280 CHAPTER 12. SPECTRAL THEORY

4. If A is the matrix of a linear transformation which rotates all vectors in R2 through30◦, explain why A cannot have any real eigenvalues.

5. If A is an n×n matrix and c is a nonzero constant, compare the eigenvalues of A andcA.

6. If A is an invertible n× n matrix, compare the eigenvalues of A and A−1. Moregenerally, for m an arbitrary integer, compare the eigenvalues of A and Am.

7. Let A,B be invertible n×n matrices which commute. That is, AB = BA. Suppose xis an eigenvector of B. Show that then Ax must also be an eigenvector for B.

8. Suppose A is an n× n matrix and it satisfies Am = A for some m a positive integerlarger than 1. Show that if λ is an eigenvalue of A then |λ | equals either 0 or 1.

9. Show that if Ax = λx and Ay = λy, then whenever a,b are scalars,

A(ax+by) = λ (ax+by) .

Does this imply that ax+by is an eigenvector? Explain.

10. Find the eigenvalues and eigenvectors of the matrix −1 −1 7−1 0 4−1 −1 5

 .

Determine whether the matrix is defective.

11. Find the eigenvalues and eigenvectors of the matrix −3 −7 19−2 −1 8−2 −3 10

 .

Determine whether the matrix is defective.

12. Find the eigenvalues and eigenvectors of the matrix −7 −12 30−3 −7 15−3 −6 14

 .

Determine whether the matrix is defective.

13. Find the eigenvalues and eigenvectors of the matrix 7 −2 08 −1 0−2 4 6

 .

Determine whether the matrix is defective.