Section 6.1

1. $λ_{1} = 2$ $λ_{1} = 2$ : $v_{1} = (1, 1)$ $v_{1} = (1, 1)$ ; $λ_{2} = 3$ $λ_{2} = 3$ : $v_{2} = (2, 1)$ $v_{2} = (2, 1)$
2. $λ_{1} = - 1$ $λ_{1} = - 1$ : $v_{1} = (1, 1)$ $v_{1} = (1, 1)$ ; $λ_{2} = 2$ $λ_{2} = 2$ : $v_{2} = (2, 1)$ $v_{2} = (2, 1)$
3. $λ_{1} = 2$ $λ_{1} = 2$ : $v_{2} = (1, 1)$ $v_{2} = (1, 1)$ ; $λ_{2} = 5$ $λ_{2} = 5$ : $v_{1} = (2, 1)$ $v_{1} = (2, 1)$
4. $λ_{1} = 1$ $λ_{1} = 1$ : $v_{2} = (1, 1)$ $v_{2} = (1, 1)$ ; $λ_{2} = 2$ $λ_{2} = 2$ : $v_{1} = (3, 2)$ $v_{1} = (3, 2)$
5. $λ_{1} = 1$ $λ_{1} = 1$ : $v_{2} = (1, 1)$ $v_{2} = (1, 1)$ ; $λ_{2} = 4$ $λ_{2} = 4$ : $v_{1} = (3, 2)$ $v_{1} = (3, 2)$
6. $λ_{1} = 2$ $λ_{1} = 2$ : $v_{2} = (1, 1)$ $v_{2} = (1, 1)$ ; $λ_{2} = 3$ $λ_{2} = 3$ : $v_{1} = (4, 3)$ $v_{1} = (4, 3)$
7. $λ_{1} = 2$ $λ_{1} = 2$ : $v_{2} = (1, 1)$ $v_{2} = (1, 1)$ ; $λ_{2} = 4$ $λ_{2} = 4$ : $v_{1} = (4, 3)$ $v_{1} = (4, 3)$
8. $λ_{1} = - 2$ $λ_{1} = - 2$ : $v_{2} = (2, 3)$ $v_{2} = (2, 3)$ ; $λ_{2} = - 1$ $λ_{2} = - 1$ : $v_{1} = (3, 4)$ $v_{1} = (3, 4)$
9. $λ_{1} = 3$ $λ_{1} = 3$ : $v_{1} = (2, 1)$ $v_{1} = (2, 1)$ ; $λ_{2} = 4$ $λ_{2} = 4$ : $v_{2} = (5, 2)$ $v_{2} = (5, 2)$
10. $λ_{1} = 4$ $λ_{1} = 4$ : $v_{1} = (2, 1)$ $v_{1} = (2, 1)$ ; $λ_{2} = 5$ $λ_{2} = 5$ : $v_{2} = (5, 2)$ $v_{2} = (5, 2)$
11. $λ_{1} = 4$ $λ_{1} = 4$ : $v_{2} = (2, 3)$ $v_{2} = (2, 3)$ ; $λ_{2} = 5$ $λ_{2} = 5$ : $v_{1} = (5, 7)$ $v_{1} = (5, 7)$
12. $λ_{1} = 3$ $λ_{1} = 3$ : $v_{2} = (3, 2)$ $v_{2} = (3, 2)$ ; $λ_{2} = 4$ $λ_{2} = 4$ : $v_{1} = (5, 3)$ $v_{1} = (5, 3)$
13. $λ_{1} = 0$ $λ_{1} = 0$ : $v_{1} = (0, - 1, 2)$ $v_{1} = (0, - 1, 2)$ ; $λ_{2} = 1$ $λ_{2} = 1$ : $v_{2} = (0, - 1, 3)$ $v_{2} = (0, - 1, 3)$ ; $λ_{3} = 2$ $λ_{3} = 2$ : $v_{3} = (1, 0, 2)$ $v_{3} = (1, 0, 2)$
14. $λ_{1} = 0$ $λ_{1} = 0$ : $v_{1} = (0, - 1, 2)$ $v_{1} = (0, - 1, 2)$ ; $λ_{2} = 2$ $λ_{2} = 2$ : $v_{2} = (0, - 1, 3)$ $v_{2} = (0, - 1, 3)$ ; $λ_{3} = 5$ $λ_{3} = 5$ : $v_{3} = (1, 0, 2)$ $v_{3} = (1, 0, 2)$
15. $λ_{1} = 0$ $λ_{1} = 0$ : $v_{1} = (1, 1, 0)$ $v_{1} = (1, 1, 0)$ ; $λ_{2} = 1$ $λ_{2} = 1$ : $v_{2} = (2, 1, 1)$ $v_{2} = (2, 1, 1)$ ; $λ_{3} = 2$ $λ_{3} = 2$ : $v_{3} = (1, 0, 2)$ $v_{3} = (1, 0, 2)$
16. $λ_{1} = 0$ $λ_{1} = 0$ : $v_{1} = (1, 1, 1)$ $v_{1} = (1, 1, 1)$ ; $λ_{2} = 1$ $λ_{2} = 1$ : $v_{2} = (1, 1, 0)$ $v_{2} = (1, 1, 0)$ ; $λ_{3} = 3$ $λ_{3} = 3$ : $v_{3} = (- 1, 0, 2)$ $v_{3} = (- 1, 0, 2)$
17. $λ_{1} = 1$ $λ_{1} = 1$ : $v_{1} = (1, 0, 1)$ $v_{1} = (1, 0, 1)$ ; $λ_{2} = 2$ $λ_{2} = 2$ : $v_{2} = (- 1, 1, 2)$ $v_{2} = (- 1, 1, 2)$ ; $λ_{3} = 3$ $λ_{3} = 3$ : $v_{3} = (1, 0, 0)$ $v_{3} = (1, 0, 0)$
18. $λ_{1} = 1$ $λ_{1} = 1$ : $v_{1} = (1, 0, 3)$ $v_{1} = (1, 0, 3)$ ; $λ_{2} = 2$ $λ_{2} = 2$ : $v_{2} = (0, - 1, 3)$ $v_{2} = (0, - 1, 3)$ ; $λ_{3} = 3$ $λ_{3} = 3$ : $v_{3} = (0, - 2, 5)$ $v_{3} = (0, - 2, 5)$
19. $λ_{1} = λ_{2} = 1$ $λ_{1} = λ_{2} = 1$ : $v_{1} = (1, 0, 1), v_{2} = (- 3, 1, 0)$ $v_{1} = (1, 0, 1), v_{2} = (- 3, 1, 0)$ ; $λ_{3} = 3, v_{3} = (1, 0, 0)$ $λ_{3} = 3, v_{3} = (1, 0, 0)$
20. $λ_{1} = λ_{2} = 1$ $λ_{1} = λ_{2} = 1$ : $v_{1} = (1, 0, 2), v_{2} = (3, 2, 0)$ $v_{1} = (1, 0, 2), v_{2} = (3, 2, 0)$ ; $λ_{3} = 2, v_{3} = (0, - 2, 5)$ $λ_{3} = 2, v_{3} = (0, - 2, 5)$
21. $λ_{1} = 1$ $λ_{1} = 1$ : $v_{1} = (1, 1, 0)$ $v_{1} = (1, 1, 0)$ ; $λ_{2} = λ_{3} = 2$ $λ_{2} = λ_{3} = 2$ : $v_{2} = (3, 2, 0), v_{3} = (- 1, 0, 2)$ $v_{2} = (3, 2, 0), v_{3} = (- 1, 0, 2)$
22. $λ_{1} = λ_{2} = - 1$ $λ_{1} = λ_{2} = - 1$ : $v_{1} = (1, 1, 0), v_{2} = (- 1, 0, 2)$ $v_{1} = (1, 1, 0), v_{2} = (- 1, 0, 2)$ ; $λ_{3} = 2, v_{3} = (1, 1, 1)$ $λ_{3} = 2, v_{3} = (1, 1, 1)$
23. $λ_{1} = 1$ $λ_{1} = 1$ : $v_{1} = (1, 0, 0, 0)$ $v_{1} = (1, 0, 0, 0)$ ; $λ_{2} = 2$ $λ_{2} = 2$ : $v_{2} = (2, 1, 0, 0)$ $v_{2} = (2, 1, 0, 0)$ ; $λ_{3} = 3$ $λ_{3} = 3$ : $v_{3} = (3, 2, 1, 0)$ $v_{3} = (3, 2, 1, 0)$ ; $λ_{4} = 4$ $λ_{4} = 4$ : $v_{4} = (4, 3, 2, 1)$ $v_{4} = (4, 3, 2, 1)$
24. $λ_{1} = λ_{2} = 1$ $λ_{1} = λ_{2} = 1$ : $v_{1} = (1, 0, 0, 0), v_{2} = (0, 1, 0, 0)$ $v_{1} = (1, 0, 0, 0), v_{2} = (0, 1, 0, 0)$ ; $λ_{3} = λ_{4} = 3$ $λ_{3} = λ_{4} = 3$ : $v_{3} = (0, 0, 0, 1), v_{4} = (2, 2, 1, 0)$ $v_{3} = (0, 0, 0, 1), v_{4} = (2, 2, 1, 0)$
25. $λ_{1} = λ_{2} = 1$ $λ_{1} = λ_{2} = 1$ : $v_{1} = (1, 0, 0, 0), v_{2} = (0, 1, 0, 0)$ $v_{1} = (1, 0, 0, 0), v_{2} = (0, 1, 0, 0)$ ; $λ_{3} = λ_{4} = 2$ $λ_{3} = λ_{4} = 2$ : $v_{3} = (0, 0, 0, 1), v_{4} = (1, 1, 1, 0)$ $v_{3} = (0, 0, 0, 1), v_{4} = (1, 1, 1, 0)$
26. $λ_{1} = - 2$ $λ_{1} = - 2$ : $v_{1} = (1, 0, 0, 2)$ $v_{1} = (1, 0, 0, 2)$ ; $λ_{2} = - 1$ $λ_{2} = - 1$ : $v_{2} = (0, 0, 1, 0)$ $v_{2} = (0, 0, 1, 0)$ ; $λ_{3} = 1$ $λ_{3} = 1$ : $v_{3} = (1, 0, 0, 1)$ $v_{3} = (1, 0, 0, 1)$ ; $λ_{4} = 2$ $λ_{4} = 2$ : $v_{4} = (0, 1, 0, 0)$ $v_{4} = (0, 1, 0, 0)$
27. $λ_{1} = - i$ $λ_{1} = - i$ : $v_{1} = (+ i, 1)$ $v_{1} = (+ i, 1)$ ; $λ_{2} = + i$ $λ_{2} = + i$ : $v_{2} = (- i, 1)$ $v_{2} = (- i, 1)$
28. $λ_{1} = - 6 i$ $λ_{1} = - 6 i$ : $v_{1} = (- i, 1)$ $v_{1} = (- i, 1)$ ; $λ_{2} = + 6 i$ $λ_{2} = + 6 i$ : $v_{2} = (+ i, 1)$ $v_{2} = (+ i, 1)$
29. $λ_{1} = - 6 i$ $λ_{1} = - 6 i$ : $v_{1} = (- i, 2)$ $v_{1} = (- i, 2)$ ; $λ_{2} = + 6 i$ $λ_{2} = + 6 i$ : $v_{2} = (+ i, 2)$ $v_{2} = (+ i, 2)$
30. $λ_{1} = - 12 i$ $λ_{1} = - 12 i$ : $v_{1} = (- i, 1)$ $v_{1} = (- i, 1)$ ; $λ_{2} = + 12 i$ $λ_{2} = + 12 i$ : $v_{2} = (+ i, 1)$ $v_{2} = (+ i, 1)$
31. $λ_{1} = - 12 i$ $λ_{1} = - 12 i$ : $v_{1} = (+ 2 i, 1)$ $v_{1} = (+ 2 i, 1)$ ; $λ_{2} = + 12 i$ $λ_{2} = + 12 i$ : $v_{2} = (- 2 i, 1)$ $v_{2} = (- 2 i, 1)$
32. $λ_{1} = - 12 i$ $λ_{1} = - 12 i$ : $v_{1} = (- i, 3)$ $v_{1} = (- i, 3)$ ; $λ_{2} = + 12 i$ $λ_{2} = + 12 i$ : $v_{2} = (+ i, 3)$ $v_{2} = (+ i, 3)$
40. We find that $Tr A = 12$ $Tr A = 12$ and $det A = 60$ $det A = 60$ , so the characteristic polynomial of the given matrix A is $p (λ) = - λ^{3} + 12 λ^{2} + c_{1} λ + 60$ $p (λ) = - λ^{3} + 12 λ^{2} + c_{1} λ + 60$ . Eigenvalues and eigenvectors: $λ_{1} = 3$ $λ_{1} = 3$ : $v_{1} = (3, 2, 1)$ $v_{1} = (3, 2, 1)$ ; $λ_{2} = 4$ $λ_{2} = 4$ : $v_{2} = (5, 7, 7)$ $v_{2} = (5, 7, 7)$ ; $λ_{3} = 5$ $λ_{3} = 5$ : $v_{3} = (- 1, 1, 2)$ $v_{3} = (- 1, 1, 2)$
41. We find that $Tr A = 8$ $Tr A = 8$ and $det A = - 60$ $det A = - 60$ , so the characteristic polynomial of the given matrix A is $p (λ) = λ^{4} - 8 λ^{3} + c_{2} λ^{2} + c_{1} λ - 60$ $p (λ) = λ^{4} - 8 λ^{3} + c_{2} λ^{2} + c_{1} λ - 60$ . Eigenvalues and eigenvectors: $λ_{1} = - 2$ $λ_{1} = - 2$ : $v_{1} = (1, 0, 1, 2)$ $v_{1} = (1, 0, 1, 2)$ ; $λ_{2} = 2$ $λ_{2} = 2$ : $v_{2} = (3, 4, 0, 3)$ $v_{2} = (3, 4, 0, 3)$ ; $λ_{3} = 3$ $λ_{3} = 3$ : $v_{3} = (3, 1, 2, 4)$ $v_{3} = (3, 1, 2, 4)$ ; $λ_{4} = 5$ $λ_{4} = 5$ : $v_{4} = (1, 1, 0, 1)$ $v_{4} = (1, 1, 0, 1)$

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Table of Contents for Section 6.1

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Table of Contents for
Section 6.1