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AP PreCalculus Practice Quiz: Matrices

Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026

Test your understanding with short quizzes. This quiz has 9 questions to check your progress.

Question 1 of 9

A matrix is described as a 5 × 3 matrix. According to the definition, how many rows and columns does this matrix have?

All Questions (9)

A matrix is described as a 5 × 3 matrix. According to the definition, how many rows and columns does this matrix have?

A) 3 rows and 5 columns

B) 5 rows and 3 columns

C) 5 rows and 5 columns

D) 3 rows and 3 columns

Correct Answer: B

The provided content states that an n × m matrix is an array consisting of n rows and m columns. Therefore, a 5 × 3 matrix has 5 rows and 3 columns.

Matrix A is a 2 × 4 matrix and Matrix B is a 4 × 3 matrix. Can the product AB be determined, and why?

A) Yes, because the number of rows in A equals the number of columns in B.

B) No, because the number of rows in A does not equal the number of rows in B.

C) Yes, because the number of columns in A equals the number of rows in B.

D) No, because the number of columns in A does not equal the number of columns in B.

Correct Answer: C

The content specifies that two matrices can be multiplied if the number of columns in the first matrix equals the number of rows in the second. Matrix A has 4 columns and Matrix B has 4 rows, so the condition is met and they can be multiplied.

If a 3 × 2 matrix is multiplied by a 2 × 5 matrix, what are the dimensions of the resulting product matrix?

A) 2 × 2

B) 3 × 5

C) 5 × 3

D) The product cannot be determined.

Correct Answer: B

The condition for multiplication is met because the first matrix has 2 columns and the second has 2 rows. The resulting matrix will have the number of rows of the first matrix (3) and the number of columns of the second matrix (5), resulting in a 3 × 5 matrix.

According to the provided content, how is the component in the i-th row and j-th column of a product matrix calculated?

A) The dot product of the i-th column of the first matrix and the j-th row of the second matrix.

B) The dot product of the i-th row of the first matrix and the j-th row of the second matrix.

C) The dot product of the i-th column of the first matrix and the j-th column of the second matrix.

D) The dot product of the i-th row of the first matrix and the j-th column of the second matrix.

Correct Answer: D

The content explicitly states: 'The resulting component is the dot product of the i-th row of the first matrix and the j-th column of the second matrix.'

Let Matrix C = [2 1] and Matrix D be a 2 × 1 matrix with elements [3] and [4]. What is the product of CD?

A) [10]

B) [6 4]

C) [6] [4]

D) The product cannot be determined.

Correct Answer: A

Matrix C is 1 × 2 and Matrix D is 2 × 1. The product is the dot product of the first (and only) row of C and the first (and only) column of D. Calculation: (2 * 3) + (1 * 4) = 6 + 4 = 10. The resulting matrix is [10].

Let Matrix P be a 2 × 3 matrix and Matrix Q be a 3 × 2 matrix. To find the element in the 2nd row and 1st column of the product PQ, what must be calculated?

A) The dot product of the 2nd row of P and the 1st column of Q.

B) The dot product of the 1st row of P and the 2nd column of Q.

C) The dot product of the 2nd column of P and the 1st row of Q.

D) The dot product of the 2nd row of P and the 1st row of Q.

Correct Answer: A

Based on the rule provided, the component at the i-th row and j-th column is the dot product of the i-th row of the first matrix and the j-th column of the second. For the 2nd row, 1st column element, this corresponds to the 2nd row of P and the 1st column of Q.

Given Matrix X (3 × 3), Matrix Y (3 × 2), and Matrix Z (2 × 3), which of the following products CANNOT be determined?

A) XY

B) YZ

C) ZX

D) YX

Correct Answer: D

To multiply two matrices, the number of columns in the first must equal the number of rows in the second. For YX, Matrix Y (3 × 2) has 2 columns, while Matrix X (3 × 3) has 3 rows. Since 2 is not equal to 3, this product cannot be determined.

The product of Matrix A and Matrix B is defined. If Matrix A is a 4 × 7 matrix, what must be true about the dimensions of Matrix B?

A) Matrix B must have 4 rows.

B) Matrix B must have 7 rows.

C) Matrix B must have 4 columns.

D) Matrix B must have 7 columns.

Correct Answer: B

For the product of two matrices to be defined, the number of columns in the first matrix must equal the number of rows in the second matrix. Since Matrix A has 7 columns, Matrix B must have 7 rows.

Let Matrix A = [[1, 0], [2, 3]] and Matrix B = [[4, 5], [0, 1]]. What is the element in the 2nd row, 2nd column of the product AB?

A) 5

B) 10

C) 13

D) 3

Correct Answer: C

To find the element in the 2nd row and 2nd column of the product, we take the dot product of the 2nd row of Matrix A and the 2nd column of Matrix B. The 2nd row of A is [2, 3] and the 2nd column of B is [5, 1]. The dot product is (2 * 5) + (3 * 1) = 10 + 3 = 13.