given a digraph $G$ with edges $e_1,..,e_n$ and a given a $n\times n$- matrix $A\in\mathbb{C}^{n\times n}$ such that $A_{ij}=0$ if the ending vertex of $e_j$ is not the starting vertex $e_i$, i want to look at the algebraic set $\mathcal{V}(A)$ given by the following polynomial: $$ P_A(x_1,...,x_n)= \det\left(\left( \begin{array}{c c c c} x_1 & & & \\ & x_2 & & \\ & &\ddots &\\ & & & x_n \end{array}\right)-A\right) $$ First, I want to show that $P_A$ is irreducible if $A$ is irreducible and invertible, moreover I would like to know more about singularities and how one could relate properties of $G$ to properties of the (then) affine variety. I unfortunately couldn't find any literature to this, not even to varieties given by these sort of polynomials.
I'd be gratefull for any hint given!