Download HELM Workbook 22 (Eigenvalues and Eigenvectors) EVS Questions

What is the determinant of
9
11
17
19
0%
19
0%
17
0%
11
0%
9
1.
2.
3.
4.
7 3 
1 2 


What is the determinant of
0
28
44
-28
0%
-2
8
0%
44
0%
28
0%
0
1.
2.
3.
4.
6 2 0 
1 4 2 


2  3 0
Which matrix represents the
following system of equations?
1.
1 4
1 7 


1 11
0%
0%
0%
3
3.
1 0 4
0 1 7 


2
1
1
2.
x=4
y=7
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.
-1
A has an eigenvalue
1

2.
(A - k) has an eigenvalue 
3.
-1
(A - kI) 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 2


4 1
1.
2.
3.
4.
λ² - 7λ + 6 = 0
λ² - 7λ + 14 = 0
λ² - 7λ - 2 = 0
None of the above
What are the eigenvalues of
4 0
0 2


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.
What are the eigenvalues of the
following matrix?
8  4
2 2 


1.
2.
3.
4.
λ = 2, 4
λ = 2, 6
λ = 2, 8
λ = 4, 8
What are the eigenvalues for the
following matrix?
 0 5  5


5 5 0
5 0 5 


1.
2.
3.
4.
λ = -5, 0, 5
λ = -5, 5, 10
λ = 0, 5, 5
λ = 5, 5, 10
What are the eigenvalues for the
following matrix?
 3 0  1


 0 3 0
  15 5 1 


1.
2.
3.
4.
λ = -2, 3, 4
λ = -6, -2, 3
λ = -2, 3, 6
λ = -6, -3, 2
Matrix A given below has eigenvalues
λ = 2, 4, 6. Without further calculation
-1
write down the eigenvalues for A .
 3 1 1


A  1 3 1
 0 0 0


1.
1, 2, 3
0%
0%
4
0%
3
0%
2
4.
1 1 1
, ,
2 4 6
1
3.
1 2
, ,1
3 3
2.
1
,1,1
3
Which of the vectors below is an
eigenvector, corresponding to the
eigenvalue λ= 7 of the matrix
 3 2


 4 5
1.
4.
0%
0%
0%
0%
4
1
 
 2
3
 2
 
1
  1
 
2
2
3.
2.
1
2
 
 1
Which of the vectors below is an
eigenvector, corresponding to the
eigenvalue λ= 3 of the matrix
1  4  2


1 
0 3
1 2

4


1.
1
 
 2
0
 
0%
0%
0%
0%
4
4.
3
 1 
 
  2
 0 
 
 2
 
1
0
 
2
3.
2.
1
  2
 
 1 
 0 
 
What are the eigenvectors of the
following matrix?
5 5 


 1  1
1.
  1  1 
  ,  
 1   5
0%
0%
4
0%
3
0%
2
4.
1   5 
  ,  
1  1 
1
3.
1  1 
  ,  
1   5 
2.
 1  1
  ,  
  1  5 
Which of the following shows the
eigenvectors for matrix A?
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
   
1, 1
0 0
   
3
3.
2.
2
 0    3
   
1,  1 
0  0 
   
1
1.
 2 0 1 
A   1 1 0 
 0 0  2
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
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?
...
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
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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
0%
4
0%
3
0%
2
0%
1
4.
 1  3 
  ,  
 1  3 
0%
0%
0%
0%
4
4.
1 4
1 1 


3
3.
3 4
1 1


2.
1 0 
1 1


2
1.
 1 0
 1 1


Obtain the modal matrix P.
1
1 0
A
.

1 2
The matrix A=
 1 3
 0 2


has eigenvalues
1 and 2 with respective eigenvectors
-
4.
 1 0
 0 2


0%
0%
0%
0%
4
3.
 1 4
 0 2


calculate P AP1.
3
2.
1 2
0 2 


1
1
2
1.
 1 4
 1 2


1
1

1
If
1 1
P1  

0
1


1
0 and
 
 2 3

A  
 4 5
What is A²?
1.
4.
0%
0%
0%
0%
4
 13 23 


 23 41
3
 16 21 


 28 37 
4 9


16 25 
2
3.
2.
1
4 6 


 8 10 
 2 3

A  
 4 5
5
What is A ?
1.
 10 15 


 20 25 
2.
 32 243 


1024 3125 

3.
 6140 8097 


10796 14237 
0%
4
0%
3
0%
2
10796 14237 


 6140 80972 
0%
1
4.
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


0%
0%
0%
4
0%
3
4.
4 6 


 1  3
2
 1  2  3


 2 3 7 
 3 7 4


1
3.
2.
1  4


4 1 
A square matrix A
is said to be orthogonal if
A A
-1
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 3
A

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?
 3 2i 3  i 


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%
Separating the variables in s  2 s gives
1.
s(t )  Ae2t
2.
s(t )  2 Ae2 s
3.
s(t )  2 Ae2t
0%
0%
4
0%
3
0%
2
s(t )  Ae
2 t
1
4.
Write in matrix form the pair of
coupled differential equations
2.
 x  2 5  x 
 y   3 1  y 
  
 
0%
0%
4
0%
3
0%
2
3.
 x  2 3   x 
 y   5  1  y 
  
 
4.
 x  2 3  x 
 y   5 1  y 
  
 
1
1.
 x  2 5   x 
 y   3  1  y 
  
 
 x  2 x  3 y

 y  5 x  y
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 )  2e t  3e3t
y(t )  3e3t
2.
x(t )  2e t  3e3t
3.
y(t )  3e3t
x(t )  2et  3et
4.
0%
0%
0%
4
0%
3
y (t )  3e
t
2
x(t )  2et  3et
1
y (t )  3et
Given r  1r   r .
What is the general solution to a system
of 2nd order differential equations for the
negative eigenvalues 1 , 2 ?
2
1
1. r  (K  L)cos  t
1
s  (M  N)sin 2 t
2.
r  Kcos1t  Lsin 1t
s  Mcos2 t  Nsin 2 t
3. r  Kcos t  Lsin  t
1
2
s  Mcos1t  Nsin 2 t
1
s  M(cos2 t  sin 2 t)
0%
0%
0%
4
0%
1
3
1
2
4. r  K(cos t  sin  t)
An elastic membrane in the x1 x2 plane
2
2
with boundary circle x1  x2  1 is
shown below.
The membrane is stretched so the point P:(x1 , x2 )
goes over the point Q:( y1 , y2 ) where
 y1 
7 4  x1 
y     Ax  
x 
y
4
7

 2 
 2
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.