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MATH1002 Quizzes

Quiz 11: Eigenvalues and eigenvectors
Question 1 Questions
Let A = 21 3 0 . Which of the following statements are correct ? (Zero or more options can be correct)
a)
v1 = 1 1 is an eigenvector of A
b)
v2 = 1 1 is an eigenvector of A
c)
v3 = 1 3 is an eigenvector of A
d)
v4 = 3 1 is an eigenvector of A

There is at least one mistake.
For example, choice (a) should be True.
A 1 1 = 3 1 1 , so v1 is a 3–eigenvector of A
There is at least one mistake.
For example, choice (b) should be False.
A 1 1 = 1 3 , so v2 is not an a eigenvector of A
There is at least one mistake.
For example, choice (c) should be True.
A 1 3 = 1 3 , so v3 is a 1–eigenvector of A
There is at least one mistake.
For example, choice (d) should be False.
A 3 1 = 5 9 , so v4 is not an eigenvector of A
Correct!
  1. True A 1 1 = 3 1 1 , so v1 is a 3–eigenvector of A
  2. False A 1 1 = 1 3 , so v2 is not an a eigenvector of A
  3. True A 1 3 = 1 3 , so v3 is a 1–eigenvector of A
  4. False A 3 1 = 5 9 , so v4 is not an eigenvector of A
What are the eigenvalues of the matrix 24 1 2 ? Exactly one option must be correct)
a)
2
b)
0, -4
c)
0, 4
d)
2, -2

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
2 λ 4 1 2 λ = (2 λ)2 4 = λ2 4λ + 4 4 = λ(λ 4) = 0. The eigenvalues are therefore λ = 0 and λ = 4.
Choice (d) is incorrect
What are the eigenvalues of the matrix 2 7 1 3 ? Exactly one option must be correct)
a)
1 + 5 2 , 1 5 2 .
b)
1 + 3 2 , 1 3 2 .
c)
1 + i3 2 , 1 i3 2 .
d)
5 + i21 2 , 5 i21 2 .

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
2 λ 7 1 3 λ = (2 λ)(3 + λ) + 7 = λ2 + λ + 1 = 0. The eigenvalues are therefore λ = 1 ±1 4 2 = 1 ± i3 2 .
Choice (d) is incorrect
What is the characteristic equation of A = 16 1 2 ? Exactly one option must be correct)
a)
λ2 3λ + 8 = 0
b)
detA = λ2 + 3λ + 4
c)
λ2 + 3λ + 4 = 0
d)
detA = λ2 3λ + 8

Choice (a) is correct!
The characteristic equation of A is det(AλI) = 0 1 λ 6 1 2 λ
= (1 λ)(2 λ) + 6 = λ2 3λ + 8 = 0.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Let A = 34 2 1 . Which of the following statements are correct? (Zero or more options can be correct)
a)
2 1 belongs to the 1–eigenspace of A
b)
2 1 belongs to the 5–eigenspace of A
c)
1 1 belongs to the 5–eigenspace of A
d)
1 1 belongs to the 1–eigenspace of A

There is at least one mistake.
For example, choice (a) should be False.
As A 2 1 = 2 3 1 2 1 , 2 1 does not belong to the 1–eigenspace of A
There is at least one mistake.
For example, choice (b) should be True.
As A 2 1 = 10 5 = 5 2 1 , 2 1 belongs to the 5–eigenspace of A
There is at least one mistake.
For example, choice (c) should be False.
As A 1 1 = 1 1 = 1 2 1 , 1 1 belongs to the 1–eigenspace of A and not the 5–eigenspace of A
There is at least one mistake.
For example, choice (d) should be True.
As A 1 1 = 1 1 = 1 1 1 , the vector 1 1 belongs to in the 1–eigenspace of A.
Correct!
  1. False As A 2 1 = 2 3 1 2 1 , 2 1 does not belong to the 1–eigenspace of A
  2. True As A 2 1 = 10 5 = 5 2 1 , 2 1 belongs to the 5–eigenspace of A
  3. False As A 1 1 = 1 1 = 1 2 1 , 1 1 belongs to the 1–eigenspace of A and not the 5–eigenspace of A
  4. True As A 1 1 = 1 1 = 1 1 1 , the vector 1 1 belongs to in the 1–eigenspace of A.
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!
A 1 1 0 = 2 2 0 = 2 1 1 0 ,A 0 1 2 = 0 2 4 = 2 0 1 2
A 1 2 1 = 1 1 5 , A 1 1 4 = 6 6 12 .
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
What are the eigenvalues of 421 2 0 1 223 ? Exactly one option must be correct)
a)
0,2,3
b)
1,2,3
c)
4,5
d)
2,3

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
4 λ2 1 2 λ 1 2 23 λ = (2λ)2(3λ) = 0 λ = 2, λ = 3.
Given that 1 is an eigenvalue of A = 1 121 2 1 0 11 , what are the other eigenvalues ? Exactly one option must be correct)
a)
1, 2
b)
2, 1
c)
1, 2
d)
1, 2

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
|AλI| = 1 λ 1 2 1 2 λ 1 0 1 1 λ = 2λ2λ2+λ3 = 0. Since (1 λ) is a factor of 2 λ 2λ2 + λ3, (1 λ)(a + bλ λ2) = 2 λ 2λ2 + λ3 hence a = 2, b = 1 and the factors of (2 + λ λ2) are (2 λ) and (1 + λ).
Choice (d) is incorrect
What is the characteristic equation of the matrix 416 2 1 6 218 ? Exactly one option must be correct)
a)
36 45λ + 13λ2 λ3 = 0
b)
det(A λI) = 36 45λ + 13λ2 λ3
c)
36 40λ + 13λ2 λ3 = 0
d)
det(A λI) = 36 40λ + 13λ2 λ3

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
4 λ 1 6 2 1 λ 6 2 1 8 λ = (4 λ)[(1 λ)(8 λ) + 6] + 2(8 λ) 12 + 6[2 2(1 λ)] = (4 λ)(2 λ)(7 λ) 10(2 λ) = (2 λ)[(4 λ)(7 λ) 10] = (2 λ)(2 λ)(9 λ) = 36 40λ + 13λ2 λ3 = 0
Choice (d) is incorrect
Let 256 1 0 0 01 0 . Which of the following statements are correct ?
(Zero or more options can be correct)
a)
4 2 1 is in the 2-eigenspace of A
b)
9 3 1 is in the 3-eigenspace of A
c)
4 2 1 is in the 2-eigenspace of A
d)
9 3 2 is in the 3-eigenspace of A

There is at least one mistake.
For example, choice (a) should be False.
A 4 2 1 = 8 4 2 , so 4 2 1 belongs to the 2–eigenspace of A (and not the 2–eigenspace)
There is at least one mistake.
For example, choice (b) should be True.
A 9 3 1 = 3 9 3 1 , so 9 3 1 belongs to the 3–eigenspace of A
There is at least one mistake.
For example, choice (c) should be True.
A 4 2 1 = 2 4 2 1 , so 4 2 1 belongs to the 2–eigenspace of A
There is at least one mistake.
For example, choice (d) should be False.
A 9 3 2 = 21 9 3 , so 9 3 2 is not an eigenvector of A
Correct!
  1. False A 4 2 1 = 8 4 2 , so 4 2 1 belongs to the 2–eigenspace of A (and not the 2–eigenspace)
  2. True A 9 3 1 = 3 9 3 1 , so 9 3 1 belongs to the 3–eigenspace of A
  3. True A 4 2 1 = 2 4 2 1 , so 4 2 1 belongs to the 2–eigenspace of A
  4. False A 9 3 2 = 21 9 3 , so 9 3 2 is not an eigenvector of A