E-Math notes · Chapter 9

Matrices: O-Level E-Math Notes

A matrix is a table of numbers that you can calculate with. By the end of this chapter you can display data as a matrix, read what it says, multiply it by a number, add and subtract matrices, multiply two matrices, and use all of this to solve shopping and sales problems.

Your outcome tracker0 / 6

Let's go! Tick an outcome once you can pass its exam check. Saved on this device.

Chapter
9

Figure 1 · Diagram for Example 9.8

4 outcomes · 28 worked examples · every step shown ✦

Display and order

The order of a matrix is rows × columns, in that order.

Interpreting data

The element in row i, column j answers: what does row i say about column j?

Scalar multiplication

k × matrix: every element gets multiplied by k.

By the end of this chapter

Display information in a matrix of any order, and state its order.
Interpret the data in a given matrix.
Multiply a matrix by a scalar.
Calculate the sum, the difference and (where defined) the product of two matrices, and solve problems with them.

Start here

Maths In Real Life

Two situations where a matrix does the bookkeeping. The scene pictures are made separately; the matrices are drawn on top by script so they match the problem exactly.

$A hawker stall counter with chicken rice, laksa and wanton noodles, with the number matrices for plates sold and prices drawn over the picture$

Maths In Real Life 1

The hawker stall's takings

A stall owner can keep track of what was sold and what it is worth by storing the day's numbers in a matrix, then multiplying by the price list.

At a hawker stall, the numbers of plates of chicken rice, laksa and wanton noodles sold on Saturday (row 1) and Sunday (row 2) are $\displaystyle \mathbf{Q} = {\small \begin{pmatrix} 40 & 25 & 15 \\ 32 & 30 & 20 \end{pmatrix}}$ The prices in dollars are $\displaystyle \mathbf{P} = \begin{pmatrix} 4.50 \\ 3.50 \\ 5.00 \end{pmatrix}$ (invented prices). Find $\mathbf{QP}$ to the nearest cent and say which day earned more.

Show full solutionHide solution

Colour keyround, estimate or compare

\displaystyle 2 \times 3 \ \text{times} \ 3 \times 1 \ \Rightarrow \ 2 \times 1

columns of A = rows of B: one takings total for each day

\displaystyle \mathbf{QP} = \begin{pmatrix} 342.50 \\ 349.00 \end{pmatrix}

Saturday took 342.50 dollars and Sunday took 349.00 dollars, so Sunday earned more.

compare the two totals

Final answer

\mathbf{QP} = \begin{pmatrix} 342.50 \\ 349.00 \end{pmatrix}

; Sunday earned more.

$A bubble tea kiosk counter with three cups, with the sales matrices added and multiplied by the price matrix drawn over the picture$

Maths In Real Life 2

Two weekends of bubble tea

Combining two weekends of orders by adding matrices, then pricing the total in one multiplication, avoids doing the same calculation twice.

A bubble-tea kiosk records how many cups of classic, fruit and milk-foam tea are ordered at two outlets (rows) on two weekends: $\displaystyle \mathbf{A} = {\small \begin{pmatrix} 24 & 10 & 15 \\ 18 & 30 & 22 \end{pmatrix}}$ and $\displaystyle \mathbf{C} = {\small \begin{pmatrix} 20 & 16 & 12 \\ 26 & 14 & 18 \end{pmatrix}}$ (numbers of cups). The prices in dollars are $\displaystyle \mathbf{P} = \begin{pmatrix} 1.50 \\ 1.00 \\ 2.50 \end{pmatrix}$ (invented). Find the total takings for each outlet over both weekends, to the nearest cent.

Show full solutionHide solution

\displaystyle \mathbf{A} + \mathbf{C} = {\small \begin{pmatrix} 44 & 26 & 27 \\ 44 & 44 & 40 \end{pmatrix}}

same order: both weekends added first

\displaystyle (\mathbf{A} + \mathbf{C})\mathbf{P} = \begin{pmatrix} 159.50 \\ 210.00 \end{pmatrix}

cups times dollars per cup gives dollars

Final answer

\begin{pmatrix} 159.50 \\ 210.00 \end{pmatrix}

: outlet 1 took 159.50 dollars and outlet 2 took 210.00 dollars.

Supports N9.1 · N9.2

Display information as a matrix; order and types

A matrix is a rectangular block of numbers arranged in rows (across) and columns (down) and written inside curved brackets. Each number is an element. Matrices store a whole table of data so you can work on all of it at once.

The Core Idea

The order of a matrix is rows × columns, in that order.

A matrix with 2 rows and 3 columns has order $2 \times 3$ (say "two by three"). A $3 \times 2$ matrix is a different shape.

Case table 1: types of matrix you must recognise
Type	What it looks like	Example	Order
Row matrix	one row only	$\begin{pmatrix} 4 & -1 & 7 \end{pmatrix}$	$1 \times 3$
Column matrix	one column only	$\begin{pmatrix} 2 \\ 9 \end{pmatrix}$	$2 \times 1$
Square matrix	as many rows as columns	$\begin{pmatrix} 3 & 0 \\ -2 & 5 \end{pmatrix}$	$2 \times 2$
Identity matrix $\mathbf{I}$	square, with 1 on the leading diagonal and 0 everywhere else	$\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$	$2 \times 2$
Null (zero) matrix $\mathbf{O}$	every element is 0	$\begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$	$2 \times 3$

SGSchoolKaki Tip

Rows run across like rows of seats in a cinema; columns run down like the columns of a building. Rows always come first: "rows × columns".

Think About It

Is a

2 \times 3

matrix the same as a

3 \times 2

matrix that holds the same six numbers?

Show the worked answerHide the worked answer

Colour keyround, estimate or compare

The first is 2 rows by 3 columns; the second is 3 rows by 2 columns.

compare the orders first

Equal matrices need the same order and the same element in every position.

the rule for equal matrices

Final answerNo. The orders differ, so the matrices are different even though they hold the same numbers.

SGSchoolKaki Tip

Do not write the columns first. A matrix with 3 rows and 2 columns is $3 \times 2$ , not $2 \times 3$ .

Example 9.1

State the order

State the order of each matrix.

(i) $\displaystyle A = {\small \begin{pmatrix} 1 & 4 & -2 \\ 0 & 3 & 5 \end{pmatrix}}$
(ii) $\displaystyle B = \begin{pmatrix} 7 \\ -1 \\ 2 \end{pmatrix}$
(iii) $\displaystyle C = {\small \begin{pmatrix} 4 & 0 & -3 & 8 \end{pmatrix}}$
(iv) $\displaystyle D = \begin{pmatrix} 2 & 5 \\ 1 & -6 \end{pmatrix}$

Show solutionHide solution

Colour keyexpand, rewrite, substitute or solve

(i)

Count the rows (they run across) and the columns (they run down).

rows first, then columns

\displaystyle A: \ 2 \text{ rows}, \ 3 \text{ columns}

\displaystyle \text{order} = 2 \times 3

write the order as rows × columns

(ii)

Count the rows (they run across) and the columns (they run down).

rows first, then columns

\displaystyle B: \ 3 \text{ rows}, \ 1 \text{ column}

\displaystyle \text{order} = 3 \times 1

write the order as rows × columns

(iii)

Count the rows (they run across) and the columns (they run down).

rows first, then columns

\displaystyle C: \ 1 \text{ row}, \ 4 \text{ columns}

\displaystyle \text{order} = 1 \times 4

write the order as rows × columns

(iv)

Count the rows (they run across) and the columns (they run down).

rows first, then columns

\displaystyle D: \ 2 \text{ rows}, \ 2 \text{ columns}

\displaystyle \text{order} = 2 \times 2

write the order as rows × columns

Final answer

(i)Order of

A

2 \times 3

(ii)Order of

B

3 \times 1

(iii)Order of

C

1 \times 4

(iv)Order of

D

2 \times 2

Your Turn 9.1

State the order of each matrix.

(i) $\displaystyle A = \begin{pmatrix} 3 & -1 \\ 2 & 0 \\ 5 & 4 \end{pmatrix}$
(ii) $\displaystyle B = {\small \begin{pmatrix} 9 & 2 & 1 \end{pmatrix}}$
(iii) $\displaystyle C = \begin{pmatrix} 6 \\ 1 \end{pmatrix}$
(iv) $\displaystyle D = {\small \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}}$

Show worked solutionHide worked solution

Colour keyexpand, rewrite, substitute or solve

(i)

Count the rows (they run across) and the columns (they run down).

rows first, then columns

\displaystyle A: \ 3 \text{ rows}, \ 2 \text{ columns}

\displaystyle \text{order} = 3 \times 2

write the order as rows × columns

(ii)

Count the rows (they run across) and the columns (they run down).

rows first, then columns

\displaystyle B: \ 1 \text{ row}, \ 3 \text{ columns}

\displaystyle \text{order} = 1 \times 3

write the order as rows × columns

(iii)

Count the rows (they run across) and the columns (they run down).

rows first, then columns

\displaystyle C: \ 2 \text{ rows}, \ 1 \text{ column}

\displaystyle \text{order} = 2 \times 1

write the order as rows × columns

(iv)

Count the rows (they run across) and the columns (they run down).

rows first, then columns

\displaystyle D: \ 3 \text{ rows}, \ 3 \text{ columns}

\displaystyle \text{order} = 3 \times 3

write the order as rows × columns

Final answer

(i)Order of

A

3 \times 2

(ii)Order of

B

1 \times 3

(iii)Order of

C

2 \times 1

(iv)Order of

D

3 \times 3

Example 9.2

Display data as a matrix

In a class test, Mei Ling scored 72 for Maths and 65 for Science; Raj scored 58 for Maths and 81 for Science; Hui Min scored 90 for Maths and 77 for Science. Write the marks as a matrix $\mathbf{M}$ with one row for each student (in the order given) and one column for each subject (Maths first).

(i)Write down the matrix $\mathbf{M}$ .
(ii)State the order of $\mathbf{M}$ .
(iii)What does the number in row 2, column 2 stand for?

Show solutionHide solution

(i)

One row per student, so the matrix has as many rows as there are students.

decide what the rows and columns mean first

\displaystyle \mathbf{M} = \begin{pmatrix} 72 & 65 \\ 58 & 81 \\ 90 & 77 \end{pmatrix}

keep the order of the students and the order of the subjects

(ii)

\displaystyle 3 \text{ rows}, \ 2 \text{ columns}

\displaystyle \text{order} = 3 \times 2

(iii)

Row 2 is Raj; column 2 is Science.

read the row label, then the column label

Final answer

(i)

\mathbf{M} = \begin{pmatrix} 72 & 65 \\ 58 & 81 \\ 90 & 77 \end{pmatrix}

(ii)

3 \times 2

(iii)Raj's mark for Science, which is

81

Your Turn 9.2

In a class test, Farid scored 68 for English and 74 for Chinese; Devi scored 81 for English and 59 for Chinese; Jun Kai scored 47 for English and 63 for Chinese; Nur scored 92 for English and 88 for Chinese. Write the marks as a matrix $\mathbf{M}$ with one row for each student (in the order given) and one column for each subject (English first).

(i)Write down the matrix $\mathbf{M}$ .
(ii)State the order of $\mathbf{M}$ .
(iii)What does the number in row 2, column 2 stand for?

Show worked solutionHide worked solution

(i)

One row per student, so the matrix has as many rows as there are students.

decide what the rows and columns mean first

\displaystyle \mathbf{M} = \begin{pmatrix} 68 & 74 \\ 81 & 59 \\ 47 & 63 \\ 92 & 88 \end{pmatrix}

keep the order of the students and the order of the subjects

(ii)

\displaystyle 4 \text{ rows}, \ 2 \text{ columns}

\displaystyle \text{order} = 4 \times 2

(iii)

Row 2 is Devi; column 2 is Chinese.

read the row label, then the column label

Final answer

(i)

\mathbf{M} = \begin{pmatrix} 68 & 74 \\ 81 & 59 \\ 47 & 63 \\ 92 & 88 \end{pmatrix}

(ii)

4 \times 2

(iii)Devi's mark for Chinese, which is

59

Example 9.3

Finding and locating an element

$\displaystyle \mathbf{P} = {\small \begin{pmatrix} 5 & -2 & 8 \\ 1 & 6 & 0 \\ -4 & 9 & 3 \end{pmatrix}}$

(i)Write down the element in row 2, column 3.
(ii)Write down the element in row 3, column 1.
(iii)In which row and column is the element $9$ ?

Show solutionHide solution

(i)

Go down to row 2, then across to column 3.

row first, column second

\displaystyle \text{element} = 0

(ii)

Go down to row 3, then across to column 1.

row first, column second

\displaystyle \text{element} = -4

(iii)

Look for

9

: it is in row 3 and column 2.

Final answer

(i)

0

(ii)

-4

(iii)Row 3, column 2.

Example 9.4

Which type of matrix?

Say whether each matrix is a row matrix, a column matrix, a square matrix, an identity matrix or a null matrix. Some matrices are more than one type.

(i) $\displaystyle A = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$
(ii) $\displaystyle B = {\small \begin{pmatrix} 0 & 0 & 0 \end{pmatrix}}$
(iii) $\displaystyle C = \begin{pmatrix} 5 \\ 2 \\ 0 \end{pmatrix}$
(iv) $\displaystyle D = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$

Show solutionHide solution

(i)

Check the order first, then look at the entries.

\displaystyle \text{order} = 2 \times 2

Every entry on the leading diagonal (top left to bottom right) is 1 and every other entry is 0.

this is what makes it an identity matrix

(ii)

Check the order first, then look at the entries.

\displaystyle \text{order} = 1 \times 3

Every entry is 0.

all zeros means a null (zero) matrix

(iii)

Check the order first, then look at the entries.

\displaystyle \text{order} = 3 \times 1

(iv)

Check the order first, then look at the entries.

\displaystyle \text{order} = 2 \times 2

Final answer

(i)

A

is a square matrix, an identity matrix.

(ii)

B

is a row matrix, a null matrix.

(iii)

C

is a column matrix.

(iv)

D

is a square matrix.

Your Turn 9.4

Say whether each matrix is a row matrix, a column matrix, a square matrix, an identity matrix or a null matrix. Some matrices are more than one type.

(i) $\displaystyle A = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$
(ii) $\displaystyle B = {\small \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}}$
(iii) $\displaystyle C = {\small \begin{pmatrix} 3 & 6 & 9 & 12 \end{pmatrix}}$
(iv) $\displaystyle D = \begin{pmatrix} 8 \\ 0 \end{pmatrix}$

Show worked solutionHide worked solution

(i)

Check the order first, then look at the entries.

\displaystyle \text{order} = 2 \times 2

Every entry is 0.

all zeros means a null (zero) matrix

(ii)

Check the order first, then look at the entries.

\displaystyle \text{order} = 3 \times 3

Every entry on the leading diagonal (top left to bottom right) is 1 and every other entry is 0.

this is what makes it an identity matrix

(iii)

Check the order first, then look at the entries.

\displaystyle \text{order} = 1 \times 4

(iv)

Check the order first, then look at the entries.

\displaystyle \text{order} = 2 \times 1

Final answer

(i)

A

is a square matrix, a null matrix.

(ii)

B

is a square matrix, an identity matrix.

(iii)

C

is a row matrix.

(iv)

D

is a column matrix.

Example 9.5

Equal matrices: finding unknown entries

$\displaystyle \begin{pmatrix} 2x + 1 & 3 \\ y - 3 & 5 \end{pmatrix} = \begin{pmatrix} 9 & 3 \\ 2 & 5 \end{pmatrix}$ Find the values of $x$ and $y$ .

Show solutionHide solution

Colour keyadd, subtract, multiply, divide or simplifyround, estimate or compare

\displaystyle \begin{pmatrix} 2x + 1 & 3 \\ y - 3 & 5 \end{pmatrix} = \begin{pmatrix} 9 & 3 \\ 2 & 5 \end{pmatrix}

equal matrices: same order, and every matching pair of entries is equal

Compare the matching entries one at a time.

compare entry by entry

\displaystyle 2x + 1 = 9

\displaystyle 2x = 9 - 1 = 8

move the number to the other side

\displaystyle x = 4

divide by the coefficient

\displaystyle y - 3 = 2

\displaystyle y = 2 + 3 = 5

move the number to the other side

Final answer

x = 4

and

y = 5

Your Turn 9.5

$\displaystyle \begin{pmatrix} 3x + 4 & 3 \\ 2y - 1 & 5 \end{pmatrix} = \begin{pmatrix} -2 & 3 \\ 5 & 5 \end{pmatrix}$ Find the values of $x$ and $y$ .

Show worked solutionHide worked solution

Colour keyadd, subtract, multiply, divide or simplifyround, estimate or compare

\displaystyle \begin{pmatrix} 3x + 4 & 3 \\ 2y - 1 & 5 \end{pmatrix} = \begin{pmatrix} -2 & 3 \\ 5 & 5 \end{pmatrix}

equal matrices: same order, and every matching pair of entries is equal

Compare the matching entries one at a time.

compare entry by entry

\displaystyle 3x + 4 = -2

\displaystyle 3x = -2 - 4 = -6

move the number to the other side

\displaystyle x = -2

divide by the coefficient

\displaystyle 2y - 1 = 5

\displaystyle 2y = 5 + 1 = 6

move the number to the other side

\displaystyle y = 3

divide by the coefficient

Final answer

x = -2

and

y = 3

Links To

Sets and matrices are treated together in the free overview: Sets & Matrices on the free E-Math revision page

Supports N9.2 · N9.1

Interpreting the data in a matrix

A matrix is only useful if you know what its rows and columns mean. Every matrix in a problem comes with labels, such as "rows are the shops, columns are the products". Once you know the labels, each element is a fact you can read off.

The Core Idea

The element in row i, column j answers: what does row i say about column j?

Say the row label first, then the column label, then the number. For example: "stall B, chicken rice: 24 portions" is one element.

SGSchoolKaki Tip

Adding along a row gives a total for that row's label. Adding down a column gives a total for that column's label.

Think About It

A matrix of sales has the shops as rows and the products as columns. If you added down a column, what would the total tell you?

Show the worked answerHide the worked answer

A column belongs to one product, and each entry in it is one shop's sales of that product.

read the labels

Final answerThe total number of that one product sold by all the shops together.

Example 9.6

Interpreting a data matrix

A canteen records the drinks sold on Monday and on Tuesday in the matrix $\displaystyle \mathbf{Q} = {\small \begin{pmatrix} 34 & 52 & 41 \\ 28 & 47 & 63 \end{pmatrix}}$ The rows are the days (Monday first) and the columns are the drinks in the order tea, coffee, Milo.

(i)State the order of $\mathbf{Q}$ .
(ii)What does the element in row 2, column 3 tell us?
(iii)Which drink sold the most on Monday?
(iv)How many drinks were sold altogether on Tuesday?

Show solutionHide solution

Colour keyadd, subtract, multiply, divide or simplify

(i)

\displaystyle 2 \times 3

two days (rows) and three drinks (columns)

(ii)

Row 2 is Tuesday and column 3 is Milo.

\displaystyle \mathbf{Q}_{23} = 63

(iii)

Compare the entries of row 1:

34

52

41

the largest entry in the row

\displaystyle \text{largest entry} = 52

(iv)

\displaystyle 28 + 47 + 63 = 138

add along row 2

Final answer

(i)

2 \times 3

(ii)On Tuesday,

63

cups of Milo were sold.

(iii)coffee (

52

cups).

(iv)

138

drinks.

Your Turn 9.6

A canteen records the drinks sold on Friday and on Saturday in the matrix $\displaystyle \mathbf{Q} = {\small \begin{pmatrix} 61 & 24 & 37 \\ 45 & 58 & 29 \end{pmatrix}}$ The rows are the days (Friday first) and the columns are the drinks in the order juice, soya milk, barley.

(i)State the order of $\mathbf{Q}$ .
(ii)What does the element in row 2, column 3 tell us?
(iii)Which drink sold the most on Friday?
(iv)How many drinks were sold altogether on Saturday?

Show worked solutionHide worked solution

Colour keyadd, subtract, multiply, divide or simplify

(i)

\displaystyle 2 \times 3

two days (rows) and three drinks (columns)

(ii)

Row 2 is Saturday and column 3 is barley.

\displaystyle \mathbf{Q}_{23} = 29

(iii)

Compare the entries of row 1:

61

24

37

the largest entry in the row

\displaystyle \text{largest entry} = 61

(iv)

\displaystyle 45 + 58 + 29 = 132

add along row 2

Final answer

(i)

2 \times 3

(ii)On Saturday,

29

cups of barley were sold.

(iii)juice (

61

cups).

(iv)

132

drinks.

Example 9.7

Totals from rows and columns

Three food stalls recorded how many portions of fried rice, laksa and wanton noodles they sold in one lunch hour: $\displaystyle \mathbf{S} = {\small \begin{pmatrix} 28 & 35 & 19 \\ 42 & 21 & 33 \\ 17 & 39 & 26 \end{pmatrix}}$ Rows are the stalls A, B, C in that order; columns are the items in the order given.

(i)How many portions did stall B sell altogether?
(ii)How many portions of wanton noodles were sold by the three stalls together?
(iii)Which stall sold the most portions, and how many?

Show solutionHide solution

Colour keyadd, subtract, multiply, divide or simplifyround, estimate or compare

(i)

\displaystyle 42 + 21 + 33 = 96

add across the row of that stall

(ii)

\displaystyle 19 + 33 + 26 = 78

add down the column of that item

(iii)

Row totals: A

82

, B

96

, C

82

compare the three row totals

Final answer

(i)

96

portions.

(ii)

78

portions.

(iii)Stall B, with

96

portions.

Your Turn 9.7

Three food stalls recorded how many portions of roti prata, nasi lemak and mee goreng they sold in one lunch hour: $\displaystyle \mathbf{S} = {\small \begin{pmatrix} 31 & 18 & 25 \\ 22 & 44 & 37 \\ 40 & 29 & 16 \end{pmatrix}}$ Rows are the stalls P, Q, R in that order; columns are the items in the order given.

(i)How many portions did stall Q sell altogether?
(ii)How many portions of mee goreng were sold by the three stalls together?
(iii)Which stall sold the most portions, and how many?

Show worked solutionHide worked solution

Colour keyadd, subtract, multiply, divide or simplifyround, estimate or compare

(i)

\displaystyle 22 + 44 + 37 = 103

add across the row of that stall

(ii)

\displaystyle 25 + 37 + 16 = 78

add down the column of that item

(iii)

Row totals: P

74

, Q

103

, R

85

compare the three row totals

Final answer

(i)

103

portions.

(ii)

78

portions.

(iii)Stall Q, with

103

portions.

Diagram for Example 9.8

Example 9.8

From a bar chart to a matrix

The bar chart shows the number of bottles of mineral water sold at a school kiosk on each day of a week.

(i)Write the sales as a column matrix $\mathbf{N}$ , one row per day in the order shown.
(ii)Say what the entry in row 3 of $\mathbf{N}$ means, and state the order of $\mathbf{N}$ .
(iii)Which day had the greatest sales, and what was the total for the week shown?

Show solutionHide solution

Colour keyadd, subtract, multiply, divide or simplify

(i)

Read each bar height and put the values in a column, in the same order as the days.

a column matrix because there is one value per day

\displaystyle \mathbf{N} = \begin{pmatrix} 35 \\ 28 \\ 42 \\ 31 \\ 50 \end{pmatrix}

(ii)

Row 3 is Wed: 42 bottles were sold on that day.

\displaystyle \text{order} = 5 \times 1

(iii)

\displaystyle \text{greatest} = 50

\displaystyle 35 + 28 + 42 + 31 + 50 = 186

add all the entries

Final answer

(i)

\mathbf{N} = \begin{pmatrix} 35 \\ 28 \\ 42 \\ 31 \\ 50 \end{pmatrix}

(ii)Row 3 is the 42 bottles sold on Wed; the order is

5 \times 1

(iii)Fri (

50

bottles); total

186

bottles.

Your Turn 9.8

The bar chart shows the number of bottles of iced tea sold at a school kiosk on each day of a week.

(i)Write the sales as a column matrix $\mathbf{N}$ , one row per day in the order shown.
(ii)Say what the entry in row 3 of $\mathbf{N}$ means, and state the order of $\mathbf{N}$ .
(iii)Which day had the greatest sales, and what was the total for the week shown?

Diagram for Your Turn 9.8

Show worked solutionHide worked solution

Colour keyadd, subtract, multiply, divide or simplify

(i)

Read each bar height and put the values in a column, in the same order as the days.

a column matrix because there is one value per day

\displaystyle \mathbf{N} = \begin{pmatrix} 24 \\ 38 \\ 30 \\ 46 \\ 33 \end{pmatrix}

(ii)

Row 3 is Wed: 30 bottles were sold on that day.

\displaystyle \text{order} = 5 \times 1

(iii)

\displaystyle \text{greatest} = 46

\displaystyle 24 + 38 + 30 + 46 + 33 = 171

add all the entries

Final answer

(i)

\mathbf{N} = \begin{pmatrix} 24 \\ 38 \\ 30 \\ 46 \\ 33 \end{pmatrix}

(ii)Row 3 is the 30 bottles sold on Wed; the order is

5 \times 1

(iii)Thu (

46

bottles); total

171

bottles.

Supports N9.3

Multiplying a matrix by a scalar

A scalar is an ordinary number. To multiply a matrix by a scalar $k$ , multiply every element by $k$ . The order of the matrix does not change.

The Core Idea

k × matrix: every element gets multiplied by k.

If $k = 2$ , each element is doubled. If $k = -1$ , each element changes sign. If $k = 0$ , you get the null matrix of the same order.

SGSchoolKaki Tip

Write each element's multiplication first (for example $3 \times (-4)$ ), then the answer. It stops sign mistakes.

SGSchoolKaki Tip

A scalar multiplies EVERY element, not just the first row or the first element.

Think About It

What happens to a matrix when you multiply it by

0

Show the worked answerHide the worked answer

Each element is multiplied by 0, and anything times 0 is 0.

apply the rule to every element

Final answerEvery element becomes zero, so you get the null matrix of the same order.

Example 9.9

Multiplying by a positive scalar

$\displaystyle \mathbf{A} = \begin{pmatrix} 2 & -1 \\ 0 & 4 \end{pmatrix}$ Find $3\mathbf{A}$ .

Show solutionHide solution

Colour keyadd, subtract, multiply, divide or simplify

Multiply every element of

\mathbf{A}

by 3.

a scalar multiplies EVERY element

\displaystyle 3\mathbf{A} = {\scriptsize \begin{pmatrix} 3 \times 2 & 3 \times (-1) \\ 3 \times 0 & 3 \times 4 \end{pmatrix}}

\displaystyle 3\mathbf{A} = \begin{pmatrix} 6 & -3 \\ 0 & 12 \end{pmatrix}

work out each product

Final answer

3\mathbf{A} = \begin{pmatrix} 6 & -3 \\ 0 & 12 \end{pmatrix}

Your Turn 9.9

$\displaystyle \mathbf{A} = {\small \begin{pmatrix} 1 & 3 & -2 \\ 4 & 0 & -6 \end{pmatrix}}$ Find $5\mathbf{A}$ .

Show worked solutionHide worked solution

Colour keyadd, subtract, multiply, divide or simplify

Multiply every element of

\mathbf{A}

by 5.

a scalar multiplies EVERY element

\displaystyle 5\mathbf{A} = {\scriptsize \begin{pmatrix} 5 \times 1 & 5 \times 3 & 5 \times (-2) \\ 5 \times 4 & 5 \times 0 & 5 \times (-6) \end{pmatrix}}

\displaystyle 5\mathbf{A} = {\small \begin{pmatrix} 5 & 15 & -10 \\ 20 & 0 & -30 \end{pmatrix}}

work out each product

Final answer

5\mathbf{A} = {\small \begin{pmatrix} 5 & 15 & -10 \\ 20 & 0 & -30 \end{pmatrix}}

Example 9.10

Negative and fractional scalars

$\displaystyle \begin{gathered} \mathbf{B} = \begin{pmatrix} 3 & -4 \\ 5 & 0 \end{pmatrix} \\ \mathbf{C} = \begin{pmatrix} 3 & -4 \\ 5 & 2 \end{pmatrix} \end{gathered}$ Find (i) $(-2)\mathbf{B}$ and (ii) $\frac{1}{2}\mathbf{C}$ .

(i) $(-2)\mathbf{B}$
(ii) $\frac{1}{2}\mathbf{C}$ (fractions are fine)

Show solutionHide solution

Colour keyexpand, rewrite, substitute or solve

(i)

A negative scalar changes the sign of every element.

negative × negative = positive

\displaystyle (-2)\mathbf{B} = {\scriptsize \begin{pmatrix} (-2) \times 3 & (-2) \times (-4) \\ (-2) \times 5 & (-2) \times 0 \end{pmatrix}}

\displaystyle (-2)\mathbf{B} = \begin{pmatrix} -6 & 8 \\ -10 & 0 \end{pmatrix}

(ii)

A fraction scalar multiplies each element in the same way.

write the answer as fractions

\displaystyle \frac{1}{2}\mathbf{C} = {\scriptsize \begin{pmatrix} \frac{1}{2} \times 3 & \frac{1}{2} \times (-4) \\ \frac{1}{2} \times 5 & \frac{1}{2} \times 2 \end{pmatrix}}

\displaystyle \frac{1}{2}\mathbf{C} = \begin{pmatrix} \frac{3}{2} & -2 \\ \frac{5}{2} & 1 \end{pmatrix}

Final answer

(i)

\begin{pmatrix} -6 & 8 \\ -10 & 0 \end{pmatrix}

(ii)

\begin{pmatrix} \frac{3}{2} & -2 \\ \frac{5}{2} & 1 \end{pmatrix}

Your Turn 9.10

$\displaystyle \begin{gathered} \mathbf{B} = {\small \begin{pmatrix} -1 & 6 & 2 \\ 4 & -5 & 0 \end{pmatrix}} \\ \mathbf{C} = \begin{pmatrix} 6 & -1 \\ 9 & 2 \end{pmatrix} \end{gathered}$ Find (i) $(-3)\mathbf{B}$ and (ii) $\frac{1}{3}\mathbf{C}$ .

(i) $(-3)\mathbf{B}$
(ii) $\frac{1}{3}\mathbf{C}$ (fractions are fine)

Show worked solutionHide worked solution

Colour keyexpand, rewrite, substitute or solve

(i)

A negative scalar changes the sign of every element.

negative × negative = positive

\displaystyle (-3)\mathbf{B} = {\scriptsize \begin{pmatrix} (-3) \times (-1) & (-3) \times 6 & (-3) \times 2 \\ (-3) \times 4 & (-3) \times (-5) & (-3) \times 0 \end{pmatrix}}

\displaystyle (-3)\mathbf{B} = {\small \begin{pmatrix} 3 & -18 & -6 \\ -12 & 15 & 0 \end{pmatrix}}

(ii)

A fraction scalar multiplies each element in the same way.

write the answer as fractions

\displaystyle \frac{1}{3}\mathbf{C} = {\scriptsize \begin{pmatrix} \frac{1}{3} \times 6 & \frac{1}{3} \times (-1) \\ \frac{1}{3} \times 9 & \frac{1}{3} \times 2 \end{pmatrix}}

\displaystyle \frac{1}{3}\mathbf{C} = \begin{pmatrix} 2 & - \frac{1}{3} \\ 3 & \frac{2}{3} \end{pmatrix}

Final answer

(i)

{\small \begin{pmatrix} 3 & -18 & -6 \\ -12 & 15 & 0 \end{pmatrix}}

(ii)

\begin{pmatrix} 2 & - \frac{1}{3} \\ 3 & \frac{2}{3} \end{pmatrix}

Example 9.11

Finding the scalar

$\displaystyle \begin{gathered} \mathbf{A} = \begin{pmatrix} 2 & -1 \\ 4 & 3 \end{pmatrix} \\ \mathbf{B} = \begin{pmatrix} 6 & -3 \\ 12 & q \end{pmatrix} \end{gathered}$ Given that $\mathbf{B} = k\mathbf{A}$ , find $k$ and the value of $q$ .

Show solutionHide solution

Compare the first elements:

2k = 6

pick an entry where both matrices are known

\displaystyle k = 6 \div 2 = 3

\displaystyle q = k \times 3 = 3 \times 3 = 9

use the same k for the unknown entry

Final answer

k = 3

q = 9

Example 9.12

A scalar in a real situation: a sale

A shop lists the prices (in dollars) of a notebook, a pen set and a bag as the matrix $\displaystyle \mathbf{P} = \begin{pmatrix} 4.50 & 3.20 & 6.80 \end{pmatrix}$ During a sale every price is reduced by $20\%$ . (This is invented shop data.) Write the sale prices as $k\mathbf{P}$ , state $k$ , and work out the new matrix. Give each price to the nearest cent.

Show solutionHide solution

Colour keyadd, subtract, multiply, divide or simplify

A reduction of

20\%

leaves

80\%

of the price.

100% minus the discount

\displaystyle k = 80 \div 100 = 0.8

\displaystyle k\mathbf{P} = 0.8 \times \begin{pmatrix} 4.50 & 3.20 & 6.80 \end{pmatrix} = \begin{pmatrix} 3.60 & 2.56 & 5.44 \end{pmatrix}

multiply every price by k

Final answer

k = 0.8

; sale prices (in dollars)

= \begin{pmatrix} 3.60 & 2.56 & 5.44 \end{pmatrix}

Your Turn 9.12

A shop lists the prices (in dollars) of a cap, a water bottle and a tote bag as the matrix $\displaystyle \mathbf{P} = \begin{pmatrix} 12.40 & 8.90 & 5.60 \end{pmatrix}$ During a sale every price is reduced by $30\%$ . (This is invented shop data.) Write the sale prices as $k\mathbf{P}$ , state $k$ , and work out the new matrix. Give each price to the nearest cent.

Show worked solutionHide worked solution

Colour keyadd, subtract, multiply, divide or simplify

A reduction of

30\%

leaves

70\%

of the price.

100% minus the discount

\displaystyle k = 70 \div 100 = 0.7

\displaystyle k\mathbf{P} = 0.7 \times \begin{pmatrix} 12.40 & 8.90 & 5.60 \end{pmatrix} = \begin{pmatrix} 8.68 & 6.23 & 3.92 \end{pmatrix}

multiply every price by k

Final answer

k = 0.7

; sale prices (in dollars)

= \begin{pmatrix} 8.68 & 6.23 & 3.92 \end{pmatrix}

Locked · Chapter 9

You've read the free 32%. Unlock the rest of Chapter 9.

Still to read in this chapter:

Sum and difference
Product of matrices
Matrix problems
Formula panel
Exercise

$9 one-time · all 19 chapters · no subscription

Worked examples, formulas and exam checks.
The whole E-Math syllabus.

ONE PRICE19chapters

Premium members get all 19 chapters included.

Notes version 1.0 · Last updated 3 Nov 2026 · Chapter 9 of 19