Diagonalization of Matrices: Interactive Learning Quizzes and Practice Questions

Question 1

Which of the following is the correct definition of an eigenvalue?

Accepted Answer

A scalar that, when multiplied by an eigenvector, results in the eigenvector multiplied by a constant.

Answer

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

Answer

A scalar that, when added to a matrix, results in the same matrix.

Answer

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

Question 2

If A is an n x n matrix, what is the maximum number of distinct eigenvalues it can have?

Accepted Answer

n

Answer

n+1

Answer

n-1

Answer

n/2

Question 3

Which of the following methods can be used to diagonalize a real symmetric matrix?

Accepted Answer

Eigenvalue decomposition

Answer

QR decomposition

Answer

LU decomposition

Answer

Gram-Schmidt orthogonalization

Question 4

If A is a diagonalizable matrix, what is the relationship between its eigenvalues and eigenvectors?

Accepted Answer

Eigenvectors are linearly independent and form a basis for the eigenspace.

Answer

Eigenvectors are linearly dependent and form a basis for the eigenspace.

Answer

Eigenvectors are linearly independent but do not form a basis for the eigenspace.

Answer

Eigenvectors are linearly dependent but do not form a basis for the eigenspace.

Question 5

Which of the following matrices is not diagonalizable?

Accepted Answer

A 3 x 3 matrix with two identical eigenvalues

Answer

A 2 x 2 matrix with distinct eigenvalues

Answer

A symmetric matrix

Answer

A matrix with a complex eigenvalue

Question 6

What is the geometric significance of the eigenvectors of a matrix?

Accepted Answer

They are the directions in which the matrix stretches or compresses space.

Answer

They are the lines along which the matrix rotates space.

Answer

They are the axes of the ellipse defined by the matrix.

Answer

They are the planes in which the matrix reflects space.

Question 7

What is the trace of a diagonal matrix?

Accepted Answer

The sum of its diagonal elements

Answer

The product of its diagonal elements

Answer

The number of its distinct eigenvalues

Answer

Zero

Question 8

What is the determinant of a diagonal matrix?

Accepted Answer

The product of its diagonal elements

Answer

The sum of its diagonal elements

Answer

The number of its distinct eigenvalues

Answer

Zero

Question 9

What is the definition of an eigenvalue of a matrix?

Accepted Answer

A scalar value λ such that A - λI = 0

Answer

A scalar value λ such that A - λI is invertible

Answer

A scalar value λ such that A * λ = λ * A

Answer

A scalar value λ such that A + λI = 0

Question 10

What is the geometric interpretation of an eigenvector?

Accepted Answer

A line that passes through the origin and is invariant under matrix multiplication

Answer

A plane that contains the origin and is invariant under matrix multiplication

Answer

A vector that is parallel to the x-axis and is invariant under matrix multiplication

Answer

A vector that is parallel to the y-axis and is invariant under matrix multiplication

Question 11

Which of the following matrices is diagonalizable?

Accepted Answer

A matrix with distinct real eigenvalues

Answer

A matrix with at least one complex eigenvalue

Answer

A matrix with all zero eigenvalues

Answer

A matrix with repeated eigenvalues

Question 12

What is the formula for the determinant of a diagonal matrix?

Accepted Answer

The product of its diagonal elements

Answer

The sum of its diagonal elements

Answer

The difference of its diagonal elements

Answer

The reciprocal of its diagonal elements

Question 13

What is the diagonal form of a matrix?

Accepted Answer

A matrix with all its diagonal elements equal to its eigenvalues

Answer

A matrix with all its non-diagonal elements equal to zero

Answer

A matrix with all its diagonal elements equal to one

Answer

A matrix with all its off-diagonal elements equal to its eigenvectors

Question 14

What is the characteristic polynomial of a matrix?

Accepted Answer

A polynomial whose roots are the eigenvalues of the matrix

Answer

A polynomial whose roots are the eigenvectors of the matrix

Answer

A polynomial whose roots are the diagonal elements of the matrix

Answer

A polynomial whose roots are the off-diagonal elements of the matrix

Question 15

Which of the following matrices is similar to a diagonal matrix?

Accepted Answer

A matrix that can be transformed into a diagonal matrix by a similarity transformation

Answer

A matrix that has all its eigenvalues equal

Answer

A matrix that has all its eigenvectors orthogonal

Answer

A matrix that has no zero eigenvalues

Question 16

What is the relationship between the eigenvalues and eigenvectors of a matrix?

Accepted Answer

Eigenvectors are scaled by their corresponding eigenvalues when multiplied by the matrix

Answer

Eigenvectors are orthogonal to their corresponding eigenvalues when multiplied by the matrix

Answer

Eigenvalues are the products of the eigenvectors of the matrix

Answer

Eigenvectors are the sums of the eigenvalues of the matrix

Question 17

What is the dimension of the eigenspace corresponding to a unique eigenvalue?

Accepted Answer

1

Answer

2

Answer

Equal to the size of the matrix

Answer

Cannot be determined

Question 18

Which of the following matrices is similar to the matrix A = [[2, 1], [0, 3]]?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 19

Which of the following is a necessary condition for a matrix to be diagonalizable?

Accepted Answer

It has n linearly independent eigenvectors.

Answer

It has distinct eigenvalues.

Answer

It is a symmetric matrix.

Answer

It is an upper triangular matrix.

Question 20

Which of the following is a common method for diagonalizing a matrix?

Accepted Answer

Eigenvalue decomposition

Answer

LU decomposition

Answer

Cholesky decomposition

Answer

Gaussian elimination

Question 21

What is the trace of a diagonalizable matrix?

Accepted Answer

Sum of its eigenvalues

Answer

Product of its eigenvalues

Answer

Determinant of its eigenvalue matrix

Answer

Inverse of its eigenvalue matrix

Question 22

Which of the following matrices is not similar to a diagonal matrix?

Accepted Answer

[1 2; 3 4]

Answer

[1 2; 0 1]

Answer

[2 1; 1 2]

Answer

[3 0; -1 3]

Question 23

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

Accepted Answer

True

Answer

False

Question 24

Which step is essential for solving the system of linear equations below using diagonalization?

2x + 3y = 5
3x + 2y = 4

Accepted Answer

Determine the eigenvalues and eigenvectors of the coefficient matrix.

Answer

Multiply both equations by a nonzero constant.

Answer

Add the two equations and simplify.

Question 25

Which of the following statements is true about eigenvalues of a matrix?

Accepted Answer

Eigenvalues can be complex numbers.

Answer

Eigenvalues are always real numbers.

Answer

Eigenvalues are always distinct.

Answer

Eigenvalues are always positive.

Question 26

What is the diagonal form of a diagonalizable matrix?

Accepted Answer

A matrix with all entries on the diagonal equal to the eigenvalues of the original matrix and zeros elsewhere.

Answer

A matrix with all non-zero entries on the diagonal and zeros elsewhere.

Answer

A matrix with all entries on the diagonal equal to 1 and zeros elsewhere.

Answer

A matrix with all entries on the diagonal equal to the eigenvectors of the original matrix.

Question 27

What is the geometric significance of eigenvectors?

Accepted Answer

They define the directions of lines that remain unchanged when the matrix is applied.

Answer

They define the directions of lines that are perpendicular to the eigenvectors of the transpose of the matrix.

Answer

They define the directions of lines that are parallel to the eigenvectors of the inverse of the matrix.

Answer

They define the directions of lines that are bisected by the eigenvectors of the adjoint of the matrix.

Question 28

What is the determinant of a diagonalizable matrix?

Accepted Answer

The product of its eigenvalues

Answer

The sum of its eigenvalues

Answer

The trace of its eigenvalue matrix

Answer

The product of its diagonal entries

Question 29

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

Accepted Answer

The eigenvalues are the roots of the characteristic polynomial.

Answer

The characteristic polynomial is the product of the eigenvalues.

Answer

The characteristic polynomial is the sum of the eigenvalues.

Answer

The characteristic polynomial is the determinant of the eigenvalue matrix.

Question 30

Which of the following matrices is NOT diagonalizable?

Accepted Answer

[1 1; 1 1]

Answer

[1 2; 0 1]

Answer

[1 1; 0 1]

Answer

[1 0; 0 2]

Question 31

Which of the following correctly defines an eigenvalue of a matrix A?

Accepted Answer

A scalar for which the corresponding eigenvector is non-zero.

Answer

A scalar that, when multiplied by A, results in the zero matrix.

Answer

A scalar that, when added to A, results in the zero matrix.

Answer

A scalar for which the corresponding eigenvector is the zero vector.

Question 32

If A is a square matrix and λ is an eigenvalue of A, which of the following is always true?

Accepted Answer

det(A - λI) = 0

Answer

det(A + λI) = 0

Answer

tr(A - λI) = 0

Answer

tr(A + λI) = 0

Question 33

Which of the following is a method commonly used for finding the eigenvalues of a matrix?

Accepted Answer

Power iteration

Answer

Gaussian elimination

Answer

Row reduction

Answer

Cramer's rule

Question 34

Which of the following is an application of diagonalization in linear algebra?

Accepted Answer

Calculating the eigenvectors and eigenvalues of a matrix.

Answer

Solving systems of linear equations.

Answer

Finding the least squares solution of a linear system.

Answer

Computing the determinant of a matrix.

Question 35

For a symmetric positive-definite matrix, what do its eigenvectors represent geometrically?

Accepted Answer

The principal axes themselves

Answer

Perpendicular lines to the principal axes

Answer

Lines parallel but distinct from the principal axes

Answer

No geometric interpretation in this context

Question 36

What is the determinant of a diagonalizable matrix with eigenvalues λ1, λ2, ..., λn?

Accepted Answer

λ1λ2...λn

Answer

λ1 + λ2 + ... + λn

Answer

1 / (λ1λ2...λn)

Answer

0

Question 37

What is the dimension of the eigenspace corresponding to an eigenvalue of a diagonalizable matrix?

Accepted Answer

The multiplicity of the eigenvalue

Answer

The trace of the matrix

Answer

The rank of the matrix

Answer

The nullity of the matrix

Question 38

Which of the following statements about eigenvectors is true?

Accepted Answer

They are linearly independent if the corresponding eigenvalues are distinct

Answer

They are always orthogonal

Answer

They span the entire vector space

Question 39

What is the diagonalization theorem?

Accepted Answer

A square matrix is diagonalizable if and only if it has n linearly independent eigenvectors.

Answer

A matrix with complex eigenvalues is not diagonalizable

Answer

A symmetric matrix is always diagonalizable

Question 40

Which of the following is a method for diagonalizing a matrix?

Accepted Answer

Finding its eigenvectors and constructing a diagonalizing matrix

Answer

Multiplying it by its inverse

Answer

Using the determinant to find its eigenvalues

Question 41

What is the purpose of diagonalizing a matrix?

Accepted Answer

To simplify its analysis and computations

Answer

To find its determinant

Answer

To solve systems of linear equations

Question 42

What is the diagonal form of a diagonalizable matrix?

Accepted Answer

A matrix with its eigenvalues on the diagonal and zeros elsewhere

Answer

A matrix with its trace on the diagonal

Answer

A matrix with its eigenvectors as columns

Question 43

If A is a diagonalizable matrix with eigenvalues λ1, λ2, and λ3, what is the determinant of A?

Accepted Answer

det(A) = λ1 * λ2 * λ3

Answer

det(A) = λ1 / λ2 / λ3

Answer

det(A) = λ1 - λ2 + λ3

Answer

det(A) = λ1 + λ2 + λ3

Question 44

State whether the following statement is true or false: Every real symmetric matrix is diagonalizable.

Accepted Answer

True

Answer

False

Answer

Not enough information provided

Answer

None of the above

Question 45

What is the trace of a diagonalizable matrix?

Accepted Answer

The sum of its eigenvalues

Answer

The number of nonzero eigenvalues

Answer

The product of its eigenvalues

Answer

The determinant of the matrix

Question 46

Which of the following is NOT an application of diagonalization?

Accepted Answer

Solving systems of linear equations

Answer

Analyzing vibrations and oscillations

Answer

Reducing quadratic forms to principal axes

Answer

Finding eigenvalues and eigenvectors

Question 47

Which of the following conditions guarantees that a matrix is diagonalizable?

Accepted Answer

The matrix has a complete set of linearly independent eigenvectors.

Answer

The matrix has distinct eigenvalues.

Answer

The matrix has only real eigenvalues.

Answer

The matrix is invertible.

Question 48

What is an eigenvector of a square matrix A?

Accepted Answer

A non-zero vector  'v' such that Av = λv, where λ is a scalar called the eigenvalue.

Answer

A vector that is perpendicular to the matrix.

Answer

A vector that spans the null space of the matrix.

Answer

A vector that is scaled by a factor of the matrix.

Question 49

What is the geometric interpretation of eigenvectors in relation to a matrix transformation?

Accepted Answer

Eigenvectors represent directions in which the transformation acts by scaling only, without rotation or reflection.

Answer

They represent the null space of the matrix transformation.

Answer

They represent the axes of symmetry of the matrix transformation.

Answer

They represent the range of the matrix transformation.

Question 50

How are the eigenvalues and eigenvectors of a matrix related?

Accepted Answer

Each eigenvalue corresponds to a unique set of linearly independent eigenvectors, forming the eigenspace for that eigenvalue.

Answer

Eigenvalues are linearly independent of their corresponding eigenvectors.

Answer

Eigenvectors are orthogonal to their corresponding eigenvalues.

Answer

All eigenvalues have the same corresponding eigenvector.

Question 51

Which of the following is NOT a common application of diagonalization in linear algebra?

Accepted Answer

Finding the determinant of a matrix.

Answer

Analyzing the stability of dynamical systems.

Answer

Simplifying matrix transformations.

Answer

Solving systems of linear differential equations.

Question 52

What is the determinant of a diagonal matrix?

Accepted Answer

The product of its diagonal entries.

Answer

The sum of its diagonal entries.

Answer

The rank of the matrix.

Answer

The trace of the matrix.

Question 53

What is the trace of a diagonal matrix?

Accepted Answer

The sum of its diagonal entries.

Answer

The rank of the matrix.

Answer

The determinant of the matrix.

Answer

The product of its diagonal entries.