Matrix arithmetic revision | Year 12

01 / LEARN

Understand the idea.

Rows, columns and matching entries

A matrix stores data in an ordered rectangle. Its order is rows by columns: $2 \times 3$ means two rows and three columns. Label those rows and columns so each entry has a definite meaning and unit.

Addition and subtraction act on corresponding entries, so the matrices must have the same order and compatible labels. Scalar multiplication changes every entry. A zero is an actual value; it should not be used to stand for unknown data.

Why multiplication uses a row and a column

In $A B$ , the columns of $A$ must match the rows of $B$ . Each result entry is one row dotted with one column: multiply matching positions, then add. The outer dimensions give the order of the result.

Check the labels as well as the dimensions. Quantity multiplied by price gives income, but unrelated categories should not be combined. Matrix multiplication generally depends on order; even when both products exist, $A B$ need not equal $B A$ .

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Combine stock changes

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

Worked solution

1
Write the supplied rows as matrices.
$A = (\begin{array}{cccccccccc} 4 & 2 \\ 1 & 3 \end{array}), B = (\begin{array}{cccccccccc} 1 & 5 \\ 2 & 0 \end{array})$
Both matrices have order $2 \times 2$ , so corresponding entries can be subtracted.
2
Multiply every entry of A by 2.
$2 A = (\begin{array}{cccccccccc} 8 & 4 \\ 2 & 6 \end{array})$
A scalar multiplies all entries, not just one row or the diagonal.
3
Subtract corresponding entries.
$\begin{aligned} 2 A - B & = (\begin{array}{cccccccccc} 8 - 1 & 4 - 5 \\ 2 - 2 & 6 - 0 \end{array}) \\ = (\begin{array}{cccccccccc} 7 & - 1 \\ 0 & 6 \end{array}) \end{aligned}$
Keep each result in the same row and column as the entries being compared.

Answer

Rows (7,−1),(0,6).

EXAMPLE 02

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.

Worked solution

1
Align the product categories.
$S = (\begin{array}{cccccccccc} 3 & 2 \\ 1 & 5 \end{array}), p = (\begin{array}{cccccccccc} 4 \\ 6 \end{array})$
Sales rows are X and Y; sales columns and price rows both follow notebook, folder order.
2
Check the product dimensions.
$(2 \times 2) (2 \times 1) → 2 \times 1$
The matching product dimension is summed out, leaving one income for each shop.
3
Calculate each row-column total.
$S p = (\begin{array}{cccccccccc} 3 (4) + 2 (6) \\ 1 (4) + 5 (6) \end{array}) = (\begin{array}{cccccccccc} 24 \\ 34 \end{array})$
Each term is quantity times AUD per item, so the incomes are AUD 24 for X and AUD 34 for Y.

Answer

Income column (24,34) dollars.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

$3 \times 4$ .

The inner dimensions in $(3 \times 2) (2 \times 4)$ agree. Each of the 3 rows combines with each of the 4 columns, so $A B$ has order $3 \times 4$ .

Keep practising

6 worksheet questions · 23 marks

Open worksheet Worked solutions

Read the questions online

For rows $A$ : $(\begin{array}{cccccccccc} 2 & - 1 \\ 3 & 4 \end{array})$ , and $B$ : $(\begin{array}{cccccccccc} 1 & 5 \\ - 2 & 0 \end{array})$ , find $2 A - B$ .
3 marks
For rows $A$ : $(\begin{array}{cccccccccc} 1 & 2 & 0 \\ - 1 & 3 & 2 \end{array})$ , and column $b = (4, 1, 2)$ , find $A b$ .
3 marks
Let $A$ have rows $(\begin{array}{cccccccccc} 1 & 1 \\ 0 & 1 \end{array})$ and $B$ have rows $(\begin{array}{cccccccccc} 2 & 0 \\ 1 & 3 \end{array})$ .
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 $A$ : $(\begin{array}{cccccccccc} 2 & - 1 \\ 3 & 4 \end{array})$ , and $B$ : $(\begin{array}{cccccccccc} 1 & 5 \\ - 2 & 0 \end{array})$ , find $2 A - B$ .
3 marks
Double $A$ to $(\begin{array}{cccccccccc} 4 & - 2 \\ 6 & 8 \end{array})$ .
Subtract corresponding entries.
Rows $(\begin{array}{cccccccccc} 3 & - 7 \\ 8 & 8 \end{array})$ .
For rows $A$ : $(\begin{array}{cccccccccc} 1 & 2 & 0 \\ - 1 & 3 & 2 \end{array})$ , 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 $(\begin{array}{cccccccccc} 6 \\ 3 \end{array})$ .
Let $A$ have rows $(\begin{array}{cccccccccc} 1 & 1 \\ 0 & 1 \end{array})$ and $B$ have rows $(\begin{array}{cccccccccc} 2 & 0 \\ 1 & 3 \end{array})$ .
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 $(\begin{array}{cccccccccc} 3 & 3 \\ 1 & 3 \end{array})$ .
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 $(\begin{array}{cccccccccc} 2 & 2 \\ 1 & 4 \end{array})$ , 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 $(\begin{array}{cccccccccc} 3 & 3 \\ 1 & 3 \end{array})$ .
  Rows $(\begin{array}{cccccccccc} 3 & 3 \\ 1 & 3 \end{array})$ .
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 $(\begin{array}{cccccccccc} 2 & 2 \\ 1 & 4 \end{array})$ , which differs from $A B$ .
  Rows $(\begin{array}{cccccccccc} 2 & 2 \\ 1 & 4 \end{array})$ ; the products differ.
a) Rows $(\begin{array}{cccccccccc} 3 & 3 \\ 1 & 3 \end{array})$ .; b) Rows $(\begin{array}{cccccccccc} 2 & 2 \\ 1 & 4 \end{array})$ ; 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 ↗

In this lesson

Understand the idea.

Rows, columns and matching entries

Why multiplication uses a row and a column

See the method in action.

Combine stock changes

Write the supplied rows as matrices.

Multiply every entry of A by 2.

Subtract corresponding entries.

Turn sales into income

Align the product categories.

Check the product dimensions.

Calculate each row-column total.

Try it yourself.

Check your understanding

Keep practising