Matrices and Linear Transformations: Interactive Learning Quizzes

Question 1

A matrix is invertible if and only if its:

Accepted Answer

Determinant is non-zero

Answer

Rank is equal to its order

Answer

Column space is equal to its row space

Question 2

The inverse of a matrix A, if it exists, is denoted as:

Accepted Answer

A^-1

Answer

A-1

Answer

1/A

Question 3

A system of linear equations is consistent if the:

Accepted Answer

Rank of the coefficient matrix is equal to the rank of the augmented matrix

Answer

Rank of the coefficient matrix is less than the rank of the augmented matrix

Answer

Rank of the coefficient matrix is greater than the rank of the augmented matrix

Question 4

Which of the following transformations is NOT considered a linear transformation?

Accepted Answer

Reflection about the y-axis

Answer

Rotation about the origin

Answer

Scaling

Answer

Projection onto a subspace

Question 5

Identify the inconsistent system of equations among the following:

Accepted Answer

x + y = 2, x - y = 4

Answer

x + y = 2, x - y = 0

Answer

x + 2y = 4, 2x + 4y = 8

Answer

x - y = 1, 2x - 2y = 2

Question 6

What is the determinant of the matrix [[2, 1], [-3, 4]]?

Accepted Answer

11

Answer

5

Answer

-11

Answer

-5

Question 7

What is the eigenvector corresponding to the eigenvalue 1 of the matrix [[3, 2], [-1, 1]]?

Accepted Answer

[-1, 1]

Answer

[1, -1]

Answer

[1, 1]

Answer

[2, 1]

Question 8

What is the matrix representation of the linear transformation f(x, y) = 2x - y?

Accepted Answer

[[2, -1]]

Answer

[[2, 1]]

Answer

[-2, 1]]

Answer

[-2, -1]]

Question 9

Which of the following is an orthonormal basis for R^2?

Accepted Answer

{(1, 0), (0, 1)}

Answer

{(1, 0), (1, 1)}

Answer

{(0, 1), (1, 1)}

Answer

{(1, 2), (2, 1)}

Question 10

Calculate the determinant of the matrix [[2, 3], [-1, 5]]

Accepted Answer

13

Answer

-13

Answer

15

Answer

17

Question 11

Determine the eigenvectors of the matrix [[1, 2], [3, 4]] corresponding to the eigenvalue 2

Accepted Answer

[[1], [1]]

Answer

[[2], [1]]

Answer

[[1], [2]]

Answer

[[2], [2]]

Question 12

Which of the following transformations is considered a linear transformation?

Accepted Answer

y = 3x + 5

Answer

y = x^2

Answer

y = sin(x)

Answer

y = e^x

Question 13

Which of the following is NOT a property of matrix multiplication?

Accepted Answer

Communicative

Answer

Associative

Answer

Distributive over addition

Answer

It has an identity matrix

Question 14

Consider the system of linear equations Ax = b, where A is a 3x3 matrix. If the matrix A is invertible, then the system has:

Accepted Answer

Exactly one solution

Answer

Exactly two solutions

Answer

Infinitely many solutions

Answer

No solution

Question 15

Find the matrix exponential of the matrix A = [0 1; -1 0].

Accepted Answer

exp(A) = [cos(t) sin(t); -sin(t) cos(t)]

Answer

exp(A) = [sin(t) cos(t); cos(t) sin(t)]

Answer

exp(A) = [cos(2t) sin(2t); -sin(2t) cos(2t)]

Answer

exp(A) = [sin(2t) cos(2t); cos(2t) sin(2t)]

Question 16

Which transformation does the matrix A = [[cos(θ), -sin(θ)], [sin(θ), cos(θ)]] represent?

Accepted Answer

Rotation by angle θ about the origin

Answer

Reflection about the x-axis

Answer

Scaling by a factor of 2

Answer

Translation by vector [1, 0]

Question 17

Which of the following operations is NOT defined for matrices of arbitrary sizes?

Accepted Answer

Division

Answer

Addition

Answer

Multiplication

Answer

Subtraction

Question 18

The eigenvalues of a square matrix are the roots of its:

Accepted Answer

Characteristic equation

Answer

Minimal polynomial

Answer

Adjugate matrix

Answer

Inverse matrix

Question 19

Which of the following is NOT a property of eigenvectors?

Accepted Answer

They are parallel to each other

Answer

They are linearly independent

Answer

They span the eigenspace

Answer

They are multiplied by the eigenvalue when multiplied by the matrix

Question 20

The matrix representation of a linear transformation is dependent on:

Accepted Answer

The choice of basis

Answer

The dimension of the vector space

Answer

The eigenvalues of the transformation

Answer

The determinant of the transformation

Question 21

Given the matrix A = [2 1; 3 4], what is the determinant of A?

Accepted Answer

5

Answer

1

Answer

-1

Answer

-5

Question 22

Which of the following is NOT a property of eigenvectors?

Accepted Answer

They exist for every square matrix.

Answer

They are linearly independent.

Answer

They are multiplied by a scalar eigenvalue in a linear transformation.

Answer

They are orthogonal to their corresponding eigenvalues.

Question 23

What is the dimension of the eigenspace corresponding to the eigenvalue λ = 2 of the matrix A = [3 1; -1 2]?

Accepted Answer

1

Answer

0

Answer

2

Answer

3

Question 24

Which of the following is NOT a valid operation for matrices?

Accepted Answer

Subtraction of matrices with different dimensions

Answer

Addition of matrices with the same dimensions

Answer

Multiplication by scalars

Answer

Inversion of matrices with non-zero determinants

Question 25

Define the null space of the matrix A = [[1, 2, 3], [4, 5, 6]]

Accepted Answer

Set of vectors x such that Ax = 0

Answer

Set of vectors x such that Ax does not equal 0

Answer

Set of vectors x such that Ax is a multiple of x

Answer

Set of vectors x such that Ax is parallel to x

Question 26

Identify the linear transformation among the following:

Accepted Answer

T(x) = 2x + 1

Answer

T(x) = x^2

Answer

T(x) = |x|

Answer

T(x) = sin(x)

Question 27

Provide the geometric interpretation of the matrix A = [[cos(θ), -sin(θ)], [sin(θ), cos(θ)]]

Accepted Answer

Rotation by angle θ

Answer

Reflection across the x-axis

Answer

Reflection across the y-axis

Answer

Shear transformation

Question 28

What is the determinant of the matrix A = [[2, 1], [3, 4]]?

Accepted Answer

7

Answer

5

Answer

9

Answer

11

Question 29

What is the null space of the matrix A = [[1, 0], [0, 1]]?

Accepted Answer

The origin

Answer

The x-axis

Answer

The y-axis

Answer

The entire plane

Question 30

Which of the following linear transformations is a rotation?

Accepted Answer

T(x, y) = (x cos(theta), y sin(theta))

Answer

T(x, y) = (x + y, x - y)

Answer

T(x, y) = (2x, 3y)

Answer

T(x, y) = (x, -y)

Question 31

Which operation is not applicable to matrices?

Accepted Answer

Exponentiation

Answer

Addition

Answer

Multiplication

Answer

Subtraction

Question 32

What is the formula for finding the determinant of a 2x2 matrix [[a, b], [c, d]]?

Accepted Answer

a*d - b*c

Answer

a + b + c + d

Answer

a - b + c - d

Answer

a*c + b*d

Question 33

What is the maximum rank a 3x3 matrix can have?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 34

Which property is NOT associated with a singular matrix?

Accepted Answer

Has no eigenvectors

Answer

Non-invertible

Answer

Determinant equals zero

Answer

Has a rank less than its order

Question 35

What is the inverse of the matrix [[1, 2], [3, 5]]?

Accepted Answer

[-3/11, 5/11]

Answer

[-5, 2]

Answer

[-3, 1]

Answer

[-5/11, 2/11]

Question 36

Which property is NOT applicable to matrices?

Accepted Answer

Matrices must be square to be invertible.

Answer

Matrices can be multiplied by scalars.

Answer

Matrices can be added and subtracted if they have the same dimensions.

Answer

Matrices can be used to represent systems of linear equations.

Question 37

Which of these matrices is invertible?

Accepted Answer

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

Answer

[[1, 2, 3], [4, 5, 6], [7, 8, 9]]

Answer

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

Answer

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