3.2 Matrix multiplication and linear systems

Matrix multiplication is the first matrix operation that feels different from the basic arithmetic in the previous note. Addition, subtraction, and scalar multiplication all work entry by entry. Matrix multiplication does not.

Instead, each output entry is built by pairing:

one row from the left matrix, and
one column from the right matrix.

That is why the size rule is stricter here.

Intuition first: the middle size has to agree

Suppose $A$ is $m × n$ and $B$ is $n × p$ . Then the product AB is defined, and the result has size $m × p$ .

The inner number n appears twice because the n entries in a row of $A$ must have exactly n partners in a column of $B$ . If those counts do not agree, there is no row-by-column pairing to carry out.

So the beginner's size rule is:

The inside sizes must match.

Definition

Matrix multiplication

Suppose $A$ is an $m × n$ matrix and $B$ is an $n × p$ matrix. The product AB is the $m × p$ matrix whose (i, j) entry is

[AB]_{ij} = \sum_{k=1}^{n} [A]_{ik}[B]_{kj}.

In words: take row i of $A$ , take column j of $B$ , multiply matching entries, then add the results.

What one entry really means

It is easy to get lost in the notation, so slow the rule down.

To compute $[AB]_{23}$ :

Go to row 2 of $A$ .
Go to column 3 of $B$ .
Multiply entry by entry.
Add the products.

That means one output entry is a compact summary of several multiplications and one final addition.

Embedded interactive moment

Use the visualizer below to keep one output cell fixed while you change the input entries. This is the quickest way to build the row-by-column habit.

Read and try

Follow one matrix product entry

The live widget updates each entry of AB as you change the entries of A and B.

Result

8	9
3	4

8 = 1×2 + 2×3

Worked example

Compute a product by the row-by-column rule

Let

A = \begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}.

Then AB is $2 × 2$ . Compute each entry:

[AB]_{11} = 1 \cdot 2 + 2 \cdot 3 = 8,

[AB]_{12} = 1 \cdot 1 + 2 \cdot 4 = 9,

[AB]_{21} = 0 \cdot 2 + 1 \cdot 3 = 3,

[AB]_{22} = 0 \cdot 1 + 1 \cdot 4 = 4.

AB = \begin{bmatrix} 8 & 9 \\ 3 & 4 \end{bmatrix}.

Why this connects to linear systems

You have already seen systems written as $Ax = b$ .

That formula is not just shorter notation. It tells you that matrix multiplication packages several linear equations into one object. Each row of $A$ creates one equation by pairing that row with the column vector x.

A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix}, \qquad x = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix},

then

Ax = \begin{bmatrix} a_{11}x_1 + a_{12}x_2 + a_{13}x_3 \\ a_{21}x_1 + a_{22}x_2 + a_{23}x_3 \end{bmatrix}.

So $Ax = b$ really means a full system of linear equations, one row at a time.

Theorem

The product order matters

Even when both AB and BA are defined, they usually are not equal.

This is a major change from ordinary number multiplication. With matrices, order carries meaning.

Common mistakes

Common mistake

Checking the wrong dimensions

For AB, compare the number of columns of $A$ with the number of rows of $B$ . Do not compare the outside numbers first.

Common mistake

Multiplying rows by rows

The rule is row of $A$ against column of $B$ , not row against row.

Common mistake

$Assuming AB = BA$

Matrix multiplication is generally not commutative. Two products with the same letters can represent different computations or even have different sizes.

Quick checks

Quick check

If $A$ is $2 × 3$ and $B$ is $3 × 4$ , what is the size of `AB`?

Use the inside-match, outside-survive rule.

Solution

Answer

Quick check

Why does $Ax = b$ represent several equations at once?

Use the word "rows" in your answer.

Solution

Answer

Exercises

Quick check

Suppose $A$ is $3 × 2$ and $B$ is $2 × 5$ . Is `BA` defined? If so, what size would it have?

Do not guess from the fact that AB is defined.

Solution

Guided solution

Quick check

Write the first entry of `Ax` when the first row of $A$ is $(4, -1, 2)$ and $x = (x_1, x_2, x_3)^T$ .

Use the row-by-column rule.

Solution

Guided solution

Review 2.2 Augmented matrices and row operations if you want to reconnect $Ax = b$ with the matrix form of a linear system.

Continue to 3.3 Transposes, symmetric matrices, and skew-symmetric matrices for the next important matrix operation.

3.2 Matrix multiplication and linear systems

MATH1030: Linear algebra I

Intuition first: the middle size has to agree

Definition

Matrix multiplication

What one entry really means

Embedded interactive moment

Follow one matrix product entry

Worked example

Compute a product by the row-by-column rule

Why this connects to linear systems

The product order matters

Common mistakes

Checking the wrong dimensions

Multiplying rows by rows

$Assuming AB = BA$

Quick checks

If $A$ is $2 × 3$ and $B$ is $3 × 4$ , what is the size of `AB`?

Answer

Why does $Ax = b$ represent several equations at once?

Answer

Exercises

Suppose $A$ is $3 × 2$ and $B$ is $2 × 5$ . Is `BA` defined? If so, what size would it have?

Guided solution

Write the first entry of `Ax` when the first row of $A$ is $(4, -1, 2)$ and $x = (x_1, x_2, x_3)^T$ .

Guided solution

Prerequisites

Key terms in this unit

Premium learning add-ons

More notes in this series

3.2 Matrix multiplication and linear systems

MATH1030: Linear algebra I

Intuition first: the middle size has to agree

Definition

Matrix multiplication

What one entry really means

Embedded interactive moment

Follow one matrix product entry

Worked example

Compute a product by the row-by-column rule

Why this connects to linear systems

The product order matters

Common mistakes

Checking the wrong dimensions

Multiplying rows by rows

AssumingAB=BAAssuming AB = BAAssumingAB=BA

Quick checks

If AAA is 2×32 × 32×3 and BBB is 3×43 × 43×4, what is the size of AB?

Answer

Why does Ax=bAx = bAx=b represent several equations at once?

Answer

Exercises

Suppose AAA is 3×23 × 23×2 and BBB is 2×52 × 52×5. Is BA defined? If so, what size would it have?

Guided solution

Write the first entry of Ax when the first row of AAA is (4,−1,2)(4, -1, 2)(4,−1,2) and x=(x1,x2,x3)Tx = (x_1, x_2, x_3)^Tx=(x1​,x2​,x3​)T.

Guided solution

Related notes

Prerequisites

Key terms in this unit

Premium learning add-ons

More notes in this series

$Assuming AB = BA$

If $A$ is $2 × 3$ and $B$ is $3 × 4$ , what is the size of `AB`?

Why does $Ax = b$ represent several equations at once?

Suppose $A$ is $3 × 2$ and $B$ is $2 × 5$ . Is `BA` defined? If so, what size would it have?

Write the first entry of `Ax` when the first row of $A$ is $(4, -1, 2)$ and $x = (x_1, x_2, x_3)^T$ .