proofs regarding the spectrum of sets.

Question

Let $A,B ∈ M_n(ℝ).$

Suppose $α \in (A)$ and $β \in (B)$. i.e. suppose they are an eigenvalue of A and B respectively.

Prove or disprove : $α+β \in (A+B).$

Answer 1

What are the spectra of the (diagonal!) matrices

$$A=\begin{pmatrix}1&0\\0&0\end{pmatrix} \ \ \text{and} \ \ B=\begin{pmatrix}0&0\\0&1\end{pmatrix} \ ?$$

and what is the spectra of $A+B$ ?

Conclusion ?