Matrix notation, order and entries | Year 11

01 / LEARN

The idea, step by step

Rows first, columns second

A matrix stores quantities in a rectangular arrangement. Its order is the number of rows followed by the number of columns. A $2 \times 3$ matrix has two horizontal rows and three vertical columns; it therefore has six entries. A row vector has one row and a column vector has one column.

The entry $a_{i j}$ is in row $i$ , column $j$ . These subscripts describe a location, not multiplication. Every row must contain the same number of entries.

Interpret the arrangement

Before using a matrix for data, give the rows and columns a meaning. A sales matrix might use rows for days and columns for products. The number in a position is useful only when the labels and units are known.

Two matrices are equal precisely when they have the same order and matching entries are equal. Compare corresponding positions to form equations; having the same entries in a different order does not make two matrices equal.

A common trap

Calling a matrix with two rows and three columns a $3 \times 2$ matrix.

Instead: State rows first, then columns. A quick sketch of the rectangular arrangement helps.

02 / WORKED EXAMPLES

See the thinking

Example 1

Locate an entry

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

There are two rows and three columns.
Row 2 is $(0, 3, 2)$ , and its third entry is 2.

Order

2 \times 3

;

a_{23} = 2

Example 2

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 $x, y$ .

Compare top-left entries: $x + 1 = 4$ .
Compare bottom-right entries: $2 y = 8$ .
Solve the two independent scalar equations.

x = 3

y = 4

Check your understanding

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

Show answer

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

There are three rows, one column, and the third-row entry is -2.

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 $(\begin{array}{cccccccccc} 1 & 2 & 3 & 4 \end{array})$ .
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 ↗

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