If you are reading this message either your browser does not support JavaScript or else JavaScript is not enabled. You will need to enable JavaScript and then reload this page before you can use this quiz.

Quiz 12: Eigenvalues and Eigenvectors

Question

Question 1

Let

A = [\begin{matrix} 4 & - 2 & 1 \\ 2 & 0 & 1 \\ 2 & - 2 & 3 \end{matrix}]

v_{1} = [\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}]

v_{2} = [\begin{matrix} 0 \\ 1 \\ 2 \end{matrix}]

v_{3} = [\begin{matrix} 1 \\ 2 \\ - 1 \end{matrix}]

and

v_{4} = [\begin{matrix} 1 \\ 1 \\ 4 \end{matrix}]

.
Which of the following statements is correct ?

Your answer is correct.

Not correct. Choice (b) is false.

A v_{3}

is not a multiple of

v_{3}

Not correct. Choice (c) is false.

A v_{3}

is not a multiple of

v_{3}

Not correct. Choice (d) is false.

A v_{3}

is not a multiple of

v_{3}

;

A v_{4}

is not a multiple of

v_{4}

Question 2

[\begin{matrix} 1 \\ 2 \\ 2 \end{matrix}]

is an eigenvector of

[\begin{matrix} 4 & - 2 & 1 \\ 2 & 0 & 1 \\ 2 & - 2 & 3 \end{matrix}]

. What is the corresponding eigenvalue? (Enter your answer into the answer box.)

Your answer is correct

Not correct. You may try again.

Multiply

[\begin{matrix} 4 & - 2 & 1 \\ 2 & 0 & 1 \\ 2 & - 2 & 3 \end{matrix}]

[\begin{matrix} 1 \\ 2 \\ 2 \end{matrix}]

. The result should be a multiple of

[\begin{matrix} 1 \\ 2 \\ 2 \end{matrix}]

Question 3

[\begin{matrix} - 5 \\ - 5 \\ - 5 \end{matrix}]

is an eigenvector of

[\begin{matrix} 4 & - 2 & 1 \\ 2 & 0 & 1 \\ 2 & - 2 & 3 \end{matrix}]

. What is the corresponding eigenvalue? (Enter your answer into the answer box.)

Your answer is correct

Not correct. You may try again.

Multiply

[\begin{matrix} 4 & - 2 & 1 \\ 2 & 0 & 1 \\ 2 & - 2 & 3 \end{matrix}]

[\begin{matrix} - 5 \\ - 5 \\ - 5 \end{matrix}]

. The result should be a multiple of

[\begin{matrix} - 5 \\ - 5 \\ - 5 \end{matrix}]

Question 4

What are the eigenvalues of

[\begin{matrix} 4 & 7 & 1 \\ 0 & - 3 & 8 \\ 0 & 0 & 2 \end{matrix}]

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 = [\begin{matrix} 2 & 5 & - 6 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}]

, find the other two eigenvalues.

Not correct. Choice (a) is false.

| A - λ I | = - λ^{3} + 2 λ^{2} + 5 λ - 6

. Now take out a factor of

(λ - 1)

Your answer is correct.

Not correct. Choice (c) is false.

| A - λ I | = - λ^{3} + 2 λ^{2} + 5 λ - 6

. Now take out a factor of

(λ - 1)

Not correct. Choice (d) is false.

| A - λ I | = - λ^{3} + 2 λ^{2} + 5 λ - 6

. Now take out a factor of

(λ - 1)

Question 6

A = [\begin{matrix} 4 & - 1 & 6 \\ 2 & 1 & 5 \\ 2 & - 1 & 0 \end{matrix}]

, which one of the following is

| A - λ I |

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 = [\begin{matrix} 1 & 2 & - 4 \\ - 1 & 4 & 8 \\ 0 & 1 & - 1 \end{matrix}]

Not correct. Choice (a) is false.

Expanding

| B - λ I |

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 - λ I |

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 - λ I |

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

[\begin{matrix} 1 & 2 & - 4 \\ - 1 & 4 & 8 \\ 0 & 1 & - 1 \end{matrix}]

, which of the following systems of equations should be solved to find the corresponding eigenvectors?

a)	$[\begin{matrix} - 4 & - 3 & - 9 \\ - 6 & - 1 & 3 \\ - 5 & - 4 & - 6 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$	b)	$[\begin{matrix} - 4 & 2 & - 4 \\ - 1 & - 1 & 8 \\ 0 & 1 & - 6 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} x \\ y \\ z \end{matrix}]$
c)	$[\begin{matrix} - 4 & 2 & - 4 \\ - 1 & - 1 & 8 \\ 0 & 1 & - 6 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$	d)	$[\begin{matrix} 1 & 2 & - 4 \\ - 1 & 4 & 8 \\ 0 & 1 & - 1 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$
e)	$[\begin{matrix} 1 & 2 & - 4 \\ - 1 & 4 & 8 \\ 0 & 1 & - 1 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 5 \\ 5 \\ 5 \end{matrix}]$

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 \times 3

matrix

A

has an eigenvalue of

- 1

. The matrix

A + I

reduces to

[\begin{matrix} 1 & 0 & - 2 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]

. Corresponding to the eigenvalue

- 1

, all the eigenvectors of

A

are non-zero vectors of the form:

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

[\begin{matrix} 0 & 0 & 3.6 \\ 0.8 & 0 & 0 \\ 0 & 0.6 & 0 \end{matrix}]

. Find the proportion of the population in each of the age groups once the population has stabilised.

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.

ABN: 15 211 513 464. CRICOS number: 00026A. Phone: +61 2 9351 2222.

Authorised by: Head, School of Mathematics and Statistics.

Contact the University | Disclaimer | Privacy | Accessibility

Sitemap

For questions or comments please contact webmaster@maths.usyd.edu.au

Questions:

right first attempt
right
wrong

a)	$v_{1}$ and $v_{2}$ are eigenvectors of $A$	b)	$v_{1}$ and $v_{3}$ are eigenvectors of $A$
c)	$v_{2}$ and $v_{3}$ are eigenvectors of $A$	d)	$v_{3}$ and $v_{4}$ are eigenvectors of $A$

a)	$1$ , $4$ , $7$ .	b)	$0$ , $4$ .
c)	$- 3$ , $2$ , $4$ .	d)	$- 3$ , $0$ , $1$ .

a)	$2$ and $- 3$ .	b)	$- 2$ and $3$ .
c)	$0$ and $3$ .	d)	$0$ and $3$ .

a)	$- λ^{3} + 5 λ^{2} + λ - 14$	b)	$- λ^{3} + 5 λ^{2} + λ - 4$
c)	$- λ^{3} + 5 λ^{2} - 4 λ$	d)	$- λ^{3} + 5 λ^{2} - 21 λ + 42$

a)	$- 1$ , $1$ , $4$ .	b)	$- 4$ , $- 1$ , $1$ .
c)	$- 5$ , $1$ , $2$ .	d)	$- 2$ , $1$ , $5$ .

a)	$[\begin{matrix} 2 t \\ 0 \\ t \end{matrix}], t \in ℝ$	b)	$[\begin{matrix} 2 t \\ s \\ t \end{matrix}], s, t \in ℝ$
c)	$[\begin{matrix} t \\ 0 \\ - 2 t \end{matrix}], t \in ℝ$	d)	$[\begin{matrix} t \\ s \\ 2 t \end{matrix}], s, t \in ℝ$

a)		Age group 1: $\frac{1}{3}$ Age group 2: $\frac{1}{3}$ Age group 3: $\frac{1}{3}$
b)		Age group 1: $\frac{1}{6}$ Age group 2: $\frac{1}{3}$ Age group 3: $\frac{1}{2}$
c)		Age group 1: $\frac{1}{4}$ Age group 2: $\frac{1}{2}$ Age group 3: $\frac{1}{4}$
d)		Age group 1: $\frac{1}{2}$ Age group 2: $\frac{1}{3}$ Age group 3: $\frac{1}{6}$

The University of Sydney - School of Mathematics and Statistics

Quiz 12: Eigenvalues and Eigenvectors

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Sitemap