Mathematics IAWorked Examples

ALGEBRA: EIGENVALUES

Produced by the Maths Learning Centre,The University of Adelaide

www.adelaide.edu.au/mathslearning/

June 13, 2012

The questions on this page have written solutions on the following pages.Click on the link with each question to go straight to the relevant page.

Questions

1. See Page 4 for worked solutions.

Let A =

−5 3 0 00 2 −2 61 0 −1

23

23

1 3 −1

. Verify that v =

310−1

is an eigenvector of

A and find the eigenvalue corresponding to v.

2. See Page 5 for worked solutions.

Find the eigenvalues of the matrix B =

3 2 42 0 24 2 3

.

3. See Page 7 for worked solutions.

The matrix C =

3 0 0 00 5 −2 00 2 1 00 −1 1 2

has eigenvalues λ = 3, 3, 3, 2. Find

the eigenspaces of C.

4. See Page 9 for worked solutions.

Find the eigenvalues and eigenvectors of the matrix A =

[1 −8−2 1

].

5. See Page 11 for worked solutions.

Let B =

8 0 01 7 40 0 3

. Find the eigenspaces and eigenvalues of B by in-

spection.

1

6. See Page 13 for worked solutions.Let A be a 3×3 matrix and suppose that (1, 0, 1)>, (0, 1, 0)> and (2, 1, 0)>

are eigenvectors of A with eigenvalues 1, 3 and 3 respectively. If x =(3, 0, 1)>, find Ax.

7. See Page 13 for worked solutions.Let λ be an eigenvalue of the n × n matrix C. Show that λ2 is aneigenvalue of C2.

8. See Page 15 for worked solutions.Show that if the n × n matrix B has eigenvalues λ1, . . . , λn, then B>

also has the eigenvalues λ1, . . . , λn.

9. See Page 16 for worked solutions.Suppose C is a 5×5 matrix all of whose eigenvalues are positive integers.If the determinant of C is 12 and the trace of C is 9, find the characteristicpolynomial of C.

10. See Page 17 for worked solutions.The 2 × 2 matrix B has trace 2 and determinant 3. Does B have anyreal eigenvalues?

11. See Page 18 for worked solutions.Suppose C is a 6× 6 matrix with eigenvalues 0, 1 and 3 of multiplicities3, 2 and 1 respectively. Find the determinant of the matrix 2I − C.

12. See Page 19 for worked solutions.

(a) Let A be a 5× 5 matrix with the following eigenspaces:span{(0, 0, 0, 5,−2)>}, span{(1, 0, 0, 1, 0)>, (0, 11, 7, 0, 0)>} and span{(1, 0, 0, 0, 0)>, (0, 3, 1, 0, 1)>}.Is A diagonalisable? If so, write down a matrix P such that P−1APis a diagonal matrix.

(b) Let B be a 5× 5 matrix with the following eigenspaces:span{(0, 0, 0, 5,−2)>}, span{(1, 0, 0, 1, 0)>} and span{(1, 0, 0, 0, 0)>, (0, 3, 1, 0, 1)>}.Is B diagonalisable? If so, write down a matrix P such that P−1BPis a diagonal matrix.

13. See Page 21 for worked solutions.

Diagonalise the matrix C =

11 −3 5−4 7 102 3 8

.

14. See Page 24 for worked solutions.The vectors (1, 0, 0)>, (5, 0, 1)> and (0,−1, 2)> are eigenvectors of the3×3 matrix A, with eigenvalues 2, 1 and 1 respectively. Find the matrixA.

15. See Page 26 for worked solutions.

Let B =

[7 −124 −7

]. Find B100 by

(a) first diagonalising B.

(b) using the Cayley-Hamilton theorem.2

16. See Page 29 for worked solutions.

Let A =

2 0 0 05 1 0 00 3 −1 0−1 0 0 −2

. Use the Cayley-Hamilton theorem to find

the inverse of A.

17. See Page 31 for worked solutions.

Let B be a 3×3 matrix. Suppose

021

,

−101

and

123

are eigenvectors

of B with eigenvalues 0, 12

and 1 respectively. If x =

100

, find B10x.

3

1. Click here to go to question list.

Let A =

−5 3 0 00 2 −2 61 0 −1

23

23

1 3 −1

. Verify that v =

310−1

is an eigenvector of

A and find the eigenvalue corresponding to v.

4

2. Click here to go to question list.

Find the eigenvalues of the matrix B =

3 2 42 0 24 2 3

.

5

6

3. Click here to go to question list.

The matrix C =

3 0 0 00 5 −2 00 2 1 00 −1 1 2

has eigenvalues λ = 3, 3, 3, 2. Find

the eigenspaces of C.

7

8

4. Click here to go to question list.

Find the eigenvalues and eigenvectors of the matrix A =

[1 −8−2 1

].

9

10

5. Click here to go to question list.

Let B =

8 0 01 7 40 0 3

. Find the eigenspaces and eigenvalues of B by in-

spection.

11

12

6. Click here to go to question list.Let A be a 3×3 matrix and suppose that (1, 0, 1)>, (0, 1, 0)> and (2, 1, 0)>

are eigenvectors of A with eigenvalues 1, 3 and 3 respectively. If x =(3, 0, 1)>, find Ax.

13

7. Click here to go to question list.Let λ be an eigenvalue of the n × n matrix C. Show that λ2 is aneigenvalue of C2.

14

8. Click here to go to question list.Show that if the n × n matrix B has eigenvalues λ1, . . . , λn, then B>

also has the eigenvalues λ1, . . . , λn.

15

9. Click here to go to question list.Suppose C is a 5×5 matrix all of whose eigenvalues are positive integers.If the determinant of C is 12 and the trace of C is 9, find the characteristicpolynomial of C.

16

10. Click here to go to question list.The 2 × 2 matrix B has trace 2 and determinant 3. Does B have anyreal eigenvalues?

17

11. Click here to go to question list.Suppose C is a 6× 6 matrix with eigenvalues 0, 1 and 3 of multiplicities3, 2 and 1 respectively. Find the determinant of the matrix 2I − C.

18

12. Click here to go to question list.

(a) Let A be a 5× 5 matrix with the following eigenspaces:span{(0, 0, 0, 5,−2)>}, span{(1, 0, 0, 1, 0)>, (0, 11, 7, 0, 0)>} and span{(1, 0, 0, 0, 0)>, (0, 3, 1, 0, 1)>}.Is A diagonalisable? If so, write down a matrix P such that P−1APis a diagonal matrix.

(b) Let B be a 5× 5 matrix with the following eigenspaces:span{(0, 0, 0, 5,−2)>}, span{(1, 0, 0, 1, 0)>} and span{(1, 0, 0, 0, 0)>, (0, 3, 1, 0, 1)>}.Is B diagonalisable? If so, write down a matrix P such that P−1BPis a diagonal matrix.

19

20

13. Click here to go to question list.

Diagonalise the matrix C =

11 −3 5−4 7 102 3 8

.

21

22

23

14. Click here to go to question list.The vectors (1, 0, 0)>, (5, 0, 1)> and (0,−1, 2)> are eigenvectors of the3×3 matrix A, with eigenvalues 2, 1 and 1 respectively. Find the matrixA.

24

25

15. Click here to go to question list.

Let B =

[7 −124 −7

]. Find B100 by

(a) first diagonalising B.

(b) using the Cayley-Hamilton theorem.

26

27

28

16. Click here to go to question list.

Let A =

2 0 0 05 1 0 00 3 −1 0−1 0 0 −2

. Use the Cayley-Hamilton theorem to find

the inverse of A.

29

30

17. Click here to go to question list.

Let B be a 3×3 matrix. Suppose

021

,

−101

and

123

are eigenvectors

of B with eigenvalues 0, 12

and 1 respectively. If x =

100

, find B10x.

31

32