6.3. EXERCISES 155

18. Find the eigenvalues and eigenvectors of the matrix

 2 1 −1

2 3 −2

2 2 −1

 . Determine

whether the matrix is defective.

19. Find the complex eigenvalues and eigenvectors of the matrix

 4 −2 −2

0 2 −2

2 0 2

 .

20. Find the eigenvalues and eigenvectors of the matrix

 9 6 −3

0 6 0

−3 −6 9

 . Determine

whether the matrix is defective.

21. Find the complex eigenvalues and eigenvectors of the matrix

 4 −2 −2

0 2 −2

2 0 2

 . De-

termine whether the matrix is defective.

22. Find the complex eigenvalues and eigenvectors of the matrix

 −4 2 0

2 −4 0

−2 2 −2

 .

Determine whether the matrix is defective.

23. Find the complex eigenvalues and eigenvectors of the matrix

 1 1 −6

7 −5 −6

−1 7 2

 .

Determine whether the matrix is defective.

24. Find the complex eigenvalues and eigenvectors of the matrix

 4 2 0

−2 4 0

−2 2 6

 . Deter-

mine whether the matrix is defective.

25. Here is a matrix. 1 a 0 0

0 1 b 0

0 0 2 c

0 0 0 2

Find values of a, b, c for which the matrix is defective and values of a, b, c for which itis nondefective.

26. Here is a matrix.  a 1 0

0 b 1

0 0 c

where a, b, c are numbers. Show this is sometimes defective depending on the choiceof a, b, c. What is an easy case which will ensure it is not defective?