Find bmatrix 1 & 0 \\ 0 & 1 bmatrixbmatrix 7 \\ -2 bmatrix. [1 point]

What size must B be for A B to be defined if A is 2 × 3? [1 point]

What are mismatched dimensions?

AB exists only when the columns of A equal the rows of B; check before multiplying.

College Board AP 2025 (style)-style practice: Section I, Part A (multiple choice, no calculator). What is the product bmatrix 2 & 0 \\ 1 & 3 bmatrixbmatrix 1 \\ 4 bmatrix? (A) bmatrix 2 \\ 13 bmatrix (B) bmatrix 2 \\ 12 bmatrix (C) bmatrix 6 \\ 13 bmatrix (D) bmatrix 3 \\ 7 bmatrix

The correct answer is (A), bmatrix 2 \\ 13 bmatrix. Multiply each row of the matrix by the column vector. Top row: 2 · 1 + 0 · 4 = 2. Bottom row: 1 · 1 + 3 · 4 = 1 + 12 = 13. The result is bmatrix 2 \\ 13 bmatrix.

College Board AP 2025 (style)-style practice: Section II (free response, no calculator). Let A = bmatrix 1 & 2 \\ 0 & 1 bmatrix and B = bmatrix 3 & 1 \\ 2 & 4 bmatrix. (a) Find A + B and 2A. (b) Find the product AB.

A 4-point question on matrix operations. (a) Sum and scalar multiple (2 points): add entrywise, A + B = bmatrix 4 & 3 \\ 2 & 5 bmatrix. Scale every entry, 2A = bmatrix 2 & 4 \\ 0 & 2 bmatrix. (b) Product (2 points): row-by-column, AB = bmatrix 13 + 22 & 11 + 24 \\ 03 + 12 & 01 + 14 bmatrix = bmatrix 7 & 9 \\ 2 & 4 bmatrix.

United StatesPrecalculusSyllabus dot point

What is a matrix, and how do you add, scale and multiply matrices?

Topic 4.10 Matrices: represent data with a matrix, and add, subtract, scale and multiply matrices, including multiplying a matrix by a vector.

A focused answer to AP Precalculus Topic 4.10, covering matrices as rectangular arrays, matrix addition and scalar multiplication, the row-by-column rule for matrix multiplication, and multiplying a matrix by a vector.

Generated by Claude Opus 4.89 min answerUpdated 2026-06-04

Reviewed by: AI editorial process; not yet individually human-reviewed

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What this topic is asking
Matrices and the simple operations
Matrix multiplication
Multiplying a matrix by a vector
Why the rule looks the way it does
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What this topic is asking

The College Board (Topic 4.10) wants you to work with matrices: rectangular arrays of numbers. You add and subtract matrices entrywise, multiply a matrix by a scalar, multiply two matrices using the row-by-column rule, and multiply a matrix by a vector, which is the operation that drives the transformations later in the unit.

Matrices and the simple operations

These operations behave like vector operations done on a grid: combine matching positions, or scale every position. They are the easy operations; the structure appears in multiplication.

Matrix multiplication

The row-by-column rule is the heart of the topic; getting the dimensions to match and pairing each row with each column carefully is what makes products correct.

Multiplying a matrix by a vector

This produces a new vector from an old one, which is exactly how matrices transform points and vectors in the plane (Topic 4.12).

Multiplying two matrices

Compute $AB$ for $A = \begin{bmatrix} 2 & 1 \\ 0 & 3 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 4 \\ 5 & 2 \end{bmatrix}$ .

step 1 Check the dimensions

$A$ is $2 \times 2$ and $B$ is $2 \times 2$ : columns of $A$ ( $2$ ) equal rows of $B$ ( $2$ ), so the product is defined and is $2 \times 2$ .

step 2 Top row of the product

Row $1$ of $A$ is $[2, 1]$ . With column $1$ of $B$ , $[1, 5]$ : $2\cdot1 + 1\cdot5 = 7$ . With column $2$ , $[4, 2]$ : $2\cdot4 + 1\cdot2 = 10$ .

step 3 Bottom row of the product

Row $2$ of $A$ is $[0, 3]$ . With column $1$ : $0\cdot1 + 3\cdot5 = 15$ . With column $2$ : $0\cdot4 + 3\cdot2 = 6$ .

step 4 Assemble the product

$AB = \begin{bmatrix} 7 & 10 \\ 15 & 6 \end{bmatrix}$ . (Computing $BA$ would give a different matrix, confirming non-commutativity.)

Why the rule looks the way it does

A point worth stating once is that the row-by-column rule is exactly what makes a product of matrices represent a composition of the transformations they encode. When a matrix multiplies a vector it produces a new vector by mixing the components according to its rows; multiplying two matrices builds the single matrix that does both transformations in sequence. That is why the dimensions must align and why the order matters, applying $A$ then $B$ is different from $B$ then $A$ . Holding onto this "matrices act on vectors, products chain the actions" picture turns the mechanical rule into something you can reason about, and it is the foundation for the inverse, determinant and transformation topics that follow.

Try this

Q1. Find $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} 7 \\ -2 \end{bmatrix}$ . [1 point]

Cue. The identity matrix leaves the vector unchanged: $\begin{bmatrix} 7 \\ -2 \end{bmatrix}$ .

Q2. What size must $B$ be for $A B$ to be defined if $A$ is $2 \times 3$ ? [1 point]

Cue. $B$ must have $3$ rows (matching $A$ 's $3$ columns); it can have any number of columns.

Exam-style practice questions

Practice questions written in the style of College Board exam questions on this dot point, with worked answer explainers. The year tag is the paper they imitate, not the source.

AP 2025 (style)1 marksSection I, Part A (multiple choice, no calculator). What is the product

\begin{bmatrix} 2 & 0 \\ 1 & 3 \end{bmatrix}\begin{bmatrix} 1 \\ 4 \end{bmatrix}

? (A)

\begin{bmatrix} 2 \\ 13 \end{bmatrix}

(B)

\begin{bmatrix} 2 \\ 12 \end{bmatrix}

(C)

\begin{bmatrix} 6 \\ 13 \end{bmatrix}

(D)

\begin{bmatrix} 3 \\ 7 \end{bmatrix}

Show worked answer →

The correct answer is (A), $\begin{bmatrix} 2 \\ 13 \end{bmatrix}$ .

Multiply each row of the matrix by the column vector. Top row: $2 \cdot 1 + 0 \cdot 4 = 2$ . Bottom row: $1 \cdot 1 + 3 \cdot 4 = 1 + 12 = 13$ . The result is $\begin{bmatrix} 2 \\ 13 \end{bmatrix}$ .

AP 2025 (style)4 marksSection II (free response, no calculator). Let

A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}

and

B = \begin{bmatrix} 3 & 1 \\ 2 & 4 \end{bmatrix}

. (a) Find

A + B

and

2A

. (b) Find the product

AB

Show worked answer →

A 4-point question on matrix operations.

(a) Sum and scalar multiple (2 points): add entrywise, $A + B = \begin{bmatrix} 4 & 3 \\ 2 & 5 \end{bmatrix}$ . Scale every entry, $2A = \begin{bmatrix} 2 & 4 \\ 0 & 2 \end{bmatrix}$ .
(b) Product (2 points): row-by-column, $AB = \begin{bmatrix} 1\cdot3 + 2\cdot2 & 1\cdot1 + 2\cdot4 \\ 0\cdot3 + 1\cdot2 & 0\cdot1 + 1\cdot4 \end{bmatrix} = \begin{bmatrix} 7 & 9 \\ 2 & 4 \end{bmatrix}$ .

Related dot points

Sources & how we know this

AP Precalculus Course and Exam Description — College Board (2023)