Nice short counterexamples relating to spectra

Question

I was wondering if there are short simple examples to show that $\sigma(AB) \not \subseteq \sigma(A)\sigma(B)$ and $\sigma(A + B) \not \subseteq \sigma(A) + \sigma(B)$.

Answer 1

Just about any two $2 \times 2$ matrices that do not commute will give you an example.

E.g. for a random-ish example take $$A = \pmatrix{0 & 2\cr 0 & 2\cr},\ B = \pmatrix{1 & 0\cr -1 & 0\cr}$$ $$ \sigma(A) \sigma(B) = \{0,2\} \cdot \{1,0\} = \{0,2\},\ \sigma(A) + \sigma(B) = \{0,1,2,3\}$$ $$ \sigma(AB) = \{0, -2\},\ \sigma(A+B) = \{(3 \pm i \sqrt{7})/2\}$$