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Quiz 12: Eigenvalues and modelling

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
The matrix A = [ ] 2 4 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 = [0 0] 0 4, D = [2 2] 1 - 1   b) T = [0 0] 0 4, D = [2 2] - 1 1
c) D = [0 0] 0 4, T = [2 2 ] 1 - 1   d) D = [0 0] 0 4, T = [2 2] - 1 1

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
For these two matrices AT = [ ] 0 8 0 4 = TD. In fact D = [ ] λ1 0 0 λ2 and T = (v1 v2).
 

Question 2

 
 
The matrix A = ⌊2 5 - 6⌋ ⌈1 0 0 ⌉ 0 1 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 = ⌊1 0 0 ⌋ ⌈0 3 0 ⌉ 0 0 - 2, D = ⌊ 1 9 4 ⌋ ⌈ 1 3 - 2⌉ 1 1 1
b) T = ⌊ 1 0 0 ⌋ ⌈ 0 3 0 ⌉ 0 0 - 2, D = ⌊ 4 9 1⌋ ⌈- 2 3 1⌉ 1 1 1
c) D = ⌊1 0 0 ⌋ ⌈0 3 0 ⌉ 0 0 - 2, T = ⌊1 9 4 ⌋ ⌈1 3 - 2⌉ 1 1 1
d) D = ⌊ ⌋ 1 0 0 ⌈0 3 0 ⌉ 0 0 - 2, T = ⌊ ⌋ 9 1 4 ⌈3 1 - 2⌉ 1 1 1

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.
AT = ⌊ ⌋ 1 27 - 8 ⌈1 9 4⌉ 1 3 - 2 = TD.
 

Question 3

 
 
Which are the matrices T and D such that AT = TD when A = [ ] 1 3 2 2 ?
a) T = [3 1] 2 1, D = [4 0] 0 1   b) T = [3 1] - 2 1, D = [- 1 0] 0 4
c) T = [ ] 3 1 - 2 1, D = [ ] 4 0 0 - 1   d) T = [ ] 3 1 2 1, D = [ ] 1 0 0 4

 

Not correct. Choice (a) is false.
Your answer 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 1 - 2 1 and D = [ ] - 1 0 0 4.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 4

 
 
The matrix A = ⌊ ⌋ ⌈ 2 2 1 ⌉ 5 2 2 - 13 - 8 - 6 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 ⌉ - 1 - 2 - 3, D = ⌊ - 1 0 0⌋ ⌈ 0 - 2 0⌉ 0 0 1
b) T = ⌊ ⌋ 1 1 0 ⌈ 1 - 1 1 ⌉ - 3 - 1 - 2, D = ⌊ ⌋ - 1 0 0 ⌈ 0 - 2 0⌉ 0 0 1
c) T = ⌊ ⌋ 1 0 1 ⌈ - 1 1 1 ⌉ - 1 - 2 - 3, D = ⌊ ⌋ 1 0 0 ⌈0 - 2 0 ⌉ 0 0 - 1
d) All of the above

 

Your answer is correct.
Av1 = -v1, Av2 = -2v2, Av3 = v3
T = ( ) v1 v2 v3, D = ⌊ ⌋ - 1 0 0 ⌈ 0 - 2 0⌉ 0 0 1.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 5

 
 
Consider the matrix A = ⌊ ⌋ 1 2 3 ⌈ 0 1 3⌉ 0 0 2. Find a matrix T such that T -1AT = D for some diagonal matrix D.
a) T = ⌊ ⌋ 1 2 9 ⌈0 0 3⌉ 0 0 1   b) T = ⌊ 0 0 9⌋ ⌈ 1 2 3⌉ 0 0 1
c) T = ⌊ ⌋ 1 2 6 ⌈ 0 0 3⌉ 0 0 1   d) No such T exists.

 

Not correct. Choice (a) is false.
It is true that AT = TD where D = ⌊ ⌋ 1 0 0 ⌈ 0 1 0⌉ 0 0 2, but det T = 0, so T -1 does not exist.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Note that T = ⌊ ⌋ 1 2 9 ⌈0 0 3⌉ 0 0 1 and D = ⌊ ⌋ 1 0 0 ⌈0 1 0⌉ 0 0 2 are such that AT = TD, but det T = 0, so T -1 does not exist.
 

Question 6

 
 
Let the Leslie matrix for a female population divided into 4 equal age groups be
⌊ ⌋ | 0 1.5 2 0.5| L = |⌈0.4 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

 

Not correct. Choice (a) is false.
Your answer is correct.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 7

 
 
Let the Leslie matrix for a female population divided into 4 equal age groups be
⌊ ⌋ 0 4.0 3.9 0.22 L = ||0.2 0 0 0 || . ⌈ 0 0.3 0 0 ⌉ 0 0 0.2 0
What is the average number of female offspring for the 2nd age group ?
a) 0.2   b) 0.22
c) 3.9   d) 4.0

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
 

Question 8

 
 
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 4.5 3.2⌋ |0.9 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 4.5 3.2⌋ ⌈ 0.9 0 0 ⌉ 0 0.7 0   d) There is not enough information provided.

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.
 

Question 9

 
 
The female population of lemmings is divided into 4 equal age groups which are described by the following statisics:
-Age group-Average 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 = ⌊ 2.4 3.1 4.9 0.9⌋ | 0.3 0 0 0 | |⌈ 0 0.7 0 0 |⌉ 0 0 0.5 0
c) L = ⌊ 2.4 0.3 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.

 

Not correct. Choice (a) is false.
Your answer is correct.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
 

Question 10

 
 
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 = [ ] 0.9 0.07 0.1 0.93   b) P = [ ] 0.9 0.07 0.93 0.1
c) P = [ ] 0.918 0.0735 0.102 0.9765   d) P = [0.918 0.0735] 0.9765 0.102

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer 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.
Not correct. Choice (d) is false.