How to prove that sub-level set of spectral radius function is not convex

Question

Let $A \in \mathbb{R}^{n \times n}$ and $\rho(A)=\{|\lambda|_\max : \lambda \text{ is an eigen-value of } A\}$. I am trying to prove that the set \begin{align*} B_\alpha=\{A: \rho(A) < \alpha \} \end{align*} need not be a convex set in general (for any $\alpha>0$). Can anyone provide a useful hint about how to proceed?

Answer 1

Note that the $2 \times 2$ matrices $\pmatrix{0 & 0\cr 1 & 0\cr}$ and $\pmatrix{0 & 1\cr 0 & 0\cr}$ have spectral radius $0$, but their average has positive spectral radius.