Page 549 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Hint See the proof of Theorem 7.1.4.

Let A be an matrix.
15.

(a) Prove that the characteristic polynomial of A has degree n.

(b) Prove that the coefficient of in the characteristic polynomial is 1.

Show that the characteristic equation of a matrix A can be expressed as , where is the
16. trace of A.

Use the result in Exercise 16 to show that if
17.

then the solutions of the characteristic equation of A are

Use this result to show that A has
(a) two distinct real eigenvalues if

(b) two repeated real eigenvalues if

(c) complex conjugate eigenvalues if

Let A be the matrix in Exercise 17. Show that if and , then eigenvectors of A corresponding to the
18. eigenvalues

are

respectively. , then

Prove: If a, b, c, and d are integers such that
19.

has integer eigenvalues—namely, and .

Prove: If is an eigenvalue of an invertible matrix A, and x is a corresponding eigenvector, then is an eigenvalue of
20. , and x is a corresponding eigenvector.