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Quiz 10: Introduction to Eigenvalues and Eigenvectors

Last unanswered question  Question  Next unanswered question
 

Question 1

 
 
Which of the following are eigenvectors of [ ] 2 1 3 0? (More than one answer may be correct.)
a) [1] 1   b) [ 1 ] - 1   c) [5] 5
d) [ ] - 2 6   e) [ ] 1 - 3   f) [ ] - 3 1

 

There is at least one mistake.
For example, choice (a) should be true.
[2 1] 3 0 [1] 1 = 3[1] 1
There is at least one mistake.
For example, choice (b) should be false.
[2 1] 3 0 [ 1] - 1 = [1] 3, which is not a multiple of [ 1 ] - 1.
There is at least one mistake.
For example, choice (c) should be true.
[2 1] 3 0 [5] 5 = 3[5] 5
There is at least one mistake.
For example, choice (d) should be true.
[ ] 2 1 3 0 [ ] - 2 6 = -[ ] - 2 6
There is at least one mistake.
For example, choice (e) should be true.
[ ] 2 1 3 0 [ ] 1 - 3 = -[ ] 1 - 3
There is at least one mistake.
For example, choice (f) should be false.
[ ] 2 1 3 0 [ ] - 3 1 = [ ] - 5 - 9, which is not a multiple of [ ] - 3 1.
Your answers are correct
  1. True. [2 1] 3 0 [1] 1 = 3[1] 1
  2. False. [2 1] 3 0 [ 1] - 1 = [1] 3, which is not a multiple of [ 1 ] - 1.
  3. True. [2 1] 3 0 [5] 5 = 3[5] 5
  4. True. [ ] 2 1 3 0 [ ] - 2 6 = -[ ] - 2 6
  5. True. [ ] 2 1 3 0 [ ] 1 - 3 = -[ ] 1 - 3
  6. False. [ ] 2 1 3 0 [ ] - 3 1 = [ ] - 5 - 9, which is not a multiple of [ ] - 3 1.
 

Question 2

 
 
Which one of the following is an eigenvector of A = ⌊ ⌋ 1 2 - 1 ⌈1 0 1 ⌉ 4 - 4 5?
a) ⌊ ⌋ - 2 ⌈ 1⌉ 5   b) ⌊ ⌋ - 2 ⌈ 1⌉ 4   c) ⌊ ⌋ ⌈ 2⌉ - 1 4

 

Not correct. Choice (a) is false.
A⌊ ⌋ - 2 ⌈ 1 ⌉ 5 = ⌊ ⌋ - 6 ⌈ 3 ⌉ 13, which is not a multiple of ⌊ ⌋ - 2 ⌈ 1⌉ 5.
Your answer is correct.
A⌊ ⌋ - 2 ⌈ 1 ⌉ 4 = 2⌊ ⌋ - 2 ⌈ 1 ⌉ 4
Not correct. Choice (c) is false.
A⌊ ⌋ ⌈ 2 ⌉ - 1 4 = ⌊ ⌋ ⌈- 4⌉ 6 32, which is not a multiple of ⌊ ⌋ ⌈ 2⌉ - 1 4.
 

Question 3

 
 
Given that [ ] 0 7 is an eigenvector of [ ] 2 0 0 - 3, what is the corresponding eigenvalue? (Enter your answer into the answer box.)

 

Your answer is correct

Not correct. You may try again.
The eigenvalue is λ such that [2 0 ] 0 - 3 [0] 7 = λ[0] 7.
 

Question 4

 
 
Given that [6] 6 is an eigenvector of [3 1] 4 0, what is the corresponding eigenvalue? (Enter your answer into the answer box.)

 

Your answer is correct

Not correct. You may try again.
The eigenvalue is λ such that [3 1] 4 0 [6] 6 = λ[6] 6.
 

Question 5

 
 
Given that ⌊- 3⌋ ⌈ 3⌉ 12 is an eigenvector of ⌊1 2 - 1⌋ ⌈1 0 1⌉ 4 - 4 5, what is the corresponding eigenvalue? (Enter your answer into the answer box.)

 

Your answer is correct

Not correct. You may try again.
The eigenvalue is λ such that ⌊ ⌋ 1 2 - 1 ⌈1 0 1 ⌉ 4 - 4 5 ⌊ ⌋ - 3 ⌈ 3 ⌉ 12 = λ⌊ ⌋ - 3 ⌈ 3 ⌉ 12.
 

Question 6

 
 
The solutions to which one of the following equations are the eigenvalues of [ 2 3] - 2 8?
a) λ2 - 10λ + 22 = 0   b) λ2 - 10λ + 10 = 0
c) λ2 - 6λ + 22 = 0   d) λ2 - 6λ + 22 = 0

 

Your answer is correct.
Not correct. Choice (b) is false.
Try again. The equation is | | ||2 - λ 3 || |- 2 8- λ| = 0.
Not correct. Choice (c) is false.
Try again. The equation is | | ||2- λ 3 || | - 2 8 - λ| = 0.
Not correct. Choice (d) is false.
Try again. The equation is | | ||2- λ 3 || | - 2 8 - λ | = 0.
 

Question 7

 
 
What are the eigenvalues of the matrix [ ] - 3 5 0 8?
a) -3, 5   b) -3, 0
c) -3, 8   d) 5, 8
e) 5, 0

 

Not correct. Choice (a) is false.
Since the matrix is triangular, the eigenvalues are the diagonal entries.
Not correct. Choice (b) is false.
Since the matrix is triangular, the eigenvalues are the diagonal entries.
Your answer is correct.
Not correct. Choice (d) is false.
Since the matrix is triangular, the eigenvalues are the diagonal entries.
Not correct. Choice (e) is false.
Since the matrix is triangular, the eigenvalues are the diagonal entries.
 

Question 8

 
 
Find the eigenvalues of the matrix [ ] - 7 3 - 6 4.
a) 1, 10   b) 2, -5
c) -7, 4   d) -2, 5

 

Not correct. Choice (a) is false.
Try again. Find λ such that | | ||- 7- λ 3 || | - 6 4- λ| = 0.
Your answer is correct.
Not correct. Choice (c) is false.
Try again. Find λ such that ||- 7 - λ 3 || || - 6 4- λ|| = 0.
Not correct. Choice (d) is false.
Try again. Find λ such that ||- 7 - λ 3 || || - 6 4- λ|| = 0.
 

Question 9

 
 
Given that 0 is an eigenvalue of [2 1] 2 1 find all the corresponding eigenvectors.
a)

[ ] 2t t , t ∈ ℝ, t ⁄= 0.
  b)
[ t] 2t , t ∈ ℝ, t ⁄= 0.
c)
[ ] - t , t ∈ ℝ, t ⁄= 0. 2t
  d)
[ ] - 2t t , t ∈ ℝ, t ⁄= 0.

 

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

Question 10

 
 
Given that -3 is an eigenvalue of [5 - 1] 8 - 4 find all the corresponding eigenvectors.
a)
[] t , t ∈ ℝ, t ⁄= 0. t
  b)
[ ] t - t , t ∈ ℝ, t ⁄= 0.
c)
[ t] 8t , t ∈ ℝ, t ⁄= 0.
  d)
[ ] 8t , t ∈ ℝ, t ⁄= 0. t

 

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