Matrix notation, order and entries | Year 11

01 / LEARN

Understand the idea.

Rows first, columns second

A matrix is a rectangular arrangement whose positions carry meaning. Count horizontal rows first, then vertical columns: a $2 \times 3$ matrix has six entries. A row vector has one row; a column vector has one column.

The subscripts in $a_{i j}$ tell you where to look: row $i$ , then column $j$ . They are labels, not factors. Trace across the selected row before choosing the entry in the selected column.

Interpret the arrangement

When a matrix records data, identify what each row and column represents and the units of its entries. Interchanging rows may change which day or shop a value describes, even when all the numbers remain present.

Matrix equality requires both the same order and equality in every corresponding position. Compare matching entries to obtain scalar equations; equal totals or the same collection of numbers do not establish matrix equality.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Locate an entry

For $A = (\begin{array}{cccccccccc} 4 & - 1 & 7 \\ 0 & 3 & 2 \end{array})$ , state its order and $a_{23}$ .

Worked solution

1
Count the rows and columns.
$rows = 2, columns = 3$
Order records the arrangement, with the row count first.
2
State the order.
$order (A) = 2 \times 3$
Two rows each contain three entries.
3
Read row 2, column 3.
$row 2 = (0, 3, 2), a_{23} = 2$
The second subscript selects the third entry of this row.

Answer

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

EXAMPLE 02

Use matrix equality

If $(\begin{array}{cccccccccc} x + 1 & 5 \\ 2 & 2 y \end{array}) = (\begin{array}{cccccccccc} 4 & 5 \\ 2 & 8 \end{array})$ , find $x, y$ .

Worked solution

1
Compare the top-left entries.
$x + 1 = 4$
Equal matrices have equal entries in row 1, column 1.
2
Solve for the first unknown.
$x = 4 - 1 = 3$
Subtracting 1 isolates the variable.
3
Compare the bottom-right entries and solve.
$2 y = 8, y = \frac{8}{2} = 4$
The row-2, column-2 entries must also match.
4
Check every position.
$(\begin{array}{cccccccccc} 3 + 1 & 5 \\ 2 & 2 (4) \end{array}) = (\begin{array}{cccccccccc} 4 & 5 \\ 2 & 8 \end{array})$
The other two entries already agree, so both matrices are equal.

Answer

$x = 3$ , $y = 4$ .

03 / YOUR TURN

Try it yourself.

Check your understanding

If $B = (\begin{array}{cccccccccc} 6 \\ 0 \\ - 2 \end{array})$ , what are its order and $b_{31}$ ?

Show answer & reasoning

$3 \times 1$ and $- 2$ .

The column vector has three rows and one column, so its order is $3 \times 1$ . Row 3, column 1 contains $- 2$ , hence $b_{31} = - 2$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

Read the questions online

State the order of $(\begin{array}{cccccccccc} 1 & 2 & 3 & 4 \end{array})$ .
2 marks
For $A = (\begin{array}{cccccccccc} 2 & 5 \\ - 1 & 8 \end{array})$ , find $a_{21}$ and $a_{12}$ .
2 marks
How many entries does a $4 \times 3$ matrix contain?
2 marks
Find $x$ if $(\begin{array}{cccccccccc} 1 & x - 2 \end{array}) = (\begin{array}{cccccccccc} 1 & 6 \end{array})$ .
2 marks
Two shops record sales of three products. Rows represent shops and columns represent products. State the order of the sales matrix.
2 marks
Are $(\begin{array}{cccccccccc} 1 & 2 \\ 3 & 4 \end{array})$ and $(\begin{array}{cccccccccc} 1 & 3 \\ 2 & 4 \end{array})$ equal? Explain.
2 marks

Read the worked solutions

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

State the order of $(\begin{array}{cccccccccc} 1 & 2 & 3 & 4 \end{array})$ .
2 marks
Count one row and four columns.
$1 \times 4$ .
For $A = (\begin{array}{cccccccccc} 2 & 5 \\ - 1 & 8 \end{array})$ , find $a_{21}$ and $a_{12}$ .
2 marks
Locate row 2 column 1.
Then locate row 1 column 2.
$a_{21} = - 1$ , $a_{12} = 5$ .
How many entries does a $4 \times 3$ matrix contain?
2 marks
Multiply the number of rows by the number of columns: $4 \times 3 = 12$ .
12 entries.
Find $x$ if $(\begin{array}{cccccccccc} 1 & x - 2 \end{array}) = (\begin{array}{cccccccccc} 1 & 6 \end{array})$ .
2 marks
Equality gives $x - 2 = 6$ .
Add 2 to both sides.
$x = 8$ .
Two shops record sales of three products. Rows represent shops and columns represent products. State the order of the sales matrix.
2 marks
There are two shop rows and three product columns.
$2 \times 3$ .
Are $(\begin{array}{cccccccccc} 1 & 2 \\ 3 & 4 \end{array})$ and $(\begin{array}{cccccccccc} 1 & 3 \\ 2 & 4 \end{array})$ equal? Explain.
2 marks
Their orders agree, but their top-right entries are 2 and 3.
Matrix equality requires every corresponding entry to agree.
No.

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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.

Rows first, columns second

Interpret the arrangement

See the method in action.

Locate an entry

Count the rows and columns.

State the order.

Read row 2, column 3.

Use matrix equality

Compare the top-left entries.

Solve for the first unknown.

Compare the bottom-right entries and solve.

Check every position.

Try it yourself.

Check your understanding

Keep practising