Linear Algebra Quizzes: Master the Fundamentals of Linear Algebra

Question 1

The determinant of a matrix A represents:

Accepted Answer

The volume of the parallelepiped spanned by its column vectors

Answer

The sum of its eigenvalues

Answer

The product of its eigenvalues

Answer

The area of the parallelogram spanned by its column vectors

Question 2

Which of the following is a linear transformation?

Accepted Answer

f(x) = 2x + 1

Answer

f(x) = x^2

Answer

f(x) = |x|

Answer

f(x) = sin(x)

Question 3

The null space of a matrix is:

Accepted Answer

The set of all vectors that are multiplied by zero under the transformation

Answer

The set of all vectors that are orthogonal to the column space

Answer

The set of all vectors that are orthogonal to the row space

Answer

The set of all vectors that are added by zero under the transformation

Question 4

The determinant of a matrix provides information about its:

Accepted Answer

Singularity

Answer

Product of eigenvalues

Answer

Trace

Answer

Order

Question 5

Eigenvalues of a matrix are obtained from the:

Accepted Answer

Roots of its characteristic polynomial

Answer

Eigenvectors

Answer

Determinants

Answer

Traces

Question 6

Find the basis for the column space of the matrix [[1, 2, 3], [4, 5, 6], [7, 8, 9]]

Accepted Answer

{(1, 4, 7), (2, 5, 8), (3, 6, 9)}

Answer

{(1, 2, 3), (0, 1, 2), (0, 0, 1)}

Answer

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

Answer

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

Question 7

Which of the following is NOT a subspace of R^3?

Accepted Answer

The set of all vectors of the form (x, y, z) where x = 2y z = 0

Answer

The set of all vectors of the form (x, 0, 0)

Answer

The set of all vectors of the form (x, y, z) where x + y + z = 0

Answer

The set of all vectors of the form (x, y, z) where x = y

Question 8

Identify which of the following sets qualifies as a vector space:

Accepted Answer

The set of all polynomials of degree less than 3

Answer

The set of all multiples of 3

Answer

The set of all 2x2 matrices

Answer

The set of all vectors with both positive and negative components

Question 9

If a 3x3 matrix has eigenvalues 1, 2, and 3, what is its determinant?

Accepted Answer

6

Answer

1

Answer

3

Question 10

Solve for *x* in the system of linear equations:  2*x* + *y* = 5 and *x* - 2*y* = 1.

Accepted Answer

3

Answer

4

Answer

2

Answer

1

Question 11

What is the relationship between the null space and the column space of a matrix?

Accepted Answer

They are orthogonal complements.

Answer

They are the same.

Answer

The null space is a subset of the column space.

Question 12

What is the dimension of the vector space containing all 3x3 symmetric matrices?

Accepted Answer

6

Answer

9

Answer

12

Answer

3

Question 13

If a matrix *A* has eigenvalues λ1 and λ2, what is its determinant?

Accepted Answer

λ1 * λ2

Answer

λ1 / λ2

Answer

λ1 + λ2

Answer

λ1 - λ2

Question 14

Which of the following transformations is NOT a linear transformation?

Accepted Answer

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

Answer

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

Answer

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

Question 15

Determine the rank of the following matrix:  [[1, 2, 3], [4, 5, 6], [7, 8, 9]]

Accepted Answer

2

Answer

1

Answer

3

Answer

0

Question 16

Which of the following sets is a subspace of R^3?

Accepted Answer

The set of all vectors of the form (x, y, 0)

Answer

The set of all vectors of the form (x, y, 1)

Answer

The set of all vectors of the form (x, y, x^2)

Answer

The set of all vectors of the form (x, y, x + y + 1)

Question 17

What is the null space of the following matrix: 
 [[1, 2], [3, 6]] ?

Accepted Answer

{(x, -x/2) | x ∈ R}

Answer

{(0, y) | y ∈ R}

Answer

{(x, x) | x ∈ R}

Answer

{(x, 0) | x ∈ R}

Question 18

If A is a 3x3 matrix with eigenvalues 2, 3, and 4, what is the trace of A?

Accepted Answer

9

Answer

12

Answer

24

Answer

6

Question 19

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

Accepted Answer

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

Answer

{(1, 1), (1, -1)}

Answer

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

Answer

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

Question 20

A system of linear equations has a unique solution if and only if...

Accepted Answer

The coefficient matrix has full rank.

Answer

The determinant of the coefficient matrix is zero.

Answer

The coefficient matrix is a diagonal matrix.

Answer

The system has more equations than unknowns.

Question 21

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

Accepted Answer

5

Answer

7

Answer

11

Answer

2

Question 22

Given a linear transformation T: R^2 -> R^2 defined by T(x, y) = (x + y, 2x - y), what is the image of the vector (1, 2) under T?

Accepted Answer

(3, 0)

Answer

(1, 2)

Answer

(2, 1)

Answer

(0, 3)

Question 23

Which of the following matrices is invertible?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 24

Find the eigenvalues of the matrix: 
 [[2, 1], [1, 2]]

Accepted Answer

1 and 3

Answer

-1 and 3

Answer

0 and 4

Answer

2 and 2

Question 25

What is the dimension of the vector space R^3?

Accepted Answer

3

Answer

1

Answer

2

Answer

4

Question 26

Which of the following is an eigenvector of the matrix: 
 [[1, 2], [2, 1]] corresponding to the eigenvalue 3?

Accepted Answer

(1, 1)

Answer

(2, 1)

Answer

(1, -1)

Question 27

The null space of a matrix is the set of all vectors that...

Accepted Answer

are mapped to the zero vector by the matrix.

Answer

are orthogonal to the column space.

Answer

have a determinant of zero.

Answer

are linearly independent.

Question 28

The rank of a matrix is equal to...

Accepted Answer

the dimension of its column space.

Answer

the number of rows.

Answer

the number of columns.

Answer

the determinant of the matrix.

Question 29

A system of linear equations is said to be consistent if...

Accepted Answer

it has at least one solution.

Answer

it has infinitely many solutions.

Answer

the determinant of the coefficient matrix is non-zero.

Answer

it has no solutions.

Question 30

What is the determinant of the following matrix?

A =  [[1, 2], [3, 4]]

Accepted Answer

-2

Answer

6

Answer

2

Question 31

A linear transformation T: R^2 -> R^2 maps the vector (1, 0) to (2, 1) and the vector (0, 1) to (3, -1). What is the image of the vector (2, 3) under T?

Accepted Answer

(11, -1)

Answer

(5, 2)

Answer

(8, 1)

Answer

(2, 3)

Question 32

Which of the following is an eigenvector of the matrix A = [[2, 1], [1, 2]]?

Accepted Answer

(1, 1)

Answer

(0, 1)

Answer

(1, -1)

Answer

(1, 0)

Question 33

Given the system of equations:

x + 2y = 5 
 3x + 4y = 11

Which of the following represents the solution in matrix form?

Accepted Answer

[[x], [y]] = [[1], [2]]

Answer

[[x], [y]] = [[1], [3]]

Answer

[[x], [y]] = [[3], [1]]

Answer

[[x], [y]] = [[2], [1]]

Question 34

What is the rank of the matrix A = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?

Accepted Answer

2

Answer

1

Answer

3

Question 35

A matrix is called invertible if and only if:

Accepted Answer

Its determinant is non-zero.

Answer

Its rank is equal to the number of rows.

Answer

It is a square matrix.

Answer

It has an inverse matrix.

Question 36

The null space of a matrix A is the set of all vectors x that satisfy:

Accepted Answer

Ax = 0

Answer

Ax = 1

Answer

x = 0

Answer

x = A

Question 37

Which of the following is NOT a subspace of R^3?

Accepted Answer

The set of all vectors with a negative x-coordinate.

Answer

The set of all vectors with a zero z-coordinate.

Answer

The set of all vectors that are scalar multiples of (1, 2, 3).

Question 38

The dimension of the vector space spanned by the vectors (1, 0, 1) and (0, 1, 1) is:

Accepted Answer

2

Answer

1

Answer

3

Answer

0

Question 39

Which of the following properties does NOT hold true for matrix multiplication?

Accepted Answer

Commutative Property: AB = BA

Answer

Left Distributive Property: A(B + C) = AB + AC

Answer

Right Distributive Property: (A + B)C = AC + BC

Answer

Associative Property: A(BC) = (AB)C

Question 40

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

Accepted Answer

10

Answer

6

Answer

-2

Answer

14

Question 41

Which of the following sets of vectors forms a basis for R^3?

Accepted Answer

{[1, 0, 0], [0, 1, 0], [0, 0, 1]}

Answer

{[1, 2, 3], [2, 4, 6], [3, 6, 9]}

Answer

{[1, 1, 0], [0, 1, 1]}

Answer

{[2, 0, 0], [0, 3, 0], [0, 0, 0]}

Question 42

A linear transformation T: R^2 -> R^2 is defined by T(x, y) = (2x + y, x - y). What is the matrix representation of T with respect to the standard basis?

Accepted Answer

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

Answer

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

Answer

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

Answer

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

Question 43

Find the eigenvalues of the matrix B = [[3, 1], [0, 2]].

Accepted Answer

λ1 = 3, λ2 = 2

Answer

λ1 = 1, λ2 = 2

Answer

λ1 = -3, λ2 = 2

Answer

λ1 = 3, λ2 = -2

Question 44

If a system of linear equations has a unique solution, what can be said about its associated augmented matrix?

Accepted Answer

It is row equivalent to a matrix in reduced row echelon form with a leading 1 in each row.

Answer

It has more columns than rows.

Answer

It is row equivalent to a matrix with a row of zeros.

Question 45

What is the rank of the matrix C = [[1, 2, 3], [4, 5, 6], [7, 8, 9]]?

Accepted Answer

2

Answer

0

Answer

3

Answer

1

Question 46

Which of the following statements is TRUE regarding eigenvectors?

Accepted Answer

Eigenvectors corresponding to distinct eigenvalues are linearly independent.

Answer

Every vector in the vector space is an eigenvector.

Answer

Eigenvectors can have a magnitude of zero.

Question 47

What is the dimension of the subspace spanned by the vectors {[1, 0, 1], [2, 1, 3], [0, 1, 1]}?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 48

If A and B are invertible matrices of the same size, then what is the inverse of the product AB?

Accepted Answer

B^-1 * A^-1

Answer

A^-1 * B^-1

Answer

(AB)^-1

Answer

A * B