Eigenvalues and Eigenvectors: A Comprehensive Guide for Mechanical Engineering Students

Question 1

Which of the following is the definition of an eigenvalue for a matrix A?

Accepted Answer

A scalar value λ such that the system of equations (A - λI)x = 0 has a non-trivial solution.

Answer

A vector x such that the system of equations (A - λI)x = 0 has a trivial solution.

Answer

A scalar value λ such that the system of equations (A - λI)x = 0 has a unique solution.

Answer

A vector x such that the system of equations (A - λI)x = 0 has a non-unique solution.

Question 2

What is the definition of an eigenvalue for a matrix **A**?

Accepted Answer

A scalar that, when multiplied by a nonzero vector **v**, results in a vector that is a multiple of **v**

Answer

A scalar that, when multiplied by a nonzero vector **v**, results in the vector **v**

Answer

A vector that, when multiplied by a nonzero scalar ( lambda ), results in the vector **v**

Answer

A vector that, when multiplied by a nonzero scalar ( lambda ), results in a vector that is a multiple of **v**

Question 3

Which of the following statements accurately defines an eigenvalue?

Accepted Answer

A scalar value associated with a non-zero vector that remains unchanged under a linear transformation of that vector.

Answer

A vector that remains unchanged in magnitude under a linear transformation.

Answer

A vector that is perpendicular to all other vectors in the vector space.

Answer

A scalar value indicating the magnitude of a linear transformation applied to a vector.

Question 4

In the context of linear algebra, which equation represents the relationship between an eigenvalue 'λ' and its corresponding eigenvector 'v'?

Accepted Answer

Av = λv

Answer

vA = λv

Answer

λv = Av

Answer

Aλ = vλ

Question 5

Given a square matrix represented as [[a, b], [c, d]], which of the following options correctly identifies its eigenvalues?

Accepted Answer

(a + d) ± √((a - d)² + 4bc)

Answer

(a + d) ± √((a - d)² - 4bc)

Answer

(a - d) ± √((a + d)² + 4bc)

Answer

(a - d) ± √((a + d)² - 4bc)

Question 6

Define an eigenvalue.

Accepted Answer

A scalar quantity associated with a linear transformation.

Answer

A vector quantity associated with a linear transformation.

Answer

A matrix quantity associated with a linear transformation.

Answer

A complex number associated with a linear transformation.

Question 7

Describe the relationship between an eigenvalue and its corresponding eigenvector.

Accepted Answer

The eigenvector is multiplied by the eigenvalue when transformed.

Answer

The eigenvector is divided by the eigenvalue when transformed.

Answer

The eigenvector is added to the eigenvalue when transformed.

Answer

The eigenvector is subtracted from the eigenvalue when transformed.

Question 8

What is a defining characteristic of eigenvalues?

Accepted Answer

They are the roots of the characteristic equation of the matrix.

Answer

They are the null vectors of a matrix.

Answer

They are always real numbers.

Answer

They correspond to a unique eigenvector.

Question 9

The eigenvalues of a matrix A are the scalar values λ that satisfy the equation det(A - λI) = 0. What do the eigenvalues represent?

Accepted Answer

The scaling factors that transform eigenvectors when multiplied

Answer

The directions along which eigenvectors lie

Answer

The magnitudes of the corresponding eigenvectors

Answer

The inverses of the corresponding eigenvectors

Question 10

Which of the following is a property of eigenvectors?

Accepted Answer

They form a basis for the eigenspace corresponding to their eigenvalue

Answer

They are always orthogonal to all other eigenvectors of the same matrix

Answer

They have a magnitude of 1

Answer

They are the same for all matrices of the same size

Question 11

Eigenvectors can be used to solve systems of linear differential equations. True or False?

Accepted Answer

True

Answer

False

Question 12

Which of the following matrices has a complex eigenvalue?

Accepted Answer

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

Answer

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

Answer

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

Answer

[[2, 1], [0, 2]]

Question 13

The eigenvectors of a real symmetric matrix are always orthogonal to each other. True or False?

Accepted Answer

True

Answer

False

Question 14

The characteristic polynomial of a matrix is the polynomial that has the eigenvalues of the matrix as its roots.

Accepted Answer

True

Answer

False

Question 15

Eigenvectors geometrically represent:

Accepted Answer

Lines passing through the origin

Answer

Planes passing through the origin

Answer

Vectors parallel to the row space

Answer

Vectors parallel to the column space

Question 16

The sum of the diagonal elements of a square matrix is referred to as its:

Accepted Answer

Trace

Answer

Determinant

Answer

Rank

Answer

Eigenvalue

Question 17

The set of vectors that the matrix transforms to zero is known as the:

Accepted Answer

Null space

Answer

Column space

Answer

Row space

Answer

Eigenvector space

Question 18

A matrix that cannot be diagonalized is one with:

Accepted Answer

Repeated eigenvalues

Answer

Distinct eigenvalues

Answer

Complex eigenvalues

Answer

Zero eigenvalues

Question 19

If a matrix has three distinct eigenvalues, what is guaranteed about its eigenvectors?

Accepted Answer

Three linearly independent eigenvectors

Answer

Two linearly independent eigenvectors

Answer

One linearly independent eigenvector

Answer

No linearly independent eigenvectors

Question 20

Which of the following is NOT a property of eigenvalues?

Accepted Answer

They are always non-negative.

Answer

They are real numbers.

Answer

They are complex numbers.

Answer

They are the roots of the characteristic polynomial.

Question 21

If the matrix A has distinct eigenvalues, then it is:

Accepted Answer

Diagonalizable

Answer

Not diagonalizable

Answer

Definite

Answer

Indefinite

Question 22

Which of the following matrices is not diagonalizable?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 23

If A is a square matrix, then the eigenvalues of A^T are:

Accepted Answer

The same as the eigenvalues of A

Answer

The inverses of the eigenvalues of A

Answer

The complex conjugates of the eigenvalues of A

Answer

The negatives of the eigenvalues of A

Question 24

What is the characteristic equation of a matrix?

Accepted Answer

The equation obtained by setting the determinant of (A-λI) to zero

Answer

The equation obtained by setting the trace of (A-λI) to zero

Answer

The equation obtained by setting the determinant of (A+λI) to zero

Answer

The equation obtained by setting the trace of (A+λI) to zero

Question 25

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

Accepted Answer

1 and 3

Answer

-1 and -3

Answer

1 and -3

Answer

-1 and 3

Question 26

A matrix A has two distinct eigenvalues. What is the maximum number of linearly independent eigenvectors that A can have?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 27

Which of the following applications of eigenvalues and eigenvectors is used in structural analysis?

Accepted Answer

Modal analysis

Answer

Graph theory

Answer

Fourier analysis

Answer

Image processing

Question 28

The number of linearly independent eigenvectors of a matrix is equal to:

Accepted Answer

The rank of the matrix

Answer

The trace of the matrix

Answer

The order of the matrix

Answer

The determinant of the matrix

Question 29

Linearly independent eigenvectors signify what property of the matrix?

Accepted Answer

Diagonalizability

Answer

Positive definiteness

Answer

Symmetry

Answer

Orthogonality

Question 30

Which matrix harbors a complex eigenvalue?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 31

For matrix A = [[1, 2], [3, 4]], determine the algebraic multiplicity of eigenvalue 2.

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 32

When the geometric multiplicity of an eigenvalue is less than its algebraic multiplicity, the matrix is termed what?

Accepted Answer

Defective

Answer

Non-defective

Answer

Diagonalizable

Answer

Triangularizable

Question 33

Identify the diagonalizable matrix from the given options:

Accepted Answer

[[1, 2], [0, 1]]

Answer

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

Answer

[[1, 0], [2, 1]]

Answer

[[1, 2], [3, 4]]

Question 34

Which matrix has complex eigenvalues?

Accepted Answer

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

Answer

[[1, 2], [0, 1]]

Answer

[[1, 0], [2, 1]]

Answer

[[1, 2], [3, 4]]

Question 35

Find the characteristic equation for the matrix [[2, 1], [-1, 2]].

Accepted Answer

(λ-2)(λ-1)=0

Answer

(λ-1)²=0

Answer

(λ+2)(λ+1)=0

Answer

(λ-2)²=0

Question 36

Suppose matrix A has eigenvalues λ₁ and λ₂, what are the eigenvalues of A²?

Accepted Answer

λ₁² and λ₂²

Answer

λ₁ and λ₂

Answer

λ₁+λ₂ and λ₁-λ₂

Answer

λ₁/λ₂ and λ₂/λ₁

Question 37

Which of the following matrices has a repeated eigenvalue?

Accepted Answer

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

Answer

[[1, 2], [0, 1]]

Answer

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

Answer

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

Question 38

If B = A - λI, where A is a matrix, I is the identity matrix, and λ is a constant, what is true about the eigenvalues of B?

Accepted Answer

λ minus the eigenvalues of A

Answer

λ plus the eigenvalues of A

Answer

Same as the eigenvalues of A

Answer

Reciprocals of the eigenvalues of A

Question 39

In a linear vibration system, the mode shapes associated with different eigenvalues exhibit what property?

Accepted Answer

Independence and orthogonality

Answer

Dependence and non-orthogonality

Answer

Identical characteristics

Answer

Mutual exclusivity

Question 40

What is the geometric representation of an eigenvector?

Accepted Answer

A line along which a transformation scales a vector without changing its direction

Answer

A polygon enclosed by the eigenvectors of a matrix

Answer

A point that remains unchanged under a transformation

Answer

A region that is invariant under a transformation

Question 41

Which of the following is not a property of an eigenvector?

Accepted Answer

It is necessarily a unit vector.

Answer

It is a non-zero vector.

Answer

It is multiplied by the eigenvalue when multiplied by the matrix.

Answer

It remains in the same direction after the transformation.

Question 42

Which of the following applications of eigenvalues and eigenvectors is commonly found in mechanical engineering?

Accepted Answer

Simulating the response of a structure to an earthquake.

Answer

Translating natural languages.

Answer

Predicting the weather.

Answer

Analyzing stock market trends.

Question 43

A system with a repeated eigenvalue is considered to be:

Accepted Answer

Defective

Answer

Non-linear

Answer

Orthogonal

Answer

Singular

Question 44

What is the characteristic polynomial of a matrix A?

Accepted Answer

det(A - λI) = 0

Answer

det(A + λI) = 0

Answer

A + λI = 0

Answer

A - λI = 0

Question 45

A matrix A has an eigenvalue of 0. What does this imply about the matrix?

Accepted Answer

The matrix is singular.

Answer

The matrix is invertible.

Answer

The matrix is symmetric.

Answer

The matrix is orthogonal.