Eigenvalues and Eigenvectors: A Comprehensive Guide for Abstract Algebra Students

Question 1

Consider the matrix [[2, 0], [0, 2]]. Which of the following is an eigenvalue of this matrix?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 2

Which of the following statements about eigenvalues is incorrect?

Accepted Answer

The product of eigenvalues is equal to the determinant of the matrix.

Answer

Eigenvalues can be real or complex.

Answer

A matrix may have repeated eigenvalues.

Answer

The sum of eigenvalues is equal to the trace of the matrix.

Question 3

Find the determinant of a diagonalizable matrix.

Accepted Answer

The product of its eigenvalues

Answer

The sum of its eigenvalues

Answer

The square of its eigenvalues

Answer

The reciprocal of the product of its eigenvalues

Question 4

Identify the geometric transformation represented by the matrix [[cos(θ), -sin(θ)], [sin(θ), cos(θ)]].

Accepted Answer

Rotation by an angle θ

Answer

Reflection about the y-axis

Answer

Translation by a vector [0, θ]

Answer

Shear mapping along the y-axis

Question 5

What is the rank of a matrix that has only zero eigenvalues?

Accepted Answer

0

Answer

1

Answer

Equal to the number of rows

Answer

Equal to the number of columns

Question 6

Which of the following is the characteristic polynomial of the matrix A?

Accepted Answer

p(λ) = det(A - λI)

Answer

p(λ) = det(A - λ)

Answer

p(λ) = det(A + λ)

Answer

p(λ) = det(A + λI)

Question 7

If λ is an eigenvalue of matrix A, what can you say about the invertibility of (A - λI)?

Accepted Answer

A - λI is not invertible.

Answer

A - λI is invertible.

Answer

A + λI is invertible.

Answer

A + λI is not invertible.

Question 8

What is the determinant of a diagonal matrix?

Accepted Answer

Product of the diagonal elements

Answer

Sum of the diagonal elements

Answer

Difference of the diagonal elements

Answer

Zero

Question 9

What is the null space of matrix A?

Accepted Answer

Set of all vectors that are annihilated by the matrix.

Answer

Set of all vectors that are perpendicular to the row space.

Answer

Set of all vectors that are orthogonal to the column space.

Answer

Set of all vectors that span the matrix.

Question 10

If a square matrix A is diagonalizable, which of the following is guaranteed to be true?

Accepted Answer

A has the same number of linearly independent eigenvectors as its dimension.

Answer

A can be written as a product of two matrices.

Answer

A is symmetric.

Answer

A is invertible.

Question 11

Which of the following is an eigenvector of the matrix A = [1 2; -1 1] corresponding to the eigenvalue λ = 1?

Accepted Answer

[1; 1]

Answer

[1; 0]

Answer

[0; 1]

Answer

[-1; 1]

Question 12

Consider the matrix A = [2 1; 0 2]. Which of the following is the diagonalized form of A?

Accepted Answer

[2 0; 0 2]

Answer

[2 1; 0 2]

Answer

[1 0; 0 2]

Answer

[0 1; 0 2]

Question 13

If the geometric multiplicity of an eigenvalue of a square matrix is less than its algebraic multiplicity, then the matrix is said to be

Accepted Answer

Defective

Answer

Diagonalizable

Answer

Invertible

Answer

Symmetric

Question 14

Which of the following is true for a matrix with complex eigenvalues?

Accepted Answer

Its eigenvalues occur in conjugate pairs.

Answer

Its eigenvectors are always real.

Answer

Its eigenvectors are always complex.

Answer

Its eigenvalues are always positive.

Question 15

For the matrix A = [2 1; -1 2], what is the eigenspace corresponding to the eigenvalue λ = 1?

Accepted Answer

The subspace spanned by [1; 1]

Answer

The subspace spanned by [1; -1]

Answer

The subspace spanned by [0; 1]

Answer

The subspace spanned by [1; 0]

Question 16

Which of the following is a defining property of diagonalizable matrices?

Accepted Answer

Their eigenvectors form a linearly independent set.

Answer

Their eigenvalues are always distinct.

Answer

They are always symmetric matrices.

Answer

They are always positive definite matrices.

Question 17

Consider the matrix A = [1 2; -1 1]. What is the characteristic equation of A?

Accepted Answer

(x - 1)(x + 1)

Answer

(x - 2)(x + 2)

Answer

(x - 1)^2

Answer

(x + 1)^2

Question 18

Find the solution to the system of differential equations dx/dt = 2x + y, dy/dt = -x + 2y using the eigenvalues and eigenvectors of the associated matrix.

Accepted Answer

x(t) = e^(2t) (c1 cos t + c2 sin t), y(t) = e^(2t) (c1 sin t - c2 cos t)

Answer

x(t) = e^t (c1 cos t + c2 sin t), y(t) = e^t (c1 sin t - c2 cos t)

Answer

x(t) = e^(-2t) (c1 cos t + c2 sin t), y(t) = e^(-2t) (c1 sin t - c2 cos t)

Question 19

Prove that the eigenvalues of an invertible matrix are non-zero.

Accepted Answer

Consider the determinant of (A - lambda * I), where lambda is an eigenvalue of A. Since A is invertible, its determinant is non-zero. By the definition of eigenvalues, (A - lambda * I) is not invertible, which implies that its determinant is zero. This contradiction establishes that lambda must be non-zero.

Answer

If A is invertible, then its inverse exists. The eigenvalues of A are the same as the eigenvalues of A^-1. Since A^-1 is also invertible, its eigenvalues are non-zero.

Answer

The eigenvalues of an invertible matrix are always positive.

Answer

The eigenvalues of an invertible matrix are always distinct.

Question 20

How do eigenvectors contribute to understanding the geometric characteristics of linear transformations represented by matrices?

Accepted Answer

Eigenvectors define the axes along which the transformation scales or rotates the coordinate space.

Answer

Eigenvectors are irrelevant to the geometric properties of linear transformations.

Answer

Eigenvectors solely determine the eigenvalues of a matrix.

Question 21

Which of the following accurately describes the nature of eigenvalues?

Accepted Answer

Eigenvalues are the roots of the characteristic polynomial of a matrix.

Answer

Eigenvalues represent the scalar multipliers associated with eigenvectors.

Answer

Eigenvalues correspond to the determinants of matrices.

Answer

Eigenvalues are always positive in value.

Question 22

In geometric terms, what does an eigenvector represent?

Accepted Answer

A linear path extending from the origin within the vector space.

Answer

A flat plane intersecting the origin within the vector space.

Answer

An isolated point within the vector space.

Answer

A subspace of the encompassing vector space.

Question 23

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

Accepted Answer

The sum of the eigenvalues equals the trace of the matrix.

Answer

The determinant of the matrix is equal to the sum of the eigenvalues.

Answer

The rank of the matrix is equivalent to the number of eigenvalues.

Answer

The nullity of the matrix is determined by the eigenvalues.

Question 24

For the matrix [2 1; -1 2], what is its characteristic polynomial?

Accepted Answer

x^2 - 4x + 3

Answer

x^2 - 3x + 4

Answer

x^2 + 4x + 3

Answer

x^2 + 3x + 4

Question 25

Determine the set of all eigenvalues for a nilpotent matrix.

Accepted Answer

{0}

Answer

All real numbers

Answer

All non-zero real numbers

Answer

All complex numbers

Question 26

What is the characteristic polynomial of a matrix A?

Accepted Answer

det(A - λI)

Answer

det(A + λI)

Answer

tr(A - λI)

Answer

tr(A + λI)

Question 27

Prove that the eigenvalues of a symmetric matrix are all real.

Accepted Answer

True: The eigenvalues of a symmetric matrix are all real.

Answer

False: The eigenvalues of a symmetric matrix can be complex.

Answer

False: The eigenvalues of a symmetric matrix can be either real or complex.

Answer

False: The eigenvalues of a symmetric matrix are always positive.

Question 28

In the context of eigenvalue-eigenvector pairs for a square matrix A, what is the name of the nonzero vector v such that Av = λv, where λ is an eigenvalue of A?

Accepted Answer

Eigenvector

Answer

Characteristic vector

Answer

Basis vector

Answer

Linearly independent vector

Question 29

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

Accepted Answer

1 and 3

Answer

1 and -3

Answer

2 and 1

Answer

2 and -1

Question 30

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

Accepted Answer

[-1, 1]^T

Answer

[1, -1]^T

Answer

[1, 1]^T

Answer

[-1, -1]^T

Question 31

Find the eigenvalues of the matrix A = [[0, 1], [-1, 0]]

Accepted Answer

i and -i

Answer

1 and -1

Answer

0 and 1

Answer

0 and -1

Question 32

Find the determinant of the matrix A = [[λ, 1], [0, λ]]

Accepted Answer

λ^2

Answer

2λ

Answer

1

Answer

λ

Question 33

Which of the following describes eigenvalues of a matrix A?

Accepted Answer

They are the roots of the characteristic polynomial of A.

Answer

They are the determinants of A

Answer

They are the diagonal elements of A.

Question 34

If a matrix A has n distinct eigenvalues, what is true about A?

Accepted Answer

A is diagonalizable.

Answer

A has a non-trivial null space.

Answer

A is symmetric

Answer

A is invertible

Question 35

The trace of a matrix A represents the sum of its:

Accepted Answer

Eigenvalues

Answer

Singular values

Answer

Eigenvectors

Answer

Determinants

Question 36

Consider a 3x3 matrix A with eigenvalues λ1 = 2, λ2 = 3, and λ3 = 4. What is the characteristic polynomial of A?

Accepted Answer

(x - 2)(x - 3)(x - 4)

Answer

(x + 2)(x + 3)(x + 4)

Answer

(x - 2)(x - 3) + (x - 4)

Question 37

What is the eigenvalue of the matrix  A = [[2, 1], [-1, 2]]?

Accepted Answer

λ = 3

Answer

λ = 1

Answer

λ = -3

Answer

λ = 0

Question 38

For a matrix to be diagonalizable, which of the following conditions must be met?

Accepted Answer

The matrix must have n linearly independent eigenvectors.

Answer

The matrix must be symmetric.

Answer

The matrix must have all distinct eigenvalues.

Question 39

Consider the matrix C = [[1, 2], [-1, 3]].  Given that the eigenvectors corresponding to eigenvalues λ1 and λ2 are [1, -1] and [2, 1], respectively, what is the diagonal matrix D that diagonalizes C?

Accepted Answer

[[λ1, 0], [0, λ2]]

Answer

[[1, 2], [-1, 3]]

Answer

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

Answer

[[λ1, λ2], [0, 0]]

Question 40

Which of the following is a practical application of eigenvalues and eigenvectors in linear algebra?

Accepted Answer

Solving systems of differential equations.

Answer

Finding the area of a triangle.

Answer

Determining the volume of a cube.

Answer

Calculating the probability of an event.

Question 41

If A is an n x n matrix with n distinct eigenvalues, what is the characteristic polynomial of A?

Accepted Answer

(λ - λ1)(λ - λ2)...(λ - λn)

Answer

λn + λn-1 + ... + λ1 + λ0

Answer

λn - λn-1 + ... - λ1 + λ0

Answer

(λ + λ1)(λ + λ2)...(λ + λn)

Question 42

What is a defining characteristic of the eigenvectors of a matrix?

Accepted Answer

They span the eigenspace corresponding to the eigenvalue.

Answer

They are always linearly dependent.

Answer

They are always orthogonal to each other.

Question 43

What is the algebraic multiplicity of an eigenvalue?

Accepted Answer

The number of times it appears as a root of the characteristic polynomial.

Answer

The sum of all the eigenvalues.

Answer

The number of linearly independent eigenvectors corresponding to it.

Question 44

If A is an invertible matrix, what can you conclude about its eigenvalues?

Accepted Answer

All its eigenvalues are nonzero.

Answer

All its eigenvectors are linearly independent.

Answer

Its trace is equal to 0.

Answer

Its determinant is equal to 0.

Question 45

An eigenvector of a square matrix A is defined as a non-zero vector v that, when multiplied by A, results in a scalar multiple of itself. Which of the following options accurately describes this relationship?

Accepted Answer

Av = λv, where λ is a scalar.

Answer

A vector that is orthogonal to all columns of A.

Answer

Any vector that is linearly independent of the columns of A.

Answer

A vector that spans the null space of A.

Question 46

If a square matrix A is diagonalizable, what can we conclude about its eigenvectors?

Accepted Answer

They form a basis for the vector space.

Answer

They are all orthogonal to each other.

Answer

They are all unit vectors.

Answer

They are all linearly dependent.

Question 47

What is the characteristic polynomial of the matrix A = [[2, 1], [1, 2]]?

Accepted Answer

λ^2 - 4λ + 3

Answer

λ^2 - λ + 1

Answer

λ^2 - 2λ + 1

Answer

λ^2 - 3λ + 2

Question 48

An eigenvalue of a matrix has algebraic multiplicity 2. What can we say about its geometric multiplicity?

Accepted Answer

It can be 1 or 2.

Answer

It must be less than 2.

Answer

It is unrelated to the algebraic multiplicity.

Answer

It must be 2.

Question 49

Which of the following statements is NOT always true about similar matrices?

Accepted Answer

They have the same eigenvectors.

Answer

They have the same trace.

Answer

They have the same eigenvalues.

Answer

They have the same determinant.

Question 50

The eigenvectors corresponding to distinct eigenvalues of a matrix are always:

Accepted Answer

Linearly independent.

Answer

Scalar multiples of each other.

Answer

Unit vectors.

Answer

Orthogonal.

Question 51

What is the geometric interpretation of diagonalizing a matrix?

Accepted Answer

Finding a new coordinate system where the linear transformation represented by the matrix is a scaling transformation along each axis.

Answer

Finding the inverse of the matrix.

Answer

Finding the determinant of the matrix.

Question 52

If a matrix A has n linearly independent eigenvectors, what can we conclude about A?

Accepted Answer

A is diagonalizable.

Answer

A is not invertible.

Answer

A has n distinct eigenvalues.

Answer

A is the identity matrix.

Question 53

Which of the following applications heavily relies on the concept of eigenvalues and eigenvectors?

Accepted Answer

Solving systems of linear differential equations

Answer

Finding the area of a triangle

Answer

Calculating the dot product of two vectors

Answer

Determining the equation of a plane