Matrix multiplication | Year 11

01 / LEARN

The idea, step by step

Inside dimensions must agree

An $m \times n$ matrix multiplies an $n \times p$ matrix because each row of the first has the same length as each column of the second. The result has order $m \times p$ . To find an entry, multiply its row–column pairs and add.

The shared category often represents products. Labels must match as well as dimensions. A store-by-product table times a product-by-price column gives a cost total per store.

One row, one column, one total

For entry $(i, j)$ , hold row $i$ of the first matrix and column $j$ of the second. Write the products before summing. Multiplying only entries in matching positions is not the matrix product used here.

In general $A B$ and $B A$ differ. Either can be undefined, or both can exist with different orders or values. Do not rearrange factors as though they were ordinary numbers.

A common trap

Multiply only entries in matching positions.

Instead: Each output sums products from a whole row and column.

02 / WORKED EXAMPLES

See the thinking

Example 1

A rectangular product

Find $A B$ for $A = (\begin{array}{cccccccccc} 1 & 2 & 0 \\ 3 & 1 & 4 \end{array})$ and $B = (\begin{array}{cccccccccc} 2 & 1 \\ 0 & 3 \\ 1 & 2 \end{array})$ .

The result is $2 \times 2$ .
Row 1: $1 (2) + 2 (0) + 0 (1) = 2$ and $1 (1) + 2 (3) + 0 (2) = 7$ .
Row 2: $3 (2) + 1 (0) + 4 (1) = 10$ and $3 (1) + 1 (3) + 4 (2) = 14$ .

A B = (\begin{array}{cccccccccc} 2 & 7 \\ 10 & 14 \end{array})

Example 2

Order matters

Find $A B$ and $B A$ for $A = (\begin{array}{cccccccccc} 1 & 2 \\ 0 & 3 \end{array})$ and $B = (\begin{array}{cccccccccc} 2 & 1 \\ 4 & 0 \end{array})$ .

Use row–column totals for each product.
For example ${(A B)}_{11} = 1 (2) + 2 (4) = 10$ but ${(B A)}_{11} = 2 (1) + 1 (0) = 2$ .

A B = (\begin{array}{cccccccccc} 10 & 1 \\ 12 & 0 \end{array})

;

B A = (\begin{array}{cccccccccc} 2 & 7 \\ 4 & 8 \end{array})

Check your understanding

Can $3 \times 2$ multiply $2 \times 4$ ? Give the order.

Show answer

Yes; $3 \times 4$ .

The inside dimensions both equal 2.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Calculate $(\begin{array}{cccccccccc} 2 & 3 \end{array}) (\begin{array}{cccccccccc} 4 \\ 5 \end{array})$ .
2 marks
Calculate $(\begin{array}{cccccccccc} 1 & 2 \\ 3 & 4 \end{array}) (\begin{array}{cccccccccc} 2 \\ 1 \end{array})$ .
2 marks
Calculate $(\begin{array}{cccccccccc} 2 & 0 \\ 1 & 3 \end{array}) (\begin{array}{cccccccccc} 1 & 4 \\ 2 & - 1 \end{array})$ .
2 marks
Does a $2 \times 4$ matrix multiply a $3 \times 2$ matrix?
2 marks
Find $x$ if $(\begin{array}{cccccccccc} x & 2 \end{array}) (\begin{array}{cccccccccc} 3 \\ 4 \end{array}) = 23$ .
2 marks
If $A$ is $2 \times 3$ and $A B$ is $2 \times 1$ , what order is $B$ ?
2 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Calculate $(\begin{array}{cccccccccc} 2 & 3 \end{array}) (\begin{array}{cccccccccc} 4 \\ 5 \end{array})$ .
2 marks
$2 (4) + 3 (5) = 8 + 15$ .
The one-entry result is 23.
23.
Calculate $(\begin{array}{cccccccccc} 1 & 2 \\ 3 & 4 \end{array}) (\begin{array}{cccccccccc} 2 \\ 1 \end{array})$ .
2 marks
Row 1 gives 4.
Row 2 gives 10.
$(\begin{array}{cccccccccc} 4 \\ 10 \end{array})$ .
Calculate $(\begin{array}{cccccccccc} 2 & 0 \\ 1 & 3 \end{array}) (\begin{array}{cccccccccc} 1 & 4 \\ 2 & - 1 \end{array})$ .
2 marks
First row gives 2 and 8.
Second gives $1 + 6 = 7$ and $4 - 3 = 1$ .
$(\begin{array}{cccccccccc} 2 & 8 \\ 7 & 1 \end{array})$ .
Does a $2 \times 4$ matrix multiply a $3 \times 2$ matrix?
2 marks
The inside dimensions are 4 and 3.
They must agree.
No.
Find $x$ if $(\begin{array}{cccccccccc} x & 2 \end{array}) (\begin{array}{cccccccccc} 3 \\ 4 \end{array}) = 23$ .
2 marks
$3 x + 8 = 23$ .
$3 x = 15$ .
$x = 5$ .
If $A$ is $2 \times 3$ and $A B$ is $2 \times 1$ , what order is $B$ ?
2 marks
It needs 3 matching rows.
One column creates one output column.
$3 \times 1$ .

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: matrices and networks. Original learning resources follow the approved Ascent Library chapter map.

Official curriculum source ↗

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