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MATH1014 Quizzes

Quiz 12: Eigenvalues and Eigenvectors
Question 1 Questions
Let A = 421 2 0 1 223 , v1 = 1 1 0 , v2 = 0 1 2 , v3 = 1 2 1 and v4 = 1 1 4 .
Which of the following statements is correct ? Exactly one option must be 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

Choice (a) is correct!
Choice (b) is incorrect
Av3 is not a multiple of v3.
Choice (c) is incorrect
Av3 is not a multiple of v3.
Choice (d) is incorrect
Av3 is not a multiple of v3; Av4 is not a multiple of v4.
1 2 2 is an eigenvector of 421 2 0 1 223 . What is the corresponding eigenvalue? (Enter your answer into the answer box.)

Correct!

Incorrect. Please try again.
Multiply 421 2 0 1 223 by 1 2 2 . The result should be a multiple of 1 2 2 .
5 5 5 is an eigenvector of 421 2 0 1 223 . What is the corresponding eigenvalue? (Enter your answer into the answer box.)

Correct!

Incorrect. Please try again.
Multiply 421 2 0 1 223 by 5 5 5 . The result should be a multiple of 5 5 5 .
What are the eigenvalues of 4 7 1 0 3 8 0 0 2 ? Exactly one option must be correct)
a)
1, 4, 7.
b)
0, 4.
c)
3, 2, 4.
d)
3, 0, 1.

Choice (a) is incorrect
The eigenvalues of a triangular matrix are the entries on the main diagonal.
Choice (b) is incorrect
The eigenvalues of a triangular matrix are the entries on the main diagonal.
Choice (c) is correct!
Choice (d) is incorrect
The eigenvalues of a triangular matrix are the entries on the main diagonal.
Given that 1 is an eigenvalue of A = 256 1 0 0 01 0 , find the other two eigenvalues. Exactly one option must be correct)
a)
2 and 3.
b)
2 and 3.
c)
0 and 3.
d)
0 and 3.

Choice (a) is incorrect
|A λI| = λ3 + 2λ2 + 5λ 6. Now take out a factor of (λ 1).
Choice (b) is correct!
Choice (c) is incorrect
|A λI| = λ3 + 2λ2 + 5λ 6. Now take out a factor of (λ 1).
Choice (d) is incorrect
|A λI| = λ3 + 2λ2 + 5λ 6. Now take out a factor of (λ 1).
If A = 416 2 1 5 210 , which one of the following is |A λI|? Exactly one option must be correct)
a)
λ3 + 5λ2 + λ 14
b)
λ3 + 5λ2 + λ 4
c)
λ3 + 5λ2 4λ
d)
λ3 + 5λ2 21λ + 42

Choice (a) is correct!
Choice (b) is incorrect
Try again.
Choice (c) is incorrect
Try again.
Choice (d) is incorrect
Try again.
Find the eigenvalues of B = 1 241 4 8 0 11 . Exactly one option must be correct)
a)
1, 1, 4.
b)
4, 1, 1.
c)
5, 1, 2.
d)
2, 1, 5.

Choice (a) is incorrect
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).
Choice (b) is incorrect
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).
Choice (c) is incorrect
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).
Choice (d) is correct!
Given that 5 is an eigenvalue of 1 241 4 8 0 11 , which of the following systems of equations should be solved to find the corresponding eigenvectors? Exactly one option must be correct)
a)
439 6 1 3 546 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)
1 241 4 8 0 11 x y z = 0 0 0
e)
1 241 4 8 0 11 x y z = 5 5 5

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Choice (e) is incorrect
A particular 3 × 3 matrix A has an eigenvalue of 1. The matrix A + I reduces to 102 0 0 0 00 0 . Corresponding to the eigenvalue 1, all the eigenvectors of A are non-zero vectors of the form: Exactly one option must be correct)
a)
2t 0 t , t
b)
2t s t , s,t
c)
t 0 2t , t
d)
t s 2t , s,t

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
An animal population with three age groups has Leslie matrix 0 0 3.60.8 0 0 0 0.6 0 . Find the proportion of the population in each of the age groups once the population has stabilised. Exactly one option must be correct)
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

Choice (a) is incorrect
Hint: Find the positive eigenvalue (1.2), and a corresponding eigenvector.
Choice (b) is incorrect
Hint: Find the positive eigenvalue (1.2), and a corresponding eigenvector.
Choice (c) is incorrect
Hint: Find the positive eigenvalue (1.2), and a corresponding eigenvector.
Choice (d) is correct!