Multiplying matrices and its properties

Warm-up

Three quick questions from earlier lessons.

Discover

To multiply a row (2, 3) by a column (4, 5), you compute 2 · 4 + 3 · 5. What is it?

23
(8, 15)

Key idea

Each entry of AB is a row of A times a column of B.

If

A

is

m \times n

and

B

is

n \times p

, then

AB

is

m \times p

: the entry in row

i

, column

j

is row $i$ of $A$ times column $j$ of $B$ (multiply pairs, then add). The inner sizes must match. Matrix multiplication is associative and distributes over addition, but in general $AB \ne BA$ .

Worked example

Compute the top-left entry of $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}\begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$ .

(1, 2) and (5, 7)

Row 1 of A, column 1 of B.

1 · 5 + 2 · 7 = 19

Multiply and add.

Your turn, guided

Your turn: the top-right entry of the same product.

22

1 · 6 + 2 · 8.

Practice ★

Let A = $\begin{pmatrix} 4 & 2 \\ 3 & -2 \end{pmatrix}$ and B = $\begin{pmatrix} 7 & 9 \\ 5 & 4 \end{pmatrix}$ . What is the entry in row 1, column 1 of AB?
Let A = $\begin{pmatrix} -2 & -4 \\ 5 & -4 \end{pmatrix}$ and B = $\begin{pmatrix} 4 & 7 \\ -5 & -1 \end{pmatrix}$ . What is the entry in row 1, column 1 of AB?

A is a 2 × 5 matrix and B is a 5 × 8 matrix. What are the dimensions of AB?
1. 2 × 8
2. 5 × 5
3. AB is not defined

Watch out!

Practice ★★

Let A = $\begin{pmatrix} 8 & 8 \\ 8 & 4 \end{pmatrix}$ and B = $\begin{pmatrix} 2 & 3 \\ 3 & 2 \end{pmatrix}$ . What is the entry in row 2, column 1 of AB?
Let A = $\begin{pmatrix} 1 & 8 \\ 2 & -2 \end{pmatrix}$ and B = $\begin{pmatrix} 2 & -5 \\ 5 & 6 \end{pmatrix}$ . What is the entry in row 2, column 1 of AB?

Boss problem. A stand sells pens and notebooks. The sales matrix S = $\begin{pmatrix} 9 & 5 \\ 3 & 7 \end{pmatrix}$ has rows Monday, Tuesday and columns pens, notebooks. The price vector is p = $\begin{pmatrix} 3 \\ 6 \end{pmatrix}$ (dollars). The product Sp gives each day's income.

Exit ticket

Three questions to check you have got it.

Let A = $\begin{pmatrix} 4 & 2 \\ 3 & -2 \end{pmatrix}$ and B = $\begin{pmatrix} 7 & 9 \\ 5 & 4 \end{pmatrix}$ . What is the entry in row 1, column 1 of AB?
Let A = $\begin{pmatrix} -2 & -4 \\ 5 & -4 \end{pmatrix}$ and B = $\begin{pmatrix} 4 & 7 \\ -5 & -1 \end{pmatrix}$ . What is the entry in row 1, column 1 of AB?
Let A = $\begin{pmatrix} 5 & 8 \\ 0 & 4 \end{pmatrix}$ and B = $\begin{pmatrix} 4 & 1 \\ -2 & 3 \end{pmatrix}$ . What is the entry in row 1, column 1 of AB?

Summary card

Multiplying matrices

(m × n)(n × p) = m × p; entry = row · column.

Watch out: AB is usually not BA.

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