Matrix multiplication | Year 11

01 / LEARN

Understand the idea.

Inside dimensions must agree

For $A B$ , the number of columns of $A$ must equal the number of rows of $B$ . These matching dimensions make every row-column pairing the same length. An $m \times n$ matrix times an $n \times p$ matrix produces an $m \times p$ result.

The shared labels must also match. Store-by-product quantities can multiply product-by-price data because each product count is paired with its own price. The resulting row-column total belongs to one store and one price scenario.

One row, one column, one total

To find entry $(i, j)$ , multiply corresponding entries of row $i$ of the first matrix and column $j$ of the second, then add the products. This row-column rule differs from multiplying entries in the same position.

Matrix multiplication generally depends on order. Reversing the factors can change the dimensions, make the product undefined, or give different values even when both products exist. Check each product separately rather than assuming $A B = B A$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A rectangular product

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

Worked solution

1
Check the multiplication dimensions.
$(2 \times 3) (3 \times 2) → 2 \times 2$
The inner dimensions match, and the outer dimensions determine the result.
2
Calculate the first result row.
$\begin{aligned} {(A B)}_{11} & = 1 (2) + 2 (0) + 0 (1) = 2 \\ {(A B)}_{12} & = 1 (1) + 2 (3) + 0 (2) = 7 \end{aligned}$
Pair row one of the first matrix with each column of the second.
3
Calculate the second result row.
$\begin{aligned} {(A B)}_{21} & = 3 (2) + 1 (0) + 4 (1) = 10 \\ {(A B)}_{22} & = 3 (1) + 1 (3) + 4 (2) = 14 \end{aligned}$
Pair row two with the same two columns in the same order.
4
Assemble the product.
$A B = (\begin{array}{cccccccccc} 2 & 7 \\ 10 & 14 \end{array})$
Each calculated entry is placed at its indicated row and column.

Answer

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

EXAMPLE 02

Order matters

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

Worked solution

1
Calculate the first row of the forward product.
$\begin{aligned} {(A B)}_{11} & = 1 (2) + 2 (4) = 10 \\ {(A B)}_{12} & = 1 (1) + 2 (0) = 1 \end{aligned}$
The first matrix supplies the row and the second supplies the column.
2
Finish the forward product.
$\begin{aligned} {(A B)}_{21} & = 0 (2) + 3 (4) = 12 \\ {(A B)}_{22} & = 0 (1) + 3 (0) = 0 \end{aligned}$
The second row of the first matrix completes the two-by-two product.
3
Calculate the first row after reversing order.
$\begin{aligned} {(B A)}_{11} & = 2 (1) + 1 (0) = 2 \\ {(B A)}_{12} & = 2 (2) + 1 (3) = 7 \end{aligned}$
Now the rows come from the original second matrix.
4
Finish the reverse product.
$\begin{aligned} {(B A)}_{21} & = 4 (1) + 0 (0) = 4 \\ {(B A)}_{22} & = 4 (2) + 0 (3) = 8 \end{aligned}$
Changing the factor order changes which entries are paired.
5
Display and compare the results.
$A B = (\begin{array}{cccccccccc} 10 & 1 \\ 12 & 0 \end{array}), B A = (\begin{array}{cccccccccc} 2 & 7 \\ 4 & 8 \end{array})$
Both products exist here, but their entries are different.

Answer

$A B = (\begin{array}{cccccccccc} 10 & 1 \\ 12 & 0 \end{array})$ ; $B A = (\begin{array}{cccccccccc} 2 & 7 \\ 4 & 8 \end{array})$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

Yes; $3 \times 4$ .

Yes. The first factor has two columns and the second has two rows. The outer dimensions give the resulting order $3 \times 4$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Inside dimensions must agree

One row, one column, one total

See the method in action.

A rectangular product

Check the multiplication dimensions.

Calculate the first result row.

Calculate the second result row.

Assemble the product.

Order matters

Calculate the first row of the forward product.

Finish the forward product.

Calculate the first row after reversing order.

Finish the reverse product.

Display and compare the results.

Try it yourself.

Check your understanding

Keep practising