Let A P and D be n times n matrices Which of the following i

Let A, P, and D be n times n matrices. Which of the following is false? If A is diagonalizable, then there are n distinctive eigenvalues of A. If PA = DP with D diagonal, then the nonzero columns of P are eigenvectors of A. If A has n distinct eigenvectors, then A is diagonalizable. If eigenvectors of A form a basis for R^n, then A is diagonalizable.

Solution

If A is diagonalizable, then there are n distinct eigenvalues of A. This statement is False. A can have some eigenvalues of multiplicity more than 1 and yet A can be diagonalizable if it has n distinct, linearly independent eigenvectors. The statement is False. The entries on the leading diagonal of D ( i.e. the eigenvalues of A) and the columns comprising the corresponding eigenvectors of A must be in the same order. The statement is False. The n eigenvectors must be linearly independent. The statement is True if A has real entries and its eigenvectors/eigenvalues are real. If the eigevectors of A form a basis for Rn, then these are distinct and linearly independent