In Exercises 1 and 2, A, B, P, and D are n x n matrices. Mark each statement True or False. Justify each answer.
1. a. A is diagonalizable if A = PDP-l for some matrix D and some invertible matrix P.
b. If Rn has a basis of eigenvectors of A, then A is diagonalizable.
c. A is diagonalizable if and only if A has n eigenvalues, counting multiplicities.
d. If A is diagonalizable, then A is invertible.
2. a. A is diagonalizable if A has n eigenvectors.
b. If A is diagonalizable, then A has n distinct eigenvalues.
c. If AP = PD, with D diagonal, then the nonzero columns of P must be eigenvectors of A.
d. If A is invertible, then A is diagonalizable.