Let A and B be n n matrices with eigenvalues and , respectively (a) Give an example to show that need not be an eigenvalue of A B (b) Give an example to show that need not be an eigenvalue of AB (c) Suppose and correspond to the same eigenvector x Show that, in this case, is an eigenvalue of A B an...

a There are many examples For instance let Then the characteristic e ...

Let A and B be n n matrices with eigenvalues and , respectively. (a) Give

Question:

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.