Page 575 - Elementary_Linear_Algebra_with_Applications_Anton__9_edition

12. matrix, then the coefficient of in the
(a) It was shown in Exercise 15 of Section 7.1 that if A is an

characteristic polynomial of A is 1. (A polynomial with this property is called monic.) Show that the matrix

has characteristic polynomial . This shows that every

monic polynomial is the characteristic polynomial of some matrix. The matrix in this

example is called the companion matrix of .

Hint Evaluate all determinants in the problem by adding a multiple of the second row to the first to introduce a

zero at the top of the first column, and then expanding by cofactors along the first column.

(b) Find a matrix with characteristic polynomial .

A square matrix A is called nilpotent if for some positive integer n. What can you say about the eigenvalues of a
13. nilpotent matrix?

Prove: If A is an matrix and n is odd, then A has at least one real eigenvalue.
14.

Find a matrix A that has eigenvalues , 1, and −1 with corresponding eigenvectors
15.

respectively. matrix A has eigenvalues , , , and .

Suppose that a
16.

(a) Use the method of Exercise 14 of Section 7.1 to find .

(b) Use Exercise 5 above to find .

Let A be a square matrix such that . What can you say about the eigenvalues of A?
17.

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