Eigenvalues and Eigenvectors: Interactive Learning Quizzes

Question 1

What is the relationship between the eigenvalues of a matrix and its characteristic equation?

Accepted Answer

The eigenvalues are the roots of the characteristic equation.

Answer

The coefficients of the characteristic equation are the eigenvalues.

Answer

The characteristic equation helps determine the eigenvectors.

Question 2

Prove that the eigenvectors of a real symmetric matrix are orthogonal to each other. Provide a formal mathematical proof.

Accepted Answer

Let A be a real symmetric matrix and let v and w be two eigenvectors of A corresponding to distinct eigenvalues λ and μ, respectively. Show that the dot product of v and w is zero.

Answer

The eigenvectors of a symmetric matrix are orthogonal only if the matrix is positive definite.

Answer

The eigenvectors of a symmetric matrix are never orthogonal.

Answer

The orthogonality of eigenvectors depends on the specific values of the eigenvalues.

Question 3

Consider the differential equation y' + 2y = 0. Find the general solution.

Accepted Answer

y = e^(-2t)

Answer

y = e^(2t)

Answer

y = 2e^(-t)

Answer

y = 2e^(t)

Question 4

Find the eigenvectors of the matrix A = [[3, 2], [-2, 3]] corresponding to the eigenvalue λ = 3.

Accepted Answer

[[1], [1]]

Answer

[[-1], [1]]

Answer

[[1], [-1]]

Answer

[[-1], [-1]]

Question 5

Find the eigenvalues of the matrix A = [[1, 2], [2, 1]]

Accepted Answer

[3, -1]

Answer

[1, 2]

Answer

[1, -2]

Answer

[3, 1]

Question 6

Find the eigenvectors of the matrix A = [[1, 2], [2, 1]] corresponding to the eigenvalue λ = 3.

Accepted Answer

[[-1], [1]]

Answer

[[1], [1]]

Answer

[[1], [-1]]

Answer

[[-1], [-1]]

Question 7

In a linear transformation, what do eigenvectors represent geometrically?

Accepted Answer

Directions where vectors are stretched or shrunk the most or least

Answer

Perpendicular directions to eigenvectors with different eigenvalues

Answer

Eigenvectors with lengths equal to their eigenvalues

Answer

Matrices with eigenvalues that sum up to their trace

Question 8

What is degeneracy, and how does it impact the number of linearly independent eigenvectors associated with an eigenvalue?

Accepted Answer

Degeneracy occurs when an eigenvalue has multiple linearly independent eigenvectors.

Answer

Degeneracy occurs when an eigenvalue is zero.

Answer

Degeneracy never occurs in matrices with real eigenvalues.

Question 9

Which of the following is NOT an eigenvalue of the matrix A = [[2, 1], [-1, 2]]?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 10

In the characteristic equation |A - λI| = 0 of a matrix, what does λ represent?

Accepted Answer

Eigenvalue

Answer

Eigenvector

Answer

Determinant

Answer

Matrix inverse

Question 11

What is the geometric interpretation of an eigenvector?

Accepted Answer

A direction in which the transformation represented by the matrix scales vectors.

Answer

A vector that is perpendicular to the subspace spanned by the eigenvectors.

Answer

A vector that is multiplied by the eigenvalue when the matrix is applied to it.

Answer

The vector with the largest magnitude among all eigenvectors.

Question 12

What is the trace of a matrix?

Accepted Answer

The sum of its diagonal elements

Answer

The sum of its eigenvalues

Answer

The product of its eigenvalues

Answer

The determinant of its cofactor matrix

Question 13

What is the relationship between the eigenvalues of a matrix and its inverse?

Accepted Answer

The eigenvalues of the inverse are the reciprocals of the eigenvalues of the original matrix.

Answer

The eigenvalues of the inverse are the same as the eigenvalues of the original matrix.

Answer

The eigenvalues of the inverse are the negatives of the eigenvalues of the original matrix.

Answer

The eigenvalues of the inverse are not related to the eigenvalues of the original matrix.

Question 14

Which of the following matrices has complex eigenvalues?

Accepted Answer

[[1, 2], [-2, 1]]

Answer

[[1, 0], [0, 1]]

Answer

[[1, 0], [0, -1]]

Answer

[[2, 1], [-1, 2]]

Question 15

In the system of differential equations x' = Ax, where A is a 2x2 matrix, how can eigenvalues and eigenvectors be used to determine the stability of the system?

Accepted Answer

By finding the eigenvalues of A and examining their real parts

Answer

By calculating the determinant of A and checking its sign

Answer

By finding the eigenvectors of A and analyzing their directions

Answer

By diagonalizing A and checking the signs of the diagonal entries

Question 16

What is the mathematical definition of an eigenvalue?

Accepted Answer

A scalar value 'λ' that satisfies the equation Av = λv for some nonzero vector 'v'

Answer

A vector 'v' that satisfies the equation Av = λv for some nonzero scalar value 'λ'

Answer

A matrix A that satisfies the equation Av = λv for some nonzero vector 'v'

Question 17

Describe the relationship between an eigenvalue and the corresponding eigenvector of a matrix.

Accepted Answer

An eigenvector is a non-zero vector that, when multiplied by the matrix, yields a vector parallel to itself, scaled by the eigenvalue.

Answer

An eigenvalue is the determinant of the corresponding eigenvector.

Question 18

Which method can you use to find the eigenvalues of a square matrix?

Accepted Answer

Solve the characteristic equation det(A - λI) = 0, where 'A' is the square matrix, 'λ' is the eigenvalue and 'I' is the identity matrix.

Answer

Use the trace of the matrix.

Answer

Find the determinant of the matrix.

Question 19

Explain how to find the eigenvectors of a matrix for a given eigenvalue.

Accepted Answer

Substitute the eigenvalue 'λ' into the equation (A - λI)v = 0, where 'A' is the matrix and 'I' is the identity matrix, and then solve for the non-trivial solutions 'v', which are the eigenvectors.

Answer

Use the adjoint of the matrix.

Answer

Find the inverse of the matrix.

Question 20

Is the matrix A = [[2, 1], [-1, 2]] diagonalizable?

Accepted Answer

Yes

Answer

Maybe

Answer

No

Answer

Cannot be determined

Question 21

What is the definition of an eigenvalue of a matrix?

Accepted Answer

A value that, when substituted into the characteristic equation, makes the determinant of the matrix zero.

Answer

A vector that, when multiplied by the matrix, results in itself.

Answer

A constant that appears in the characteristic equation.

Question 22

What is the characteristic equation of a matrix?

Accepted Answer

An equation that results from setting the determinant of the matrix minus lambda times the identity matrix to zero.

Answer

An equation that describes the eigenvectors of the matrix.

Answer

An equation that calculates the eigenvalues of the matrix.

Question 23

Determine if the following statement is true or false: Eigenvalues can be complex numbers

Accepted Answer

True

Answer

False

Answer

Only for matrices with complex entries

Question 24

Define an eigenvector.

Accepted Answer

A nonzero vector that, when transformed by a matrix, results in a scalar multiple of itself.

Answer

A vector with a length of 1.

Answer

A vector orthogonal to the subspace spanned by the eigenvectors.

Question 25

Name an application of eigenvalues and eigenvectors.

Accepted Answer

Solving systems of differential equations

Answer

Calculating the volume of a sphere

Answer

Determining the probability of an event

Answer

Finding the area of a triangle

Question 26

Which property applies to eigenvectors?

Accepted Answer

Eigenvectors corresponding to distinct eigenvalues are linearly independent.

Answer

The sum of all eigenvectors of a matrix is the zero vector.

Answer

Eigenvectors corresponding to the same eigenvalue are always orthogonal.

Question 27

Which of the following is the characteristic equation for a matrix A?

Accepted Answer

det(A - λI) = 0

Answer

λdet(A) = 0

Answer

λ - det(A) = 0

Answer

det(A + λI) = 0

Question 28

A matrix with distinct eigenvalues is:

Accepted Answer

Diagonalizable

Answer

Non-singular

Answer

Orthogonal

Answer

Symmetric

Question 29

Which of the following methods can be used to find eigenvalues of a matrix?

Accepted Answer

Characteristic polynomial method

Answer

LU decomposition

Answer

Gram-Schmidt process

Answer

Conjugate gradient method

Question 30

If λ is an eigenvalue of a matrix A, then A - λI is:

Accepted Answer

Singular

Answer

Symmetric

Answer

Orthogonal

Answer

Invertible

Question 31

The geometric interpretation of an eigenvector is:

Accepted Answer

A direction in which a transformation scales a vector by a factor of the eigenvalue

Answer

A vector that is orthogonal to all other eigenvectors

Answer

A vector that lies on the diagonal of the matrix

Question 32

Which of the following matrices has 0 as an eigenvalue?

Accepted Answer

[[0, 1], [-1, 0]]

Answer

[[1, 0], [0, 1]]

Answer

[[2, 1], [-1, 2]]

Question 33

If A is an invertible matrix and λ is an eigenvalue of A, then 1/λ is an eigenvalue of:

Accepted Answer

A^-1

Answer

A^2

Answer

A - I

Answer

A^T

Question 34

Which of the following statements about eigenvectors and eigenvalues is incorrect?

Accepted Answer

Eigenvalues are always real numbers for symmetric matrices

Answer

The number of eigenvectors is equal to the number of eigenvalues

Answer

Eigenvectors can be used to construct a basis for the vector space

Answer

Eigenvectors are linearly independent

Question 35

If A is an n x n matrix and λ is an eigenvalue of A, then:

Accepted Answer

λ is a root of the characteristic polynomial of A

Answer

λ is the determinant of A

Answer

λ is the rank of A

Answer

λ is the trace of A

Question 36

If A is an invertible matrix, then its eigenvalues are:

Accepted Answer

Non-zero

Answer

Positive

Answer

Zero

Answer

Negative

Question 37

If the eigenvalues of a matrix A are complex, then the matrix A is:

Accepted Answer

Non-real

Answer

Symmetric

Answer

Orthogonal

Answer

Real

Question 38

Which property describes eigenvectors?

Accepted Answer

They scale the matrix when multiplied by the corresponding eigenvalue.

Answer

They are always nonzero.

Answer

They are independent of the matrix.

Answer

They are always perpendicular to each other.

Question 39

True or False: The eigenvectors of a symmetric matrix are always orthogonal.

Accepted Answer

True

Answer

Only if the matrix is positive definite

Answer

False

Answer

Only if the eigenvalues are distinct

Question 40

An eigenvalue of a square matrix A is a scalar λ for which there exists a non-zero vector x satisfying the equation Ax = λx. Is this statement true?

Accepted Answer

True

Answer

Not always true

Answer

Only if A is symmetric

Answer

False

Question 41

Determine an eigenvector corresponding to the eigenvalue λ = 2 for the matrix A = [[2, 1], [-1, 2]].

Accepted Answer

[[-1], [1]]

Answer

[[1], [1]]

Answer

[[0], [2]]

Answer

[[2], [-1]]

Question 42

Identify the matrix from the following options that has a repeated eigenvalue.

Accepted Answer

A = [[1, 2], [0, 1]]

Answer

D = [[0, 1], [-1, 0]]

Answer

C = [[2, 0], [1, 1]]

Answer

B = [[2, 1], [-1, 2]]

Question 43

Find the eigenvectors of the matrix A = [[2, 1], [-1, 2]] corresponding to the eigenvalue λ = 1.

Accepted Answer

[[1], [-1]]

Answer

[[2], [-1]]

Answer

[[0], [2]]

Answer

[[1], [1]]

Question 44

Which of the following is a direct application of eigenvalues and eigenvectors?

Accepted Answer

Solving systems of differential equations

Answer

Finding the area of a triangle

Answer

Computing the volume of a sphere

Answer

Calculating the probability of an event

Question 45

If A is a real symmetric matrix, what can you say about its eigenvalues?

Accepted Answer

All eigenvalues are real.

Answer

All eigenvalues are complex.

Answer

All eigenvalues are negative.

Answer

All eigenvalues are positive.

Question 46

Which of the following is an eigenvalue of a matrix?

Accepted Answer

A scalar that makes the difference between the matrix and the identity matrix singular

Answer

A vector that is multiplied by the matrix to give itself

Answer

A matrix that commutes with the given matrix

Question 47

Which of the following is an eigenvector of a matrix?

Accepted Answer

A non-zero vector that is multiplied by the matrix to give a multiple of itself

Answer

A vector that is orthogonal to all eigenvectors of the matrix

Answer

A matrix that has the same eigenvectors as the given matrix

Question 48

What is the sum of the eigenvalues of a matrix?

Accepted Answer

The trace of the matrix

Answer

The determinant of the matrix

Answer

The rank of the matrix

Answer

The nullity of the matrix

Question 49

What is the product of the eigenvalues of a matrix?

Accepted Answer

The determinant of the matrix

Answer

The nullity of the matrix

Answer

The rank of the matrix

Answer

The trace of the matrix

Question 50

Which of the following matrices has an eigenvalue of 3?

Accepted Answer

[[3, 0], [0, 2]]

Answer

[[0, 1], [1, 0]]

Answer

[[2, 0], [0, 3]]

Answer

[[1, 0], [0, 4]]

Question 51

Which of the following vectors is an eigenvector of the matrix [[3, 0], [0, 2]] corresponding to the eigenvalue 3?

Accepted Answer

[1, 0]

Answer

[0, 1]

Answer

[1, 1]

Question 52

Which of the following is a characteristic equation of a matrix?

Accepted Answer

The polynomial whose roots are the eigenvalues of the matrix

Answer

The polynomial whose roots are the eigenvectors of the matrix

Answer

The polynomial whose roots are the traces of the matrix

Question 53

Which of the following is a property of eigenvalues?

Accepted Answer

They are invariant under similarity transformations

Answer

They are always real

Answer

They are always positive