School of Mathematics and Statistics
Junior
The University of Sydney
Maths & Stats Home / Teaching program / Junior / MATH1014 / Quizzes / Quiz 12
The University of Sydney
Usyd Home
MyUni
MyUni
Library
Sitemap
spcr
   Questions




right first
    attempt
right
wrong

Quiz 12: Eigenvalues and Eigenvectors

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Let A = ⌊ ⌋ 4 - 2 1 ⌈2 0 1⌉ 2 - 2 3, v1 = ⌊ ⌋ 1 ⌈1⌉ 0, v2 = ⌊ ⌋ 0 ⌈ 1⌉ 2, v3 = ⌊ ⌋ 1 ⌈ 2 ⌉ - 1 and v4 = ⌊ ⌋ 1 ⌈1⌉ 4.
Which of the following statements is correct ?
a) v1 and v2 are eigenvectors of A   b) v1 and v3 are eigenvectors of A
c) v2 and v3 are eigenvectors of A   d) v3 and v4 are eigenvectors of A

 

Your answer is correct.
Not correct. Choice (b) is false.
Av3 is not a multiple of v3.
Not correct. Choice (c) is false.
Av3 is not a multiple of v3.
Not correct. Choice (d) is false.
Av3 is not a multiple of v3; Av4 is not a multiple of v4.
 

Question 2

 
 
⌊ ⌋ 1 ⌈2⌉ 2 is an eigenvector of ⌊ ⌋ 4 - 2 1 ⌈2 0 1⌉ 2 - 2 3. What is the corresponding eigenvalue? (Enter your answer into the answer box.)

 

Your answer is correct

Not correct. You may try again.
Multiply ⌊ ⌋ ⌈4 - 2 1⌉ 2 0 1 2 - 2 3 by ⌊ ⌋ ⌈1⌉ 2 2. The result should be a multiple of ⌊ ⌋ ⌈1⌉ 2 2.
 

Question 3

 
 
⌊ ⌋ ⌈- 5⌉ - 5 - 5 is an eigenvector of ⌊ ⌋ ⌈4 - 2 1⌉ 2 0 1 2 - 2 3. What is the corresponding eigenvalue? (Enter your answer into the answer box.)

 

Your answer is correct

Not correct. You may try again.
Multiply ⌊ 4 - 2 1⌋ ⌈ 2 0 1⌉ 2 - 2 3 by ⌊- 5⌋ ⌈- 5⌉ - 5. The result should be a multiple of ⌊- 5⌋ ⌈- 5⌉ - 5.
 

Question 4

 
 
What are the eigenvalues of ⌊4 7 1⌋ ⌈0 - 3 8⌉ 0 0 2?
a) 1, 4, 7.   b) 0, 4.
c) -3, 2, 4.   d) -3, 0, 1.

 

Not correct. Choice (a) is false.
The eigenvalues of a triangular matrix are the entries on the main diagonal.
Not correct. Choice (b) is false.
The eigenvalues of a triangular matrix are the entries on the main diagonal.
Your answer is correct.
Not correct. Choice (d) is false.
The eigenvalues of a triangular matrix are the entries on the main diagonal.
 

Question 5

 
 
Given that 1 is an eigenvalue of A = ⌊ ⌋ ⌈2 5 - 6⌉ 1 0 0 0 1 0, find the other two eigenvalues.
a) 2 and -3.   b) -2 and 3.
c) 0 and 3.   d) 0 and 3.

 

Not correct. Choice (a) is false.
|A-λI| = -λ3 + 2λ2 + 5λ- 6. Now take out a factor of (λ- 1).
Your answer is correct.
Not correct. Choice (c) is false.
|A - λI| = -λ3 + 2λ2 + 5λ - 6. Now take out a factor of (λ - 1).
Not correct. Choice (d) is false.
|A - λI| = -λ3 + 2λ2 + 5λ - 6. Now take out a factor of (λ - 1).
 

Question 6

 
 
If A = ⌊4 - 1 6⌋ ⌈2 1 5⌉ 2 - 1 0, which one of the following is |A - λI|?
a) -λ3 + 5λ2 + λ - 14   b) -λ3 + 5λ2 + λ - 4
c) -λ3 + 5λ2 - 4λ   d) -λ3 + 5λ2 - 21λ + 42

 

Your answer is correct.
Not correct. Choice (b) is false.
Try again.
Not correct. Choice (c) is false.
Try again.
Not correct. Choice (d) is false.
Try again.
 

Question 7

 
 
Find the eigenvalues of B = ⌊ ⌋ 1 2 - 4 ⌈- 1 4 8 ⌉ 0 1 - 1.
a) -1, 1, 4.   b) -4, -1, 1.
c) -5, 1, 2.   d) -2, 1, 5.

 

Not correct. Choice (a) is false.
Expanding |B - λI| by the first column gives

(1- λ)[(4 - λ)(- 1- λ)- 8]+ 2(1- λ).
Now take out a factor of (1 -λ).
Not correct. Choice (b) is false.
Expanding |B -λI| by the first column gives
(1- λ)[(4 - λ)(- 1- λ)- 8]+ 2(1- λ).
Now take out a factor of (1 - λ).
Not correct. Choice (c) is false.
Expanding |B - λI| by the first column gives
(1- λ)[(4 - λ)(- 1- λ)- 8]+ 2(1- λ).
Now take out a factor of (1 - λ).
Your answer is correct.
 

Question 8

 
 
Given that 5 is an eigenvalue of ⌊ ⌋ 1 2 - 4 ⌈- 1 4 8 ⌉ 0 1 - 1, which of the following systems of equations should be solved to find the corresponding eigenvectors?
a) ⌊ - 4 - 3 - 9⌋ ⌈ - 6 - 1 3 ⌉ - 5 - 4 - 6 ⌊ x⌋ ⌈ y⌉ z = ⌊ 0⌋ ⌈ 0⌉ 0   b) ⌊ - 4 2 - 4⌋ ⌈ - 1 - 1 8 ⌉ 0 1 - 6 ⌊x ⌋ ⌈ y⌉ z = ⌊ x⌋ ⌈ y⌉ z
c) ⌊- 4 2 - 4⌋ ⌈- 1 - 1 8 ⌉ 0 1 - 6 ⌊ x⌋ ⌈ y⌉ z = ⌊ 0⌋ ⌈ 0⌉ 0   d) ⌊ 1 2 - 4⌋ ⌈- 1 4 8 ⌉ 0 1 - 1 ⌊ x⌋ ⌈ y⌉ z = ⌊ 0⌋ ⌈ 0⌉ 0
e) ⌊ 1 2 - 4⌋ ⌈- 1 4 8 ⌉ 0 1 - 1 ⌊ x⌋ ⌈ y⌉ z = ⌊5⌋ ⌈5⌉ 5

 

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

Question 9

 
 
A particular 3 × 3 matrix A has an eigenvalue of -1. The matrix A + I reduces to ⌊1 0 - 2⌋ ⌈0 0 0 ⌉ 0 0 0. Corresponding to the eigenvalue -1, all the eigenvectors of A are non-zero vectors of the form:
a) ⌊ ⌋ 2t ⌈0 ⌉ t, t ∈ ℝ   b) ⌊ ⌋ 2t ⌈ s⌉ t, s, t ∈ ℝ
c) ⌊ ⌋ t ⌈ 0 ⌉ - 2t, t ∈ ℝ   d) ⌊ ⌋ t ⌈s ⌉ 2t, s, t ∈ ℝ

 

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

Question 10

 
 
An animal population with three age groups has Leslie matrix ⌊ ⌋ 0 0 3.6 ⌈0.8 0 0 ⌉ 0 0.6 0. Find the proportion of the population in each of the age groups once the population has stabilised.
a) Age group 1: 1 3    Age group 2: 1 3    Age group 3: 1 3
b) Age group 1: 1 6    Age group 2: 1 3    Age group 3: 1 2
c) Age group 1: 1 4    Age group 2: 1 2    Age group 3: 1 4
d) Age group 1: 1 2    Age group 2: 1 3    Age group 3: 1 6

 

Not correct. Choice (a) is false.
Hint: Find the positive eigenvalue (1.2), and a corresponding eigenvector.
Not correct. Choice (b) is false.
Hint: Find the positive eigenvalue (1.2), and a corresponding eigenvector.
Not correct. Choice (c) is false.
Hint: Find the positive eigenvalue (1.2), and a corresponding eigenvector.
Your answer is correct.