Question: Let \(A, B \in M_{n} .\) Suppose that \(A\) and \(B\) are diagonalizable and commute. Let \(\lambda_{1}, \ldots, \lambda_{n}\) be the eigenvalues of \(A\) and let \(\mu_{1}, \ldots, \mu_{n}\) be the eigenvalues of \(B\). (a) Show that the eigenvalues of \(A+B\) are \(\lambda_{1}+\mu_{i_{1}}, \lambda_{2}+\mu_{i_{2}}, \ldots, \lambda_{n}+\mu_{i_{n}}\), for some permutation \(i_{1}, \ldots, i_{n}\) of \(1, \ldots, n .\) (b) If \(B\) is nilpotent, explain why \(A\) and \(A+B\) have the same eigenvalues. (c) What are the eigenvalues of $A B ?$

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