Page 573 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

Chapter 7

Supplementary Exercises

1. , then
(a) Show that if

has no eigenvalues and consequently no eigenvectors.
(b) Give a geometric explanation of the result in part (a).

Find the eigenvalues of
2.

3.
(a) Show that if D is a diagonal matrix with nonnegative entries on the main diagonal, then there is a matrix S such

that .

(b) Show that if A is a diagonalizable matrix with nonnegative eigenvalues, then there is a matrix S such that .

(c) Find a matrix S such that , if

Prove: If A is a square matrix, then A and have the same characteristic polynomial.
4.

Prove: If A is a square matrix and is the characteristic polynomial of A, then the coefficient of

5. in is the negative of the trace of A.

Prove: If , then
6.