```What is the determinant of
3

2
9
11
17
19
0%
19
0%
17
0%
11
0%
9
1.
2.
3.
4.
7

1
What is the determinant of
2
4
3
0

2

0 
0
28
44
-28
0%
-2
8
0%
44
0%
28
0%
0
1.
2.
3.
4.
6

1

 2
Which matrix represents the
following system of equations?
x=4
y=7
1.
1

1
4

7
2.
1
11 
1
3.
0%
0%
0%
3
1
4

7
2
0
1
1

0
What is the solution to the following
system of equations?
x 1 + x2 = 3
2x1 - 6x2 = -10
s.
..
e
e
er
e
ar
ar
Th
0%
no
an
x2
=½
Th
an
d
=1
0%
i..
.
0%
er
e
x2
=2
0%
an
d
x2
=5
=1
x1
x1
=4
an
=9/
8
0%
an
d
d
x2
...
0%
x1
x1=-9/8 and x2=-30/8
x1=4 and x2=5
x1=1 and x2=2
x1=1 and x2=½
There are an infinite
number of solutions
6. There are no solutions
x1
1.
2.
3.
4.
5.
What is the solution to the following
system of equations?
6x1 + 9x2 = 3
2x1 + 3x2 = 5
an
d
=1
x1
0%
0%
0%
0%
x2
==1.
5/
..
2
an
d
Th
x2
er
...
e
ar
e
Th
an
er
i..
e
.
ar
e
no
s.
..
/..
.
0%
x2
=1
an
d
=0
x1
=2
an
d
x2
=0
0%
x1
x1=2 and x2=0
x1=0 and x2=1/3
x1=1 and x2=-1/3
x1=-5/2 and x2=2
There are an infinite
number of solutions
6. There are no solutions
x1
1.
2.
3.
4.
5.
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
0%
(A
),
(B
)a
nd
(..
.
(..
.
0%
nd
(..
.
(B
)a
nd
B
B
0%
ot
h
(A
)a
nd
(..
.
0%
ot
h
(A
)a
nl
y
ot
h
O
B
nl
y
O
0%
(C
)
0%
(B
)
(A
)
0%
nl
y
Only (A)
Only (B)
Only (C)
Both (A) and (B)
Both (A) and (C)
Both (B) and (C)
(A), (B) and (C)
O
1.
2.
3.
4.
5.
6.
7.
Which of the following statements
is true?
1.
A
-1
has an eigenvalue
1

2.
(A - k) has an eigenvalue

3.
(A - kI)
-1
has an eigenvalue
1

4.
0%
4
0%
3
0%
2
0%
1
None of the above are true
Dominant eigenvalue
=
eigenvalue with largest magnitude
1. True
2. False
3. Don’t Know
0%
’t
Kn
o
ls
e
w
0%
D
on
Fa
Tr
ue
0%
What is the characteristic equation
of the following matrix?
6

4
1.
2.
3.
4.
λ² - 7λ + 6 = 0
λ² - 7λ + 14 = 0
λ² - 7λ - 2 = 0
None of the above
2

1
What are the eigenvalues of
4

0
2 and 4
0 and 4
0 and 2
6 and 8
6
an
d
8
0%
2
an
d
0
an
d
0
an
d
0%
4
0%
4
0%
2
1.
2.
3.
4.
0

2
What are the eigenvalues of the
following matrix?
8

2
1.
2.
3.
4.
λ = 2, 4
λ = 2, 6
λ = 2, 8
λ = 4, 8
 4

2 
What are the eigenvalues for the
following matrix?
 0

 5
5

1.
2.
3.
4.
λ = -5, 0, 5
λ = -5, 5, 10
λ = 0, 5, 5
λ = 5, 5, 10
5
5
0
 5

0 
5 
What are the eigenvalues for the
following matrix?
 3

 0
  15

1.
2.
3.
4.
λ = -2, 3, 4
λ = -6, -2, 3
λ = -2, 3, 6
λ = -6, -3, 2
0
3
5
 1

0 
1 
Matrix A given below has eigenvalues
λ = 2, 4, 6. Without further calculation
-1
write down the eigenvalues for A .
3

A  1
0

1.
1
3
0
1

1
0 
1, 2 , 3
2.
1
4.
0%
0%
0%
4
0%
3
1 1 1
, ,
2 4 6
2
1 2
, ,1
3 3
3
1
3.
, 1, 1
Which of the vectors below is an
eigenvector, corresponding to the
eigenvalue λ= 7 of the matrix
3

4
1.
4.
0%
0%
0%
0%
4
1
 
2
3
2
 
1
  1


 2 
2
3.
2.
1
 2 


  1
2

5
Which of the vectors below is an
eigenvector, corresponding to the
eigenvalue λ= 3 of the matrix
1

0
1

1.
2.
2
 
1
0
 
1
 
2
0
 
0%
0%
0%
0%
4
4.
3
 1 



2


 0 


2
2
3.
3
 2

1 
4 
1
 2


 1 
 0 


4
What are the eigenvectors of the
following matrix?
5 5 


 1  1
1.
  1  1 

 ,  
 1  5
2.
3.
 1  1

 ,  
  1  5 
1  1 
  , 

1   5 
4.
0%
4
0%
3
0%
2
0%
1
1   5 
  , 

1  1 
Which of the following shows the
eigenvectors for matrix A?
2.
0 3
   
1, 1
0 0
   
4.
0 3 1
     
1, 1, 1
0 0 0
     
0%
0%
0%
0%
4
0   3 1
  
  
1,  1 , 1
0  0  0
  
  
0
3
3.
1
1 

0

 2 
2
0   3
  

1,  1 
0  0 
  

0
1
1.
 2

A 1

 0
Which set of vectors is linearly
independent?
0%
se
th
e
ot
h
B
on
e
of
of
th
e
,2
),
(3
,
ab
...
...
0%
4)
0%
(1
,6
),
(3
,
18
)
0%
N
(1,6), (3,18)
(1,2), (3,4)
None of the above
Both of the sets
(1
1.
2.
3.
4.





1


2


 3 
 1 


 3 








4.
2 6 


4 2 
2 2 


0%
0%
0%
0%
4
3.

3


8

 1

2
 1

 2 2
3

3


2


2 

 1 


2


2.
2
1.
Normalise the eigenvector X.
1
 3 


X   2 
 1 


Diagonalization means which of the
following?
0%
0%
ab
...
a.
..
th
e
of
e
on
sf
or
m
in
g
N
M
ul
tip
ly
in
g
th
.
..
di
a.
th
e
in
g
dd
A
0%
..
0%
Tr
an
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?
...
e
th
es
th
e
fin
d
of
e
on
N
ot
h
of
to
sy
Ea
0%
...
0%
se
e.
w
e.
..
po
tin
g
om
pu
C
0%
..
0%
B
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.
1. True
2. False
3. Don’t Know
0%
’t
Kn
o
ls
e
w
0%
D
on
Fa
Tr
ue
0%
Below are eigenvectors of four 2x2
matrices. Which matrix is definitely
diagonalizable?
1.
 0  0

 ,  
  1  3 
2.
1 0
  ,  
0 3
3.
  1  1 

 , 

 3    3
4.
0%
4
0%
3
0%
2
0%
1
1  3 
  ,  
1  3 
1
A
1
0
.
2
Obtain the modal matrix P.
1.
1

 1
0

1
2.
1

1
0 

 1
3.
4.
0%
0%
0%
0%
4
4

1
3
1

1
2
4

1
1
3

1
The matrix A=
 1

 0
3

2
has eigenvalues
1 and 2 with respective eigenvectors
-
If
1.
calculate
1
P1 AP1 .
2.
4

2
3.
1

0
2

2
4.
0

2
0%
0%
0%
0%
4
 1

0
3
4

2
2
 1

0
1

1
1
 1

 1
1
P1  
0
1 
1
  and  
0 
1
2
A  
4
3

5
What is A²?
1.
 4

 16
4.
 13

 23
23 

41 
0%
0%
0%
0%
4
21 

37 
9 

25 
3
3.
 16

 28
2.
2
6 

10 
1
4

8
2
A  
4
3

5
What is A ?
5
1.
243 

3125 
8097 

14237 
4.
 10796

 6140
14237 

80972 
0%
0%
0%
0%
4
 6140

 10796
 32

 1024
3
3.
2.
2
15 

25 
1
 10

 20
The eigenvalues of a symmetric
matrix with real elements are...
1. Always complex
2. Always real
3. Either complex or real
rc
th
e
Ei
A
lw
ay
s
om
pl
ex
re
a
m
pl
ex
co
s
ay
lw
A
0%
...
0%
l
0%
Which of the following is a
symmetric matrix?
1.
 1

 5
 2

5
3
7
 2

7 
 2 
2.
7
4.
4

1
6 

 3
0%
0%
0%
0%
4
3
 3

7 
4 
3
2
2
 1

 2
 3

 4

1 
1
3.
1

4
A square matrix A
is said to be orthogonal if
A
-1
A
T
1. True
2. False
3. Don’t Know
no
w
0%
Do
n’
tK
se
0%
Fa
l
Tr
ue
0%
Two n x 1 column vectors X and Y
are orthogonal if XY=0
1. True
2. False
3. Don’t Know
0%
’t
Kn
o
ls
e
w
0%
D
on
Fa
Tr
ue
0%
The eigenvalues of a symmetric matrix A are λ=0 and λ=10
9
A
3
3

1
X and Y are the eigenvectors for λ=0 and λ=10 respectively.
Are X and Y orthogonal?
1. Yes
2. No
3. Don’t Know
no
w
0%
Do
n’
tK
0%
No
Ye
s
0%
An Hermitian matrix is one satisfying
A A
T
1. True
2. False
3. Don’t Know
no
w
0%
Do
n’
tK
se
0%
Fa
l
Tr
ue
0%
Is the following matrix Hermitian?
2i 3  i 
 3


A  2i
0
1


 3  i 1
5 
1. Yes
2. No
3. Don’t Know
0%
w
o
0%
D
on
’t
Kn
o
N
Ye
s
0%
s   2 s
Separating the variables in
gives
1.
s ( t )  Ae
 2t
2.
s ( t )  2 Ae
2 s
3.
2t
0%
0%
0%
4
0%
3
s ( t )  Ae
2t
2
4.
1
s ( t )  2 Ae
Write in matrix form the pair of
coupled differential equations
 x  2 x  3 y

 y  5 x  y
1.
5 x
 
 1  y 
 x   2
 
 y   3
3.
5  x 
 
1  y 
3 x
 
 1  y 
3  x 
 
1  y 
0%
0%
0%
4
 x   2
 
 y   5
0%
3
4.
2
 x   2
 
 y   5
2.
1
 x   2
 
 y   3
Find the solution of the coupled
differential equations
 x   x  4 y

 y  3 y
with initial conditions x(0)=1 and y(0)=3
1.
x (t )  2 e
t
y (t )  3e
3t
 3e
3t
2.
x (t )   2 e
y (t )  3e
3.
t
4.
x (t )   2 e  3e
t
y (t )  3e
t
t
0%
0%
0%
0%
4
t
3t
3
y (t )  3e
3t
2
t
 3e
1
x (t )  2 e  3e
t
2


Given r  1 r    1 r .
What is the general solution to a system
of 2nd order differential equations for the
negative eigenvalues  1 ,  2 ?
1.
r  (K  L)cos  1 t
s  ( M  N)sin  2 t
2.
r  Kcos  1 t  Lsin  1 t
s  Mcos  2 t  Nsin  2 t
r  Kcos  1 t  Lsin  2 t
s  Mcos  1 t  Nsin  2 t
r  K(cos  1 t  sin  1 t)
0%
0%
0%
0%
4
3
s  M(cos  2 t  sin  2 t)
2
4.
1
3.
An elastic membrane in the x1 x 2 plane
2
2
with boundary circle x1  x 2  1 is
shown below.
The membrane is stretched so the point P:(x1 , x 2 )
goes over the point Q:( y 1 , y 2 ) where
 y1 
7
y     Ax  
4
 y2 
4   x1 
 
7   x2 
Find the amount that the principle directions are
stretched by
’t
K
no
D
on
7
ct
or
s
fa
y
0%
w
..
a.
..
0%
a.
ct
or
s
4
a.
fa
y
B
B
y
fa
ct
or
s
7
a.
3
ct
or
s
fa
y
0%
..
0%
..
0%
B
By factors 3 and 11.
By factors 7 and 4.
By factors 4 and 4.
By factors 7 and 7.
Don’t Know
B
1.
2.
3.
4.
5.
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