12.5. EXERCISES 283
27. Find the complex eigenvalues and eigenvectors of the matrix 4 2 0−2 4 0−2 2 6
.
Determine whether the matrix is defective.
28. Let A be a real 3×3 matrix which has a complex eigenvalue of the form a+ ib whereb ̸= 0. Could A be defective? Explain. Either give a proof or an example.
29. Let T be the linear transformation which reflects vectors about the x axis. Find amatrix for T and then find its eigenvalues and eigenvectors.
30. Let T be the linear transformation which rotates all vectors in R2 counterclockwisethrough an angle of π/2. Find a matrix of T and then find eigenvalues and eigenvec-tors.
31. Let A be the 2×2 matrix of the linear transformation which rotates all vectors in R2
through an angle of θ . For which values of θ does A have a real eigenvalue?
32. Let T be the linear transformation which reflects all vectors in R3 through the xyplane. Find a matrix for T and then obtain its eigenvalues and eigenvectors.
33. Find the principal direction for stretching for the matrix
139
215
√5 8
45
√5
215
√5 6
5415
845
√5 4
156145
.
The eigenvalues are 2 and 1.
34. Find the principal directions for the matrix52 − 1
2 0
− 12
52 0
0 0 1
35. Suppose the migration matrix for three locations is .5 0 .3
.3 .8 0
.2 .2 .7
.
Find a comparison for the populations in the three locations after a long time.