What is the determinant of

9 11 17 19

0% 0%0%0%

21

37

1. 9

2. 11

3. 17

4. 19

What is the determinant of

0 28 44 -28

0% 0%0%0%

032

241

026

1. 0

2. 28

3. 44

4. -28

Which matrix represents the following system of equations?

x = 4y = 7

1 2 3

0% 0%0%

71

411.

11112.

710

4013.

What is the solution to the following system of equations?

x1 + x2 = 3 2x1 - 6x2 = -10

x1=

-9/8

and x

2...

x1=

4 an

d x2=

5

x1=

1 an

d x2=

2

x1=

1 an

d x2=

½

Ther

e ar

e an

i...

Ther

e ar

e no

s...

0% 0% 0%0%0%0%

1. x1=-9/8 and x2=-30/8

2. x1=4 and x2=5

3. x1=1 and x2=2

4. x1=1 and x2=½

5. There are an infinite number of solutions

6. There are no solutions

What is the solution to the following system of equations?

6x1 + 9x2 = 32x1 + 3x2 = 5

x1=

2 an

d x2=

0

x1=

0 an

d x2=

1/...

x1=

1 an

d x2=

-1...

x1=

-5/2

and x

2...

Ther

e ar

e an

i...

Ther

e ar

e no

s...

0% 0% 0%0%0%0%

1. x1=2 and x2=0

2. x1=0 and x2=1/3

3. x1=1 and x2=-1/3

4. x1=-5/2 and x2=2

5. There are an infinite number of solutions

6. There are no solutions

Which of the following statements are true?(A) sum of eigenvalues = sum diagonal elements (trace)

(B) product of eigenvalues = determinant of square matrix A(C) distinct eigenvalues = linearly dependent eigenvectors

Only

(A)

Only

(B)

Only

(C)

Both

(A) a

nd (..

.

Both

(A) a

nd (..

.

Both

(B) a

nd (..

.

(A),

(B) a

nd (...

0% 0% 0% 0%0%0%0%

1. Only (A)

2. Only (B)

3. Only (C)

4. Both (A) and (B)

5. Both (A) and (C)

6. Both (B) and (C)

7. (A), (B) and (C)

Which of the following statements is true?

1 2 3 4

0% 0%0%0%

1

eigenvaluean has A 1-

1.

eigenvaluean has k)-(A 2.

1

eigenvaluean has kI)-(A 1-

3.

trueare above theof None4.

Dominant eigenvalue =

eigenvalue with largest magnitude

Tru

e

Fal

se

Don’t

Know

0% 0%0%

1. True

2. False

3. Don’t Know

What is the characteristic equation of the following matrix?

None

of the

ab.

..

0% 0%0%0%

14

26

1. λ² - 7λ + 6 = 0

2. λ² - 7λ + 14 = 0

3. λ² - 7λ - 2 = 0

4. None of the above

What are the eigenvalues of

2 a

nd 4

0 a

nd 4

0 a

nd 2

6 a

nd 8

0% 0%0%0%

20

04

1. 2 and 4

2. 0 and 4

3. 0 and 2

4. 6 and 8

What are the eigenvalues of the following matrix?

0% 0%0%0%

22

48

1. λ = 2, 4

2. λ = 2, 6

3. λ = 2, 8

4. λ = 4, 8

What are the eigenvalues for the following matrix?

0% 0%0%0%

505

055

550

1. λ = -5, 0, 5

2. λ = -5, 5, 10

3. λ = 0, 5, 5

4. λ = 5, 5, 10

What are the eigenvalues for the following matrix?

0% 0%0%0%

1515

030

103

1. λ = -2, 3, 4

2. λ = -6, -2, 3

3. λ = -2, 3, 6

4. λ = -6, -3, 2

Matrix A given below has eigenvalues λ = 2, 4, 6. Without further calculation

write down the eigenvalues for .

1 2 3 4

0% 0%0%0%

-1A

000

131

113

A

3,2,11.

1,1,3

12.

1,3

2,

3

13.

6

1,

4

1,

2

14.

Which of the vectors below is an eigenvector, corresponding to the

eigenvalue λ= 7 of the matrix

1 2 3 4

0% 0%0%0%

54

23

1

2

1.

2

1

2.

1

2

3.

2

1

4.

Which of the vectors below is an eigenvector, corresponding to the

eigenvalue λ= 3 of the matrix

1 2 3 4

0% 0%0%0%

421

130

241

0

1

21.

0

1

22.

0

2

13.

0

2

14.

What are the eigenvectors of the following matrix?

1 2 3 4

0% 0%0%0%

11

55

5

1,

1

11.

5

1,

1

12.

5

1,

1

13.

1

5,

1

14.

Which of the following shows the eigenvectors for matrix A?

1 2 3 4

0% 0%0%0%

200

011

102

A

0

1

3

,

0

1

01.

0

1

3

,

0

1

02.

0

1

1

,

0

1

3

,

0

1

03.

0

1

1

,

0

1

3

,

0

1

04.

Which set of vectors is linearly independent?

(1,6

), (3

,18)

(1,2

), (3

,4)

None

of the

ab.

..

Both

of t

he se

...

0% 0%0%0%

1. (1,6), (3,18)

2. (1,2), (3,4)

3. None of the above

4. Both of the sets

Normalise the eigenvector X.

1 2 3 4

0% 0%0%0%

1

2

3

X

2

122

3

1.

22

12

18

32.

3

13

21

3.

22

24

624.

Diagonalization means which of the following?

Addin

g th

e dia

...

Multi

plyi

ng th...

Tra

nsform

ing

a...

None

of the

ab.

..

0% 0%0%0%

1. Adding the diagonal elements of a matrix.

2. Multiplying the diagonal elements of a matrix.

3. Transforming a non-diagonal matrix.

4. None of the above.

Why might we want to diagonalize a matrix?

Com

puting p

owe...

Eas

y to

find e

...

Both

of t

hese

...

None

of the

se ..

.

0% 0%0%0%

1. Computing powers of the matrix becomes easy.

2. Easy to find eigenvalues of a diagonal matrix.

3. Both of these reasons.

4. None of these reasons

You can always diagonalize an n x n matrix with n distinct

eigenvalues.

Tru

e

Fal

se

Don’t

Know

0% 0%0%

1. True

2. False

3. Don’t Know

Below are eigenvectors of four 2x2 matrices. Which matrix is definitely

diagonalizable?

1 2 3 4

0% 0%0%0%

3

0,

1

01.

3

0,

0

12.

3

1,

3

13.

3

3,

1

14.

Obtain the modal matrix P.

1 2 3 4

0% 0%0%0%

.21

01

A

11

011.

11

012.

11

433.

11

414.

The matrix A= has eigenvalues

-1 and 2 with respective eigenvectors

If calculate .

0% 0%0%0%

20

31

1

1 and

0

1

10

111P

11

1 APP

21

411.

20

212.

20

413.

20

014.

What is A²?

1 2 3 4

0% 0%0%0%

54

32A

108

641.

2516

942.

3728

21163.

4123

23134.

What is ?

1 2 3 4

0% 0%0%0%

54

32A 5A

2520

15101.

31251024

243322.

1423710796

809761403.

809726140

14237107964.

The eigenvalues of a symmetric matrix with real elements are...

Alw

ays

com

plex

Alw

ays

real

Eith

er c

omple

x...

0% 0%0%

1. Always complex

2. Always real

3. Either complex or real

Which of the following is a symmetric matrix?

1 2 3 4

0% 0%0%0%

272

735

2511.

14

412.

473

732

3213.

31

644.

A square matrix A is said to be orthogonal if

True

False

Don’t K

now

0% 0%0%

1. True

2. False

3. Don’t Know

T-1 AA

Two n x 1 column vectors X and Y are orthogonal if XY=0

Tru

e

Fal

se

Don’t

Know

0% 0%0%

1. True

2. False

3. Don’t Know

The eigenvalues of a symmetric matrix A are λ=0 and λ=10

X and Y are the eigenvectors for λ=0 and λ=10 respectively. Are X and Y orthogonal?

0% 0%0%

1. Yes

2. No

3. Don’t Know

13

39A

An Hermitian matrix is one satisfying

True

False

Don’t K

now

0% 0%0%

AAT

1. True

2. False

3. Don’t Know

Is the following matrix Hermitian?

Yes N

o

Don’t

Know

0% 0%0%

1. Yes

2. No

3. Don’t Know

513

102

323

i

i

ii

A

Separating the variables in gives

1 2 3 4

0% 0%0%0%

ss 2

tAets 2)( 1.

sAets 22)( 2.

tAets 22)( 3.

tAets 2)( 4.

Write in matrix form the pair of coupled differential equations

1 2 3 4

0% 0%0%0%

yxy

yxx

5

32

y

x

y

x

13

52

1.

y

x

y

x

13

52

2.

y

x

y

x

15

32

3.

y

x

y

x

15

32

4.

Find the solution of the coupled differential equations

with initial conditions x(0)=1 and y(0)=3

0% 0%0%0%

yy

yxx

3

4

t

tt

ety

eetx3

3

3)(

32)(

1.

t

tt

ety

eetx3

3

3)(

32)(

2.

t

tt

ety

eetx

3)(

32)(

3.

t

tt

ety

eetx

3)(

32)(

4.

Given . What is the general solution to a system of 2nd order differential equations for the

negative eigenvalues ?

1 2 3 4

0% 0%0%0%

21,

tN)sinM(s

tL)cos(Kr

2

1

1.

tNsintMcoss

tLsintKcosr

22

11

2.

tNsintMcoss

tLsintKcosr

21

21

3.

t)sintM(coss

t)sintK(cosr

22

11

4.

rrr 211

An elastic membrane in the plane with boundary circle is

shown below.

21xx12

221 xx

princ

iple

direc

tionprinciple

direction

The membrane is stretched so the point P:( ) goes over the point Q:( ) where

Find the amount that the principle directions are

0% 0% 0%0%0%

21, xx

21, yy

2

1

2

1

74

47

x

xAx

y

yy

stretched by

1. By factors 3 and 11.

2. By factors 7 and 4.

3. By factors 4 and 4.

4. By factors 7 and 7.

5. Don’t Know