n Exercises 21 and 22 A B P and D are n x n matrices Mark ea

n Exercises 21 and 22, A, B, P, and D are n x n matrices. Mark each statement True cr False. Justify each answer. (Study Theorems 5 and 6 and the examples in this section carefully before you try these exercises.) a. A is diagonalizable if A has n eigenvectors. b. If A is diagonalizable.then A has n distinct eigenvalues. If AP with D diagonal, then the nonzero columns PD c. of P must be eigenvectors of A. d. If A is invertible, then A is diagonalizable.

Solution

a. false

because A is diagonalizable iff A has n linearly independent eigen vectors.

in this statement linearly independent is not mentioned.

if these are linearly independent then it is true.

b.

false

It may have repeated eigenvalues as long as the basis of each eigenspace is equal to the multiplicity of that eigenvalue. The converse is true however.

c.

true

Each column of PD is a column of P times A and is equal to the corresponding
entry in D times the vector P.This satisfies the eigenvector definition as long as the column is nonzero.

d.

false

these are not related.