)a, p and d are n×n matrices. check the true statements below: a. if a is diagonalizable, then a has n distinct eigenvalues. b. if ap=pd, with d diagonal, then the nonzero columns of p must be eigenvectors of a. c. if a is invertible, then a is diagonalizable. d. a is diagonalizable if a has n distinct eigenvectors.

A. False. Consider the identity matrix, which is diagonalizable (it's already diagonal) but all its eigenvalues are the same (1).

b. True. Suppose is the matrix of the eigenvectors of , and is the diagonal matrix of the eigenvalues of :





Then



In other words, the columns of are , which are identically , and these are the columns of .

c. False. A counterexample is the matrix



which is nonsingular, but it has only one eigenvalue.

d. False. Consider the matrix



with eigenvalue and eigenvector , where . But the matrix can't be diagonalized.