Matrix organisation and order | Year 11

01 / LEARN

The idea, step by step

Matrices organise information

A matrix is a rectangular array. Its order counts horizontal rows first and vertical columns second. A $3 \times 2$ matrix has three rows and two columns. Every row must have the same number of entries.

An entry is meaningful through its position. Sales rows might identify stores while columns identify products. Preserve these labels when calculating: reversing them changes the meaning.

Find the named entry

The notation $a_{i j}$ means row $i$ , column $j$ . Start at the named row, then move across to the column. Subscripts describe position, not multiplication. A row matrix has one row, a column matrix one column, and a square matrix equally many rows and columns.

Check units and meaning. Zero is a recorded value, not automatically missing information. Negative values can describe losses or temperatures, but not the number of people attending.

A common trap

A $2 \times 3$ matrix has three rows.

Instead: The first number counts rows.

02 / WORKED EXAMPLES

See the thinking

Example 1

Rows before columns

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

There are two rows and three columns.
The third entry of row 2, $(4, 9, 1)$ , is 1.

Order

2 \times 3

;

a_{23} = 1

Example 2

Read the context

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

21 is in row 2, column 2.
Those labels are South and pens.

South sold 21 pens.

Check your understanding

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

Show answer

$3 \times 1$ .

There are three rows, each containing one entry.

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF 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 ↗

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