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H1002 Quizzes

Quiz 12: Eigenvalues and modelling
Question 1 Questions
The matrix A = 24 1 2 has eigenvalues λ1 = 0, λ2 = 4 with corresponding eigenvectors v1 = 2 1 and v2 = 2 1 .
Which of the following matrices T and D satisfy the equation AT = TD ?
a)
T = 00 0 4 , D = 2 2 1 1
 b)
T = 00 0 4 , D = 2 21 1
c)
D = 00 0 4 , T = 2 2 1 1
 d)
D = 00 0 4 , T = 2 21 1

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
For these two matrices AT = 08 0 4 = TD. In fact D = λ1 0 0 λ2 and T = v1v2 .
The matrix A = 256 1 0 0 01 0 has eigenvalues λ1 = 1, λ2 = 3, λ3 = 2 with corresponding eigenvectors v1 = 1 1 1 , v2 = 9 3 1 and v3 = 4 2 1 .
Which of the following matrices T and D are such that AT = TD ?
a)
T = 10 0 0 3 0 002 , D = 19 4 1 3 2 11 1
b)
T = 10 0 0 3 0 002 , D = 4 912 3 1 1 11
c)
D = 10 0 0 3 0 002 , T = 19 4 1 3 2 11 1
d)
D = 10 0 0 3 0 002 , T = 91 4 3 1 2 11 1

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
AT = 1278 1 9 4 1 3 2 = TD.
Which are the matrices T and D such that AT = TD when A = 13 2 2 ?
a)
T = 31 2 1 , D = 40 0 1
 b)
T = 3 12 1 , D = 10 0 4
c)
T = 3 12 1 , D = 4 0 0 1
 d)
T = 31 2 1 , D = 10 0 4

Choice (a) is incorrect
Choice (b) is correct!
The eigenvalues of A are λ1 = 1, λ2 = 4 and the corresponding eigenvectors are v1 = 3 2 and v2 = 1 1 . Hence T = 3 12 1 and D = 10 0 4 .
Choice (c) is incorrect
Choice (d) is incorrect
The matrix A = 2 2 1 5 2 2 1386 has eigenvectors v1 = 1 1 1 , v2 = 0 1 2 and v3 = 1 1 3 .
Which are the matrices T and D such that AT = TD ?
a)
T = 1 0 1 1 1 1 123 , D = 1 0 0 0 2 0 0 0 1
b)
T = 1 1 0 1 1 1 312 , D = 1 0 0 0 2 0 0 0 1
c)
T = 1 0 1 1 1 1 123 , D = 1 0 0 0 2 0 0 0 1
d)
All of the above

Choice (a) is correct!
Av1 = v1, Av2 = 2v2, Av3 = v3
T = v1v2v3 , D = 1 0 0 0 2 0 0 0 1 .
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Consider the matrix A = 123 0 1 3 002 . Find a matrix T such that T1AT = D for some diagonal matrix D.
a)
T = 129 0 0 3 001
 b)
T = 009 1 2 3 001
c)
T = 126 0 0 3 001
 d)
No such T exists.

Choice (a) is incorrect
It is true that AT = TD where D = 100 0 1 0 002 , but det T = 0, so T1 does not exist.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Note that T = 129 0 0 3 001 and D = 100 0 1 0 002 are such that AT = TD, but det T = 0, so T1 does not exist.
Let the Leslie matrix for a female population divided into 4 equal age groups be
L = 0 1.5 2 0.50.4 0 0 0 0 0.5 0 0 0 0 0.3 0
What is the fraction of the second age group which survives to the next generation ?
a)
0.4
 b)
0.5
c)
0.3
 d)
1.5

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Let the Leslie matrix for a female population divided into 4 equal age groups be
L = 0 4.03.90.220.2 0 0 0 0 0.3 0 0 0 0 0.2 0 .
What is the per capita average number of female offspring for the 2nd age group?
a)
0.2
 b)
0.22
c)
3.9
 d)
4.0

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
We are given that a female population is divided into 3 equal age groups and that the average no. of female offspring for the first, second and third age groups are 1,4.5 and 3.2 respectively. If the proportion of the first and second age groups which survive to the next age group is 0.9 and 0.7 respectively, what is the Leslie matrix which models this behaviour ?
a)
L = 0 1 4.53.20.9 0 0 0 0 0.7 0 0 0 0 0.7 0
 b)
L = 0.9 0 1 0 0.7 4.5 0 0 3.2
c)
L = 1 4.53.20.9 0 0 0 0.7 0
 d)
There is not enough information provided.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
The female population of lemmings is divided into 4 equal age groups which are described by the following statistics: Age groupAverage No. offspring% surviving to next generation ̲ ̲ ̲ 1 2.4 30% 2 3.1 70% 3 4.9 50% 4 0.9 0% What is the Leslie matrix which models this behaviour ?
a)
L = 1 0.3 0 0 2.4 2 0.7 0 0 3.1 3 0.5 0 4.9 4 0
 b)
L = 2.43.14.90.9 0.3 0 0 0 0 0.7 0 0 0 0 0.5 0
c)
L = 2.40.3 0 0 3.1 0 0.7 0 4.9 0 0 0.5 0.9 0 0 0
 d)
Not enough information provided to answer the question.

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Two countries, A and B, have open door immigration policies between them. Let the population of countries A and B at the end of year n be xn and yn respectively. The population of country A grows by 2% per year, excluding migration and the population of country B grows by 5% per year excluding migration. It is observed that 10% of the increased population of country A migrates to country B and 7% of the increased population of country B migrates to country A each year. If xn+1 yn+1 = P xn yn , what is the matrix P ?
a)
P = 0.90.07 0.1 0.93
 b)
P = 0.90.070.93 0.1
c)
P = 0.9180.0735 0.102 0.9765
 d)
P = 0.9180.07350.9765 0.102

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
xn+1 = 0.9(1.02xn) + 0.07(1.05yn) = 0.918xn + 0.0735yn. yn+1 = 0.1(1.02xn) + 0.93(1.05yn) = 0.102xn + 0.9765yn.
Choice (d) is incorrect