Matrix arithmetic revision | Year 12

01 / LEARN

The idea, step by step

Rows, columns and matching entries

A matrix is an ordered rectangular array. Its order is rows by columns: two rows and three columns give order $2 \times 3$ . Before calculating, label what each row and column represents. The entry in row 2, column 1 need not mean the same thing as the entry in row 1, column 2.

Addition and subtraction compare corresponding entries, so both matrices must have the same order. Scalar multiplication multiplies every entry. A zero is a value, not missing data. Keep units in the interpretation, such as dollars of income or numbers of items.

Why multiplication uses a row and a column

For $A B$ to exist, the number of columns of $A$ must equal the number of rows of $B$ . Every result entry is a sum of products from one complete row and one complete column. In a sales model the terms can be quantity times price; in a transition model they can be proportion times population.

Matrix multiplication usually depends on order. Even if $A B$ and $B A$ both exist, they may differ. Check meaning as well as dimensions: an arithmetically possible product may combine labels in an inappropriate way.

A common trap

Multiplying only corresponding matrix entries.

Instead: A matrix product combines each complete row with each complete column.

02 / WORKED EXAMPLES

See the thinking

Example 1

Combine stock changes

Matrices have rows $A$ : (4,2),(1,3), and $B$ : (1,5),(2,0). Find $2 A - B$ .

Double $A$ to obtain (8,4),(2,6).
Subtract corresponding entries: (8−1,4−5),(2−2,6−0).

Rows (7,−1),(0,6).

Example 2

Turn sales into income

Shop X sells 3 notebooks and 2 folders; Y sells 1 notebook and 5 folders. Prices are AUD 4 and AUD 6 respectively. Find each income.

Use sales rows (3,2),(1,5) and price column (4,6).
X earns $3 (4) + 2 (6) = 24$ ; Y earns $1 (4) + 5 (6) = 34$ .

Income column (24,34) dollars.

Check your understanding

If $A$ is $3 \times 2$ and $B$ is $2 \times 4$ , state the order of $A B$ .

Show answer

$3 \times 4$ .

The inner dimensions agree; the outer dimensions give the result.

03 / WORKSHEET

Your turn

6 questions · 23 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

For rows $A$ : (2,−1),(3,4), and $B$ : (1,5),(−2,0), find $2 A - B$ .
3 marks
For rows $A$ : (1,2,0),(−1,3,2), and column $b = (4, 1, 2)$ , find $A b$ .
3 marks
Let $A$ have rows (1,1),(0,1) and $B$ have rows (2,0),(1,3).
6 marks
1. (a)
  Find $A B$ .
  3 marks
2. (b)
  Find $B A$ and compare.
  3 marks
A $4 \times 3$ matrix multiplies a $3 \times 2$ matrix. State the result order. Is the reverse product defined?
3 marks
Equal matrices have rows $(x + 2, 5), (4, y - 1)$ and $(9, 5), (4, 6)$ . Find $x, y$ .
3 marks
A bakery sells muffins at AUD 3 and rolls at AUD 2. Branch X sells (12,20); Y sells (8,30).
5 marks
1. (a)
  State the sales-matrix and price-column orders.
  2 marks
2. (b)
  Find and interpret the income column.
  3 marks

Read the worked solutions

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

For rows $A$ : (2,−1),(3,4), and $B$ : (1,5),(−2,0), find $2 A - B$ .
3 marks
Double $A$ to (4,−2),(6,8).
Subtract corresponding entries.
Rows (3,−7),(8,8).
For rows $A$ : (1,2,0),(−1,3,2), and column $b = (4, 1, 2)$ , find $A b$ .
3 marks
First row: $1 (4) + 2 (1) + 0 (2) = 6$ .
Second row: $- 1 (4) + 3 (1) + 2 (2) = 3$ .
Column (6,3).
Let $A$ have rows (1,1),(0,1) and $B$ have rows (2,0),(1,3).
6 marks
a) Row 1: $1 (2) + 1 (1) = 3$ and $1 (0) + 1 (3) = 3$ .
a) Row 2: $0 (2) + 1 (1) = 1$ and $0 (0) + 1 (3) = 3$ .
a) Thus $A B$ has rows (3,3),(1,3).
b) Row 1: $2 (1) + 0 (0) = 2$ and $2 (1) + 0 (1) = 2$ .
b) Row 2: $1 (1) + 3 (0) = 1$ and $1 (1) + 3 (1) = 4$ .
b) Thus $B A$ has rows (2,2),(1,4), which differs from $A B$ .
1. (a)
  Find $A B$ .
  3 marks
  Row 1: $1 (2) + 1 (1) = 3$ and $1 (0) + 1 (3) = 3$ .
  Row 2: $0 (2) + 1 (1) = 1$ and $0 (0) + 1 (3) = 3$ .
  Thus $A B$ has rows (3,3),(1,3).
  Rows (3,3),(1,3).
2. (b)
  Find $B A$ and compare.
  3 marks
  Row 1: $2 (1) + 0 (0) = 2$ and $2 (1) + 0 (1) = 2$ .
  Row 2: $1 (1) + 3 (0) = 1$ and $1 (1) + 3 (1) = 4$ .
  Thus $B A$ has rows (2,2),(1,4), which differs from $A B$ .
  Rows (2,2),(1,4); the products differ.
a) Rows (3,3),(1,3).; b) Rows (2,2),(1,4); the products differ.
A $4 \times 3$ matrix multiplies a $3 \times 2$ matrix. State the result order. Is the reverse product defined?
3 marks
Inner dimensions match in the first product.
For the reverse product, 2 and 4 do not match.
$4 \times 2$ ; the reverse product is not defined.
Equal matrices have rows $(x + 2, 5), (4, y - 1)$ and $(9, 5), (4, 6)$ . Find $x, y$ .
3 marks
Matching entries give $x + 2 = 9$ and $y - 1 = 6$ .
$x = 7, y = 7$ .
A bakery sells muffins at AUD 3 and rolls at AUD 2. Branch X sells (12,20); Y sells (8,30).
5 marks
a) Rows represent branches and columns products.
b) Calculate $12 (3) + 20 (2) = 76$ and $8 (3) + 30 (2) = 84$ .
1. (a)
  State the sales-matrix and price-column orders.
  2 marks
  Rows represent branches and columns products.
  $2 \times 2$ and $2 \times 1$ .
2. (b)
  Find and interpret the income column.
  3 marks
  Calculate $12 (3) + 20 (2) = 76$ and $8 (3) + 30 (2) = 84$ .
  (76,84): X earns AUD 76 and Y earns AUD 84.
a) $2 \times 2$ and $2 \times 1$ .; b) (76,84): X earns AUD 76 and Y earns AUD 84.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics matrix-models option. This is the offered fifth-topic option, alongside the four compulsory topics; a school-designed Open Topic is an alternative, not prescribed content in this chapter.

Official curriculum source ↗

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