Suppose that the $n$ Gersgorin discs of $A \in M_n$ are mutually disjoint. (a) If $A$ is real, show that every eigenvalue of $A$ is real. (b) If $A \in M_n$ has real main diagonal entries and its characteristic polynomial has only real coefficients, show that every eigenvalue of $A$ is real.
The fact I tried to use is that if the union of $k$ of the $n$ discs that comprise $G(A)$, the union of Gersgorin discs, forms a set $G_k(A)$ that is disjoint from the remaining $n-k$ discs, then $G_k(A)$ contains exactly $k$ eigenvalues of $A$. However, I don't think this fact leads to the result I'm trying to show. I'm not sure how to approach this problem. Any solutions, hints, or suggestions would be appreciated.
The proof of both statements is the same. If the characteristic polynomial has all real coefficients, then any complex root $\lambda$ is accompanied by its conjugate $\overline{\lambda}$. But if a disk is centered on the real axis, then it contains $\lambda$ if and only if it contains $\overline{\lambda}$. Derive a contradiction of the second part of Gerschgorin's theorem from this fact.