Relationship between a matrix, its characteristic polynomial and the convex hull formed by its roots?

Question

Suppose I have a matrix $\mathcal{A} \in \mathbb{R}^{n \times n}$, whose row sum is zero. The matrix is non-symmetric and has a simple zero as its eigenvalue. Can I comment about the convex hull formed by the roots of the characteristic polynomial of $\mathcal{A}$. Does there exists some relationship between the matrix, its characteristic polynomial and the convex hull formed by its roots?(Like when can it form a 'n'-sided polygon)