Matrix multiplication and non-commutativity | Year 11

01 / LEARN

Understand the idea.

Check dimensions before calculating

For $A$ of order $m \times n$ and $B$ of order $n \times p$ , each row of $A$ and column of $B$ have the same length. Thus $A B$ exists and has order $m \times p$ .

To find entry $(i, j)$ , pair every entry of row $i$ with the corresponding entry of column $j$ , multiply the pairs, and add the products. Repeat for each output position; matching-position multiplication is a different operation.

Order is part of the operation

Reversing the factors changes which rows meet which columns. Consequently, $A B$ and $B A$ can differ, or one may be undefined. Never swap factors merely to simplify arithmetic; order is part of the calculation.

Associativity allows $(A B) C = A (B C)$ for compatible matrices without changing their order. One unequal entry disproves $A B = B A$ ; to establish equality of two particular products, every corresponding entry must agree.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A rectangular product

Calculate $(\begin{array}{cccccccccc} 1 & 2 & - 1 \\ 0 & 3 & 4 \end{array}) (\begin{array}{cccccccccc} 2 & 1 \\ 1 & - 2 \\ 3 & 0 \end{array})$ .

Worked solution

1
Check the product dimensions.
$(2 \times 3) (3 \times 2) → 2 \times 2$
The inner dimensions agree, and the outer dimensions give the result size.
2
Calculate the first output row.
$\begin{aligned} c_{11} & = 1 (2) + 2 (1) + (- 1) (3) = 1 \\ c_{12} & = 1 (1) + 2 (- 2) + (- 1) (0) = - 3 \end{aligned}$
Use row 1 of the left matrix with each column of the right matrix.
3
Calculate the second output row.
$\begin{aligned} c_{21} & = 0 (2) + 3 (1) + 4 (3) = 15 \\ c_{22} & = 0 (1) + 3 (- 2) + 4 (0) = - 6 \end{aligned}$
Use row 2 with the same two columns, preserving signs.
4
Place the entries in the result.
$A B = (\begin{array}{cccccccccc} 1 & - 3 \\ 15 & - 6 \end{array})$
Each calculated entry belongs in its indexed row and column.

Answer

$(\begin{array}{cccccccccc} 1 & - 3 \\ 15 & - 6 \end{array})$ .

EXAMPLE 02

Demonstrate non-commutativity

Let $A = (\begin{array}{cccccccccc} 1 & 1 \\ 0 & 1 \end{array})$ , $B = (\begin{array}{cccccccccc} 2 & 0 \\ 0 & 3 \end{array})$ . Compare $A B$ and $B A$ .

Worked solution

1
Multiply in the stated order.
$A B = (\begin{array}{cccccccccc} 1 (2) + 1 (0) & 1 (0) + 1 (3) \\ 0 (2) + 1 (0) & 0 (0) + 1 (3) \end{array})$
Rows now come from $A$ and columns from $B$ .
2
Simplify the first product.
$A B = (\begin{array}{cccccccccc} 2 & 3 \\ 0 & 3 \end{array})$
Each row-column dot product supplies one entry.
3
Reverse the factors and calculate.
$\begin{aligned} B A & = (\begin{array}{cccccccccc} 2 (1) + 0 (0) & 2 (1) + 0 (1) \\ 0 (1) + 3 (0) & 0 (1) + 3 (1) \end{array}) \\ = (\begin{array}{cccccccccc} 2 & 2 \\ 0 & 3 \end{array}) \end{aligned}$
The reversed order uses rows of $B$ and columns of $A$ .
4
Compare corresponding entries.
${(A B)}_{12} = 3 \neq 2 = {(B A)}_{12}, A B \neq B A$
A single unequal corresponding entry rules out matrix equality.

Answer

$A B \neq B A$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

If $A$ is $2 \times 3$ and $B$ is $3 \times 4$ , what is the order of $A B$ ?

Show answer & reasoning

$2 \times 4$ .

In $(2 \times 3) (3 \times 4)$ the shared inner dimension is 3, so multiplication is defined. The two row choices and four column choices give a $2 \times 4$ product.

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

Find $(\begin{array}{cccccccccc} 2 & 1 \end{array}) (\begin{array}{cccccccccc} 3 \\ - 4 \end{array})$ .
2 marks
Find $(\begin{array}{cccccccccc} 1 & 2 \\ 3 & 0 \end{array}) (\begin{array}{cccccccccc} 2 & 1 \\ 1 & 4 \end{array})$ .
2 marks
$A$ is $3 \times 2$ and $B$ is $2 \times 5$ . State the order of $A B$ and explain whether $B A$ exists.
2 marks
Find $(\begin{array}{cccccccccc} 1 \\ 2 \end{array}) (\begin{array}{cccccccccc} 3 & 4 \end{array})$ .
2 marks
For $A = (\begin{array}{cccccccccc} 0 & 1 \\ 1 & 0 \end{array})$ , calculate $A^{2}$ .
2 marks
A shop sells 6 notebooks at 4 dollars each and 3 folders at 5 dollars each. Express the total as a row-column matrix product and evaluate it.
2 marks

Read the worked solutions

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

Find $(\begin{array}{cccccccccc} 2 & 1 \end{array}) (\begin{array}{cccccccccc} 3 \\ - 4 \end{array})$ .
2 marks
The product is $2 (3) + 1 (- 4) = 2$ .
$2$ .
Find $(\begin{array}{cccccccccc} 1 & 2 \\ 3 & 0 \end{array}) (\begin{array}{cccccccccc} 2 & 1 \\ 1 & 4 \end{array})$ .
2 marks
Rows dotted with columns give $(2 + 2, 1 + 8)$ and $(6 + 0, 3 + 0)$ .
$(\begin{array}{cccccccccc} 4 & 9 \\ 6 & 3 \end{array})$ .
$A$ is $3 \times 2$ and $B$ is $2 \times 5$ . State the order of $A B$ and explain whether $B A$ exists.
2 marks
The $A B$ inner dimensions are both 2.
For $B A$ the inner dimensions would be 5 and 3, which do not match.
$A B$ is $3 \times 5$ ; $B A$ is not defined.
Find $(\begin{array}{cccccccccc} 1 \\ 2 \end{array}) (\begin{array}{cccccccccc} 3 & 4 \end{array})$ .
2 marks
A $2 \times 1$ by $1 \times 2$ product has order $2 \times 2$ .
Multiply each column-vector entry by the row.
$(\begin{array}{cccccccccc} 3 & 4 \\ 6 & 8 \end{array})$ .
For $A = (\begin{array}{cccccccccc} 0 & 1 \\ 1 & 0 \end{array})$ , calculate $A^{2}$ .
2 marks
Each output column is found by applying $A$ twice.
Direct multiplication gives diagonal entries 1 and off-diagonal entries 0.
$I = (\begin{array}{cccccccccc} 1 & 0 \\ 0 & 1 \end{array})$ .
A shop sells 6 notebooks at 4 dollars each and 3 folders at 5 dollars each. Express the total as a row-column matrix product and evaluate it.
2 marks
Quantities form a row and prices form a column.
$6 (4) + 3 (5) = 39$ .
$(\begin{array}{cccccccccc} 6 & 3 \end{array}) (\begin{array}{cccccccccc} 4 \\ 5 \end{array}) = 39$ dollars.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: matrices and transformations; original lessons follow the approved school-sized chapter structure.

Official curriculum source ↗

In this lesson

Understand the idea.

Check dimensions before calculating

Order is part of the operation

See the method in action.

A rectangular product

Check the product dimensions.

Calculate the first output row.

Calculate the second output row.

Place the entries in the result.

Demonstrate non-commutativity

Multiply in the stated order.

Simplify the first product.

Reverse the factors and calculate.

Compare corresponding entries.

Try it yourself.

Check your understanding

Keep practising