Matrix organisation and order | Year 11

01 / LEARN

Understand the idea.

Matrices organise information

A matrix is a rectangular arrangement of entries. Its order is rows first, then columns: $3 \times 2$ means three horizontal rows, with two entries in each. Unequal row lengths do not form a matrix.

Row and column labels give the numbers meaning. In a store-by-product matrix, moving an entry to another row changes the store it describes. Record the label order beside the matrix and preserve it in later calculations.

Find the named entry

The entry $a_{i j}$ is found in row $i$ , column $j$ . Read the first subscript to choose a row, then count across to the second subscript. These subscripts locate an entry; they do not mean multiply $i$ by $j$ .

A one-row matrix is a row matrix; a one-column matrix is a column matrix. A square matrix has equal row and column counts. Interpret zeros and negative entries using context: zero sales is recorded information, while a negative physical attendance count needs investigation.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Rows before columns

For $A = (\begin{array}{cccccccccc} 7 & 2 & 5 \\ 4 & 9 & 1 \end{array})$ , state the order and find $a_{23}$ .

Worked solution

1
Count the rows and columns.
$rows = 2, columns = 3$
The array has two horizontal lists, each containing three numbers.
2
Write the order.
$2 \times 3$
Matrix order always places the row count first.
3
Locate the requested entry.
$row 2 = (4, 9, 1), a_{23} = 1$
The first subscript selects row two and the second selects its third entry.

Answer

Order $2 \times 3$ ; $a_{23} = 1$ .

EXAMPLE 02

Read the context

Rows are North, South and West stores; columns are notebooks and pens. Interpret 21 in $(\begin{array}{cccccccccc} 12 & 30 \\ 8 & 21 \\ 15 & 18 \end{array})$ .

Worked solution

1
Locate the number twenty-one.
$a_{22} = 21$
It appears in the second row and second column.
2
Read the labels.
Row two represents South and column two represents pens.
3
Interpret the entry in context.
South sold twenty-one pens; the number is a sales count for that store and product.

Answer

South sold 21 pens.

03 / YOUR TURN

Try it yourself.

Check your understanding

State the order of $(\begin{array}{cccccccccc} 4 \\ 7 \\ 2 \end{array})$ .

Show answer & reasoning

$3 \times 1$ .

The entries are stacked into three rows with one column, so the order is $3 \times 1$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

State the order of $A = (\begin{array}{cccccccccc} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{array})$ .
2 marks
Find $a_{12}$ and $a_{31}$ in $A = (\begin{array}{cccccccccc} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{array})$ .
2 marks
Rows are Monday/Tuesday and columns adult/child tickets. Interpret 31 in $(\begin{array}{cccccccccc} 24 & 10 \\ 31 & 8 \end{array})$ .
2 marks
How many entries are in a $4 \times 3$ matrix?
2 marks
Can lists $(2, 5)$ and $(1, 4, 7)$ form the two rows of one matrix?
2 marks
A profit matrix includes $- 40$ . Must it be invalid?
2 marks

Read the worked solutions

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

State the order of $A = (\begin{array}{cccccccccc} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{array})$ .
2 marks
Count 3 rows.
Each contains 2 entries.
$3 \times 2$ .
Find $a_{12}$ and $a_{31}$ in $A = (\begin{array}{cccccccccc} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{array})$ .
2 marks
Row 1 column 2 is 4.
Row 3 column 1 is 3.
$a_{12} = 4$ , $a_{31} = 3$ .
Rows are Monday/Tuesday and columns adult/child tickets. Interpret 31 in $(\begin{array}{cccccccccc} 24 & 10 \\ 31 & 8 \end{array})$ .
2 marks
31 is in row 2, column 1.
Read the matching labels.
31 adult tickets were sold on Tuesday.
How many entries are in a $4 \times 3$ matrix?
2 marks
Four rows have three entries each.
$4 (3) = 12$ .
12.
Can lists $(2, 5)$ and $(1, 4, 7)$ form the two rows of one matrix?
2 marks
Their lengths differ.
A matrix must be rectangular.
No.
A profit matrix includes $- 40$ . Must it be invalid?
2 marks
Matrix entries may be negative.
Its context determines whether that makes sense.
No; it could represent a 40 dollars loss.

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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.

Matrices organise information

Find the named entry

See the method in action.

Rows before columns

Count the rows and columns.

Write the order.

Locate the requested entry.

Read the context

Locate the number twenty-one.

Read the labels.

Interpret the entry in context.

Try it yourself.

Check your understanding

Keep practising