Page 550 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Prove: If is an eigenvalue of A, x is a corresponding eigenvector, and s is a scalar, then is an eigenvalue of ,
21. and x is a corresponding eigenvector.

Find the eigenvalues and bases for the eigenspaces of
22.

Then use Exercises 20 and 21 to find the eigenvalues and bases for the eigenspaces of
(a)
(b)
(c)

23.
(a) Prove that if A is a square matrix, then A and have the same eigenvalues.

Hint Look at the characteristic equation .

(b) Show that A and need not have the same eigenspaces.
Hint Use the result in Exercise 18 to find a matrix for which A and have different eigenspaces.

Indicate whether each statement is always true or sometimes false. Justify your answer by giving
24. a logical argument or a counterexample. In each part, A is an matrix.

(a) If for some nonzero scalar , then x is an eigenvector of A.

(b) If is not an eigenvalue of A, then the linear system has only the trivial
solution.

(c) is an eigenvalue of A, then is singular.

(d) If the characteristic polynomial of A is , then A is invertible.

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