Master Linear Algebra: Online Quizzes and Practice Questions

Question 1

Given matrices A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], compute A*B.

Accepted Answer

[[19, 22], [43, 50]]

Answer

[[11, 14], [25, 30]]

Answer

[[5, 10], [15, 20]]

Answer

[[2, 4], [6, 8]]

Question 2

Which of the following is a vector in three-dimensional space?

Accepted Answer

[1, 2, 3]

Answer

[1, 2]

Answer

[1; 2; 3]

Answer

1 + 2i

Question 3

What is the span of the vectors [1, 2] and [3, 4]?

Accepted Answer

The line y = 2x

Answer

The x-axis

Answer

The y-axis

Answer

The plane z = 0

Question 4

Which operation is NOT applicable to matrices?

Accepted Answer

Division of one matrix by another.

Answer

Addition of matrices with the same dimensions.

Answer

Multiplication of a matrix by a scalar.

Answer

Finding the transpose of a matrix by switching rows and columns.

Question 5

Calculate the inverse of the matrix C = [[2, -1], [-3, 2]] if it exists.

Accepted Answer

[[1/5, -1/5], [-3/5, 2/5]]

Answer

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

Answer

[[1/5, 1/5], [3/5, 2/5]]

Answer

C has no inverse

Question 6

Which property do linearly independent sets of vectors exhibit?

Accepted Answer

No vector in the set can be written as a linear combination of the other vectors

Answer

The set contains the zero vector

Answer

The set contains a finite number of vectors

Answer

The set is closed under addition and subtraction

Question 7

What is the rank of the matrix

$$
\begin{bmatrix} 1 & 2 \ 3 & 6 end{bmatrix}

$$?

Accepted Answer

2

Answer

1

Answer

3

Answer

4

Question 8

Which set of vectors exhibits linear independence?

Accepted Answer

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

Answer

{(1, 2, 3), (2, 4, 6), (3, 6, 9)}

Answer

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

Answer

{(1, 2, 3), (2, 2, 4), (4, 6, 8)}

Question 9

Determine the span of the vectors {(1, 0), (0, 1)}:

Accepted Answer

Entire plane

Answer

x-axis

Answer

y-axis

Answer

y = x line

Question 10

Identify the linear combination of {(1, 2), (3, 4)} that produces:

Accepted Answer

(5, 6)

Answer

(2, 4)

Answer

(1, 3)

Answer

(7, 8)

Question 11

Find the inverse of the matrix [[1, 2], [3, 4]]:

Accepted Answer

[[1/2, -1/2], [-3/4, 1/4]]

Answer

[[-1/2, 1/2], [3/4, -1/4]]

Answer

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

Answer

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

Question 12

Which characteristic ensures a unique solution for a system of linear equations?

Accepted Answer

The coefficient matrix has a non-zero determinant.

Answer

The number of equations equals the number of variables.

Answer

The rank of the coefficient matrix is one less than the number of variables.

Answer

The system is consistent.

Question 13

Find the inverse of the matrix [[1, 2], [3, 4]] (if it exists).

Accepted Answer

[-2, 1], [3/2, -1/2]]

Answer

[2, -1], [-3/2, 1/2]]

Answer

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

Answer

No inverse exists

Question 14

Which matrix is an identity matrix?

Accepted Answer

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

Answer

[[1, 2, 3], [0, 1, 0], [0, 0, 1]]

Answer

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

Answer

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

Question 15

Given vectors u = (1, 2) and v = (3, -1), find a vector that is orthogonal to both u and v.

Accepted Answer

(2, -1)

Answer

(1, 1)

Answer

(3, 1)

Answer

(-2, 1)

Question 16

Which of the following operations cannot be performed on matrices?

Accepted Answer

Addition of a matrix and a vector

Answer

Multiplication of a matrix by a scalar

Answer

Determinant of a matrix

Answer

Transpose of a matrix

Question 17

Identify the vector in 3-dimensional space:

Accepted Answer

(2, -1, 5)

Answer

(2, -1)

Answer

(5, 2)

Answer

5

Question 18

Calculate the dot product of the vectors: (1, 2, 3) and (4, -1, 7)

Accepted Answer

17

Answer

-17

Answer

13

Answer

-13

Question 19

Which of the following represents a system of linear equations in matrix form?

Accepted Answer

[2 1 0][x] = [3]   [0 3 -1][y] = [5]

Answer

2x + y = 0   3z - x = 5

Answer

x + y = 3   3y - z = 5

Answer

(2, 1, 0)x + (0, 3, -1)y = (3, 5)

Question 20

Which of the following sets of vectors is linearly independent?

Accepted Answer

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

Answer

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

Answer

{(1, 2, 3), (2, 4, 6), (3, 6, 9)}

Answer

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

Question 21

What is the span of the set of vectors {(1, 0), (0, 1)}?

Accepted Answer

The x-y plane

Answer

The z-axis

Answer

The x-axis

Answer

The y-axis

Question 22

Determine whether the set of vectors {(1, 0), (0, 1)} in the x-y plane is linearly independent.

Accepted Answer

Yes, the vectors are linearly independent

Answer

Not enough information

Answer

Cannot be determined

Answer

No, the vectors are linearly dependent

Question 23

Determine the transpose of the matrix
A = [2 1]
[3 4]]

Accepted Answer

[2 3]
[1 4]

Answer

[4 3]
[2 1]

Answer

[3 1]
[4 2]

Answer

[1 2]
[3 4]

Question 24

Compute the determinant of the matrix:
A = [2 -1]
[1 3]]

Accepted Answer

7

Answer

-7

Answer

0

Answer

1

Question 25

Identify a property of the span of a set of vectors?

Accepted Answer

It forms a subspace of the vector space.

Answer

It is always a plane

Answer

It is always a hyperplane

Answer

It is always a line