Page 563 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Let .
22.

Show that:

(a) A is diagonalizable if

(b) A is not diagonalizable if .

Hint See Exercise 17 of Section 7.1.

In the case where the matrix A in Exercise 22 is diagonalizable, find a matrix P that diagonalizes A.
23.

Hint See Exercise 18 of Section 7.1.

Prove that if A is a diagonalizable matrix, then the rank of A is the number of nonzero eigenvalues of A.
24.

Indicate whether each statement is always true or sometimes false. Justify your answer by giving
25. a logical argument or a counterexample.

(a) A square matrix with linearly independent column vectors is diagonalizable.

(b) If A is diagonalizable, then there is a unique matrix P such that is a diagonal
matrix.

(c) If , , and come from different eigenspaces of A, then it is impossible to express
as a linear combination of and .

(d) If A is diagonalizable and invertible, then is diagonalizable.

(e) If A is diagonalizable, then is diagonalizable.

Suppose that the characteristic polynomial of some matrix A is found to be
26. . In each part, answer the question and explain your reasoning.