For each part, determine if you believe the statement is true or false, and justify your answers as appropriate.(linear algebra)

Question

For each part, determine if you believe the statement is true or false, and justify your answers as appropriate.

(a) Let k ≥ 1 be an integer. If λ is an eigenvalue of a square matrix A, then λ^k is an eigenvalue of A^k .

(b) Let A be an n × n matrix A. If A is invertible, then it is diagonalizable.

(c) Let A be an n × n matrix A. If A is diagonalizable, then it is invertible

Reminder: An appropriate justification for one of the true/false problems is always either a short proof, or a counterexample along with an explanation of why that counterexample works. It’s up to you to determine which of those is appropriate for each question, but your answer must be of one of those two types.

Answer 1

a) This one, I would say, try this out! Set up the equation for eigenvector x associated with eigenvalue λ:

y = A^k x

Then, go through one by one converting (Ax) -> (λx) (since we can always rearrange (Aλx) -> (λAx) since λ is scalar). What is y? If it's (λ^kx) you have your answer. But be careful, because if λ^k is not an eigenvalue here, that does not disprove the statement.

b/c) Try to think of counterexamples for both of these. I can help more if necessary. (b) might require a little more thought, but (c) you only need a zero on the diagonal of D for A = PDP^-1 to make things complicated for invertibility. Try it out! You only need a 2x2 matrix to prove either.