Eigenvalues and Eigenvectors: A Comprehensive Guide for Linear Algebra Students

Question 1

What is the definition of an eigenvalue?

Accepted Answer

A vector that transforms into a multiple of itself when multiplied by a square matrix.

Answer

A scalar that, when multiplied by a square matrix, produces the matrix itself.

Answer

A matrix that has a real eigenvalue.

Answer

A transformation that preserves the shape of all vectors in a vector space.

Question 2

Determine whether the following statement is true or false: Every square matrix has at least one eigenvalue.

Accepted Answer

True

Answer

False

Question 3

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

Accepted Answer

A scalar lambda such that the system of linear equations **(A - lambda * I) * x = 0** has at least one nonzero solution for **x**.

Answer

A nonzero vector **x** such that **A * x = lambda * x**.

Question 4

If λ is an eigenvalue of a square matrix **A**, then which of the following is always an eigenvalue of **A^k**?

Accepted Answer

λ^k

Answer

kλ

Answer

λ

Question 5

Which of the following matrices has λ = 0 as an eigenvalue?

Accepted Answer

```[0 1; 0 0]```

Answer

```[1 2; 0 1]```

Answer

```[1 2; 2 4]```

Answer

```[1 0; 0 1]```

Question 6

What is the geometric interpretation of the eigenvector corresponding to the largest eigenvalue of a symmetric matrix **A**?

Accepted Answer

The direction of the greatest spread of a set of data points.

Answer

The direction of the line that best fits a set of data points.

Question 7

What is the algebraic multiplicity of an eigenvalue λ of a square matrix **A**?

Accepted Answer

The dimension of the eigenspace corresponding to the eigenvalue λ.

Answer

The number of times the eigenvalue λ appears on the diagonal of the matrix **A**.

Question 8

An eigenvalue of a matrix is a scalar that, when multiplied by the matrix:

Accepted Answer

Produces a scalar multiple of itself.

Answer

Produces the same matrix.

Answer

Produces the zero matrix.

Answer

Produces the identity matrix.

Question 9

The characteristic equation of a matrix is the equation whose roots are the:

Accepted Answer

Eigenvalues of the matrix.

Answer

Eigenvectors of the matrix.

Answer

Coefficients of the matrix.

Answer

Trace of the matrix.

Question 10

The geometric interpretation of eigenvectors is that they represent the directions in which the matrix:

Accepted Answer

Stretches or shrinks vectors.

Answer

Rotates vectors.

Answer

Reflects vectors.

Answer

Projects vectors.

Question 11

The determinant of a matrix with three distinct eigenvalues is:

Accepted Answer

The product of the eigenvalues

Answer

The sum of the eigenvalues

Answer

The difference of the eigenvalues

Answer

Zero

Question 12

What is the definition of the characteristic equation of a matrix?

Accepted Answer

An equation whose roots are the eigenvalues of the matrix.

Answer

A system of equations whose solutions are the eigenvectors of the matrix.

Answer

An equation that determines the rank of the matrix.

Answer

An equation that determines the determinant of the matrix.

Question 13

Can a matrix have complex eigenvalues?

Accepted Answer

Yes

Answer

No

Answer

Only if the matrix is symmetric

Answer

Only if the matrix is diagonalizable

Question 14

What is the relationship between an eigenvalue and its corresponding eigenvector?

Accepted Answer

The eigenvector is a scalar multiple of the eigenvalue.

Answer

The eigenvalue is the magnitude of the eigenvector.

Answer

The eigenvector is perpendicular to the eigenvalue.

Question 15

What is the key difference between an eigenvalue and an eigenvector?

Accepted Answer

An eigenvalue is a scalar, while an eigenvector is a vector.

Answer

An eigenvector is a scalar, while an eigenvalue is a vector.

Answer

Eigenvalues and eigenvectors are both scalars.

Answer

Eigenvalues and eigenvectors are both vectors.

Question 16

Define an eigenvalue.

Accepted Answer

A scalar that, when substituted into the characteristic equation of a matrix, makes it true.

Answer

A constant that, when subtracted from the elements of a matrix, simplifies it.

Answer

A vector that, when multiplied by a matrix, scales it by a constant.

Answer

A scalar that, when multiplied by a matrix, produces the zero matrix.

Question 17

What is the characteristic equation of a square matrix A?

Accepted Answer

det(A - λI) = 0

Answer

det(A + λI) = 0

Answer

det(λA - I) = 0

Answer

det(I - λA) = 0

Question 18

What is the geometric interpretation of an eigenvector?

Accepted Answer

A line along which a linear transformation scales vectors by their eigenvalue.

Answer

A plane that is invariant under a linear transformation.

Answer

A point that is mapped to itself under a linear transformation.

Answer

A subspace that is spanned by the eigenvectors of a matrix.

Question 19

How can eigenvalues and eigenvectors be used to solve systems of linear equations?

Accepted Answer

To find the general solution of a homogeneous system.

Answer

To find the particular solution of a non-homogeneous system.

Answer

To find the rank of a matrix.

Answer

To determine if a matrix is invertible.

Question 20

If A is an n x n matrix with distinct eigenvalues, then A is:

Accepted Answer

Diagonalizable

Answer

Non-diagonalizable

Answer

Symmetric

Answer

Orthogonal

Question 21

Which of the following statements about eigenvectors is false?

Accepted Answer

Every nonzero vector can be an eigenvector of a matrix.

Answer

Eigenvectors of the same matrix are linearly independent if their eigenvalues are distinct.

Answer

Eigenvectors can be used to construct the eigenbasis of a matrix.

Answer

Eigenvectors can be used to find the eigenvalues of a matrix by substituting them into the characteristic equation.

Question 22

Find the eigenvalues and corresponding eigenvectors of the matrix A = [1 + i, -2; 2, 1 - i], where i is the imaginary unit.

Accepted Answer

Eigenvalues: λ1 = 1 + i, λ2 = 1 - i
Eigenvectors: v1 = (1, -i), v2 = (1, i)

Answer

Eigenvalues: λ1 = 1, λ2 = -1
Eigenvectors: v1 = (1, 0), v2 = (0, 1)

Answer

Eigenvalues: λ1 = 2, λ2 = -2
Eigenvectors: v1 = (1, -1), v2 = (1, 1)

Answer

Eigenvalues: λ1 = 1 + 2i, λ2 = 1 + 2i
Eigenvectors: v1 = (1, 2), v2 = (1, -2)

Question 23

For matrix A = [2 1, -1 3], identify the eigenvector corresponding to the eigenvalue λ = 1.

Accepted Answer

[1, 1]

Answer

[1, 0]

Answer

[0, 1]

Answer

[-1, 1]

Question 24

Which of the following represents the characteristic equation for matrix A?

Accepted Answer

det(A - λI) = 0

Answer

||A - λI|| = 0

Answer

tr(A - λI) = 0

Answer

rank(A - λI) = 0

Question 25

If λ is an eigenvalue of matrix A, what condition must be satisfied?

Accepted Answer

Ax = λx for some nonzero vector x

Answer

λ is a real number

Answer

Ax = 0 for all vectors x

Answer

A is invertible

Question 26

When does an n x n matrix A with n distinct eigenvalues become diagonalizable?

Accepted Answer

When A has a complete set of linearly independent eigenvectors

Answer

When A is invertible

Answer

When A has a dominant eigenvalue

Answer

When A is symmetric

Question 27

If λ is an eigenvalue of matrix A with algebraic multiplicity m, what is the dimension of its associated eigenspace?

Accepted Answer

m

Answer

m - 1

Answer

m + 1

Answer

n - m

Question 28

What is a property of the eigenvalues of a symmetric matrix?

Accepted Answer

They are all real numbers

Answer

They are all zero

Answer

They are all positive

Answer

They are always distinct

Question 29

What is a characteristic of the eigenvectors of a matrix?

Accepted Answer

They form a basis for the vector space

Answer

They are always linearly dependent

Answer

They are always real

Answer

They are always orthogonal

Question 30

Under what condition can a matrix have complex eigenvalues?

Accepted Answer

If the matrix has real coefficients but non-real eigenvalues

Answer

If the matrix has zero eigenvalues

Answer

If the matrix is symmetric

Question 31

Find the matrix P that diagonalizes the matrix [[2, 1], [1, 2]].

Accepted Answer

[[1/√2, -1/√2], [1/√2, 1/√2]]

Answer

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

Answer

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

Answer

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

Question 32

Provide an application of eigenvalues and eigenvectors.

Accepted Answer

Solving systems of differential equations

Answer

Finding the area of a triangle

Answer

Finding the derivative of a function

Answer

Calculating the volume of a sphere

Question 33

Which of the following accurately defines an eigenvalue?

Accepted Answer

A scalar that, when multiplied by the corresponding eigenvector, produces another vector aligned with the original eigenvector

Answer

A value that determines the determinant of a matrix

Question 34

Explain the geometric interpretation of eigenvectors.

Accepted Answer

Directions along which a linear transformation scales vectors by a constant factor

Answer

Lines passing through the matrix's center

Answer

Vectors orthogonal to the matrix

Question 35

Which condition ensures a matrix is diagonalizable?

Accepted Answer

Possessing n distinct eigenvalues

Answer

Having a non-zero determinant

Answer

Having the same number of rows and columns

Question 36

If A is an invertible matrix, what can be said about its eigenvalues?

Accepted Answer

They are all non-zero

Answer

They are all positive

Answer

They are all negative

Answer

They are all complex

Question 37

Identify an application of eigenvalues and eigenvectors.

Accepted Answer

Solving systems of differential equations

Answer

Inverting a matrix

Answer

Finding the volume of a parallelepiped

Answer

Calculating the cross-product of vectors

Question 38

What characterizes orthogonal matrices?

Accepted Answer

Their eigenvectors form an orthonormal basis

Answer

Their determinant is unity

Answer

They have equal row and column counts

Question 39

Which property is true for eigenvectors corresponding to distinct eigenvalues?

Accepted Answer

They are linearly independent.

Answer

They always form a basis for the eigenspace of the corresponding eigenvalue.

Answer

They are always orthogonal to each other.

Question 40

If a square matrix has three distinct real eigenvalues, what is the conclusion about the matrix?

Accepted Answer

It is diagonalizable.

Answer

It is invertible.

Answer

It has at least one repeated eigenvalue.

Answer

It is not diagonalizable.

Question 41

What is the geometric interpretation of eigenvectors of a matrix?

Accepted Answer

They represent the directions in which the matrix scales space.

Answer

They are the vectors that are orthogonal to the null space of the matrix.

Answer

They are the points where the matrix has a zero eigenvalue.

Question 42

Find the eigenvalues and corresponding eigenvectors of the matrix A = [[4, 2], [-2, 1]]

Accepted Answer

Eigenvalues: 2, 5 Eigenvectors: v1 = [1, 1], v2 = [-2, 1]

Answer

Eigenvalues: 3, 4 Eigenvectors: v1 = [1, -1], v2 = [-2, 1]

Answer

Eigenvalues: 2, 5 Eigenvectors: v1 = [1, -1], v2 = [-2, 1]

Answer

Eigenvalues: 3, 4 Eigenvectors: v1 = [1, 1], v2 = [-2, 1]

Question 43

What is the relationship between an eigenvalue and its corresponding eigenvector?

Accepted Answer

Eigenvectors are scalar multiples of their corresponding eigenvalues.

Answer

Eigenvectors are always orthogonal to their corresponding eigenvalues.

Answer

Eigenvectors always have a value of 1.

Answer

Eigenvectors and eigenvalues have the same magnitude.

Question 44

Which matrix has a complex conjugate pair of eigenvalues?

Accepted Answer

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

Answer

[[2, 3], [4, 5]]

Answer

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

Answer

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

Question 45

What is the geometric multiplicity of the eigenvalue 2 for the matrix A = [[2, 1], [0, 2]]?

Accepted Answer

1

Answer

2

Answer

3

Answer

4

Question 46

Which of the following matrices is similar to [[1, 0], [0, 2]]?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 47

Which of the following is an application of eigenvalues and eigenvectors?

Accepted Answer

Solving systems of differential equations

Answer

Simplifying trigonometric expressions

Answer

Finding the area of a triangle

Answer

Calculating the mean of a dataset