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Let A and B be n × n matrices with eigenvalues λ and μ, respectively.

(a) Give an example to show that λ + μ need not be an eigenvalue of A + B.

(b) Give an example to show that λμ need not be an eigenvalue of AB.

(c) Suppose λ and μ correspond to the same eigenvector x. Show that, in this case, λ + μ is an eigenvalue of A + B and λμ is an eigenvalue of AB.

(a) Give an example to show that λ + μ need not be an eigenvalue of A + B.

(b) Give an example to show that λμ need not be an eigenvalue of AB.

(c) Suppose λ and μ correspond to the same eigenvector x. Show that, in this case, λ + μ is an eigenvalue of A + B and λμ is an eigenvalue of AB.

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