Sec 4 E-Math Matrices worksheet with answers

A free Sec 4 E-Math worksheet on matrices: 8 questions, 30 marks, short questions first and then structured ones, in the school exam style. The answers and full worked solutions are on the last page.

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Sec 4 E-Math worksheet: Matrices

Total: 30 marksSuggested time: 45 minutes

Name: ______________________________Class: ____________ Date: ____________

Answer all questions. Show all your working. Give non-exact answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless the question says otherwise.

Section A

A = \begin{pmatrix} -2 & 4 \\ -2 & 9 \end{pmatrix}

and

B = \begin{pmatrix} -3 & 7 \\ -3 & -5 \end{pmatrix}

. Find

A + B

Answer[2]

(a)The matrix

P = \begin{pmatrix} 7 & 6 & 9 \\ 7 & 7 & 13 \end{pmatrix}

shows the numbers of small, medium and large cups of juice sold at Stall A and Stall B (one row for each stall). The matrix

Q = \begin{pmatrix} 1.5 \\ 2 \\ 2.5 \end{pmatrix}

shows the prices, in dollars, of a small, medium and large cup. Using

PQ

, find the total amount, in dollars, collected at all the stalls.

Answer (a)[1]

(b)The matrix

P = \begin{pmatrix} 12 & 4 & 5 \\ 8 & 6 & 4 \end{pmatrix}

shows the numbers of wins, draws and losses for the Lions and the Tigers in a league (one row for each team). The matrix

Q = \begin{pmatrix} 3 \\ 1 \\ 0 \end{pmatrix}

shows the points given for a win, a draw and a loss. Using

PQ

, find the total number of points scored by all the teams.

Answer (b)[1]

A = \begin{pmatrix} -3 & 6 & 2 \end{pmatrix}

and

B = \begin{pmatrix} -1 & -1 \\ -1 & 6 \\ 5 & -3 \end{pmatrix}

. Find

AB

Answer[2]

(a)State the order of the matrix

\begin{pmatrix} 7 & 1 \end{pmatrix}

Answer (a)[2]

(b)

A = \begin{pmatrix} 12 & -4 \\ 4 & 10 \end{pmatrix}

. Write down the element in the first row and first column of

A

Answer (b)[1]

A = \begin{pmatrix} -2 & 5 \\ 2 & 1 \end{pmatrix}

and

B = \begin{pmatrix} 1 & -3 \\ -3 & 1 \end{pmatrix}

. Find

AB

Answer[3]

Section B

The matrix

P = \begin{pmatrix} 2 & 10 \\ 6 & 2 \end{pmatrix}

shows the numbers of loaves and buns sold by a bakery on Monday and Tuesday (one row for each day). The matrix

Q = \begin{pmatrix} 3.5 \\ 1.2 \end{pmatrix}

shows the prices, in dollars, of a loaf and a bun.

(a)Find

PQ

Answer (a)[2]

(b)State what the elements of

PQ

represent.

[1]

(c)Find the total amount, in dollars, the bakery collected over these days.

Answer (c)[1]

A bakery sells loaves, cakes and pies. Before a sale, a loaf costs $20, a cake costs $10 and a pie costs $9. These prices are

P = \begin{pmatrix} 20 \\ 10 \\ 9 \end{pmatrix}

. During the sale, the prices are reduced by 10%, 10% and 20% respectively. The table shows the numbers sold on two days of the sale; they are the matrix

Q = \begin{pmatrix} 23 & 28 & 21 \\ 23 & 28 & 27 \end{pmatrix}

	Loaves	Cakes	Pies
Saturday	23	28	21
Sunday	23	28	27

(a)Write down the

3 \times 3

diagonal matrix

D

such that

DP

gives the sale prices.

Answer (a)[1]

(b)Find

DP

Answer (b)[1]

(c)Using your answer to part (b), find

Q(DP)

Answer (c)[2]

(d)State what the elements of

Q(DP)

represent.

Answer (d)[1]

(e)On a third day of the sale, the shop sold

x

loaves,

6

cakes and

12

pies, and took $212.40. Using your answer to part (b), find

x

Answer (e)[2]

A bakery has two outlets, North and South. The matrix

S = \begin{pmatrix} 26 & 8 & 16 \\ 43 & 21 & 30 \end{pmatrix}

shows the numbers of muffins, croissants and tarts sold on Saturday, with the first row for North and the second row for South. The table shows the numbers sold on Sunday.

	Muffins	Croissants	Tarts
North	44	32	24
South	17	29	10

Sunday

(a)Write down a

2 \times 3

matrix

U

to represent the Sunday sales, in the same form as

S

Answer (a)[1]

(b)Find

W = S + U

Answer (b)[1]

(c)The bakery expects each of its sales next weekend to be 20% less than this weekend's. Using your answer to part (b), find the matrix of the expected sales next weekend.

Answer (c)[2]

(d)A muffin costs $1.50, a croissant costs $2.20 and a tart costs $3. Write down a

3 \times 1

matrix

P

for these prices, and find

WP

Answer (d)[2]

(e)Using your answer to part (d), find how much more money South collected than North from these sales over the weekend.

Answer (e)$[1]

END OF PAPER

Answers and worked solutions

Question 1

Answer:

A + B = \begin{pmatrix} -5 & 11 \\ -5 & 4 \end{pmatrix}

[2]

Column 1:

-2 + (-3) = -5

Column 2:

4 + 7 = 11

Column 1:

-2 + (-3) = -5

Column 2:

9 + (-5) = 4

A + B = \begin{pmatrix} -5 & 11 \\ -5 & 4 \end{pmatrix}

Question 2

(a)Answer:

\text{Total} = 102

[1]

PQ = \begin{pmatrix} 45 \\[8pt] 57 \end{pmatrix}

\text{Total} = 102

(b)Answer:

\text{Total} = 70

[1]

PQ = \begin{pmatrix} 40 \\[8pt] 30 \end{pmatrix}

\text{Total} = 70

Question 3

Answer:

AB = \begin{pmatrix} 7 & 33 \end{pmatrix}

[2]

(1 \times 3)(3 \times 2) \rightarrow 1 \times 2

Column 1:

(-3) \times (-1) + 6 \times (-1) + 2 \times 5 = 7

Column 2:

(-3) \times (-1) + 6 \times 6 + 2 \times (-3) = 33

AB = \begin{pmatrix} 7 & 33 \end{pmatrix}

Question 4

(a)Answer: Order

= 1 \mathbin{\times} 2

[2]

\text{Rows} = 1

\text{Columns} = 2

Order

= 1 \mathbin{\times} 2

(b)Answer:

\text{Element} = 12

[1]

\text{Row } 1: \; 12, \; -4

\text{Element} = 12

Question 5

Answer:

AB = \begin{pmatrix} -17 & 11 \\ -1 & -5 \end{pmatrix}

[3]

(2 \times 2)(2 \times 2) \rightarrow 2 \times 2

Column 1:

(-2) \times 1 + 5 \times (-3) = -17

Column 2:

(-2) \times (-3) + 5 \times 1 = 11

Column 1:

2 \times 1 + 1 \times (-3) = -1

Column 2:

2 \times (-3) + 1 \times 1 = -5

AB = \begin{pmatrix} -17 & 11 \\ -1 & -5 \end{pmatrix}

Question 6

(a)Answer:

\begin{pmatrix} 19 \\ 23.4 \end{pmatrix}

[2]

\text{Row } 1: 2 \times 3.5 + 10 \times 1.2 = 19

\text{Row } 2: 6 \times 3.5 + 2 \times 1.2 = 23.4

PQ = \begin{pmatrix} 19 \\ 23.4 \end{pmatrix}

(b)[1]

The top element is the total amount, in dollars, the bakery collected on Monday.The bottom element is the total amount, in dollars, the bakery collected on Tuesday.

(c)Answer:

42.4

[1]

19 + 23.4 = 42.4

Question 7

(a)Answer:

D = \begin{pmatrix} 0.9 & 0 & 0 \\ 0 & 0.9 & 0 \\ 0 & 0 & 0.8 \end{pmatrix}

[1]

A price reduced by

10\%

0.9

of the price, and so on.

D = \begin{pmatrix} 0.9 & 0 & 0 \\ 0 & 0.9 & 0 \\ 0 & 0 & 0.8 \end{pmatrix}

(b)Answer:

\begin{pmatrix} 18 \\ 9 \\ 7.2 \end{pmatrix}

[1]

DP = \begin{pmatrix} 0.9 & 0 & 0 \\ 0 & 0.9 & 0 \\ 0 & 0 & 0.8 \end{pmatrix}\begin{pmatrix} 20 \\ 10 \\ 9 \end{pmatrix}

= \begin{pmatrix} 18 \\ 9 \\ 7.2 \end{pmatrix}

(c)Answer:

\begin{pmatrix} 817.2 \\ 860.4 \end{pmatrix}

[2]

\text{Row } 1: 23 \times 18 + 28 \times 9 + 21 \times 7.2 = 817.2

\text{Row } 2: 23 \times 18 + 28 \times 9 + 27 \times 7.2 = 860.4

Q(DP) = \begin{pmatrix} 817.2 \\ 860.4 \end{pmatrix}

(d)Answer: The total amount of money taken on Saturday and on Sunday, at the sale prices.[1]

Each row of

Q

is a day's numbers sold;

DP

holds the sale prices, so each element is a day's takings after the discounts.

(e)Answer:

x = 4

[2]

\begin{pmatrix} x & 6 & 12 \end{pmatrix}\begin{pmatrix} 18 \\ 9 \\ 7.2 \end{pmatrix} = 212.4

18x + 54 + 86.4 = 212.4

18x = 72

x = 4

Question 8

(a)Answer:

U = \begin{pmatrix} 44 & 32 & 24 \\ 17 & 29 & 10 \end{pmatrix}

[1]

U = \begin{pmatrix} 44 & 32 & 24 \\ 17 & 29 & 10 \end{pmatrix}

(b)Answer:

\begin{pmatrix} 70 & 40 & 40 \\ 60 & 50 & 40 \end{pmatrix}

[1]

W = \begin{pmatrix} 26 & 8 & 16 \\ 43 & 21 & 30 \end{pmatrix} + \begin{pmatrix} 44 & 32 & 24 \\ 17 & 29 & 10 \end{pmatrix}

= \begin{pmatrix} 70 & 40 & 40 \\ 60 & 50 & 40 \end{pmatrix}

(c)Answer:

\begin{pmatrix} 56 & 32 & 32 \\ 48 & 40 & 32 \end{pmatrix}

[2]

100\% - 20\% = 80\%

0.8W = 0.8\begin{pmatrix} 70 & 40 & 40 \\ 60 & 50 & 40 \end{pmatrix}

= \begin{pmatrix} 56 & 32 & 32 \\ 48 & 40 & 32 \end{pmatrix}

(d)Answer:

\begin{pmatrix} 313 \\ 320 \end{pmatrix}

[2]

P = \begin{pmatrix} 1.5 \\ 2.2 \\ 3 \end{pmatrix}

\text{Row } 1: 70 \times 1.5 + 40 \times 2.2 + 40 \times 3 = 313

\text{Row } 2: 60 \times 1.5 + 50 \times 2.2 + 40 \times 3 = 320

WP = \begin{pmatrix} 313 \\ 320 \end{pmatrix}

(e)Answer: $7[1]

The elements of

WP

are the amounts, in dollars, collected at North and at South.

320 - 313 = 7

Practice questions by skill

Sec 4 E-Math Matrices worksheet with answers

A free Sec 4 E-Math worksheet on matrices: 8 questions, 30 marks, short questions first and then structured ones, in the school exam style. The answers and full worked solutions are on the last page.

In the print window, choose Save as PDF to download it.

Need the formulas first? Read the Matrices notes. Or practise it online with step-by-step help when you get one wrong.

Section B

The matrix

P = \begin{pmatrix} 2 & 10 \\ 6 & 2 \end{pmatrix}

shows the numbers of loaves and buns sold by a bakery on Monday and Tuesday (one row for each day). The matrix

Q = \begin{pmatrix} 3.5 \\ 1.2 \end{pmatrix}

shows the prices, in dollars, of a loaf and a bun.

(a)Find

PQ

Answer (a)[2]

(b)State what the elements of

PQ

represent.

[1]

(c)Find the total amount, in dollars, the bakery collected over these days.

Answer (c)[1]

Loaves

Cakes

Pies

Saturday

Sunday

Muffins

Croissants

Tarts

North

South

Answers and worked solutions

Question 1

Answer:

A + B = \begin{pmatrix} -5 & 11 \\ -5 & 4 \end{pmatrix}

[2]

Column 1:

-2 + (-3) = -5

Column 2:

4 + 7 = 11

Column 1:

-2 + (-3) = -5

Column 2:

9 + (-5) = 4

A + B = \begin{pmatrix} -5 & 11 \\ -5 & 4 \end{pmatrix}

Question 2

(a)Answer:

\text{Total} = 102

[1]

PQ = \begin{pmatrix} 45 \\[8pt] 57 \end{pmatrix}

\text{Total} = 102

(b)Answer:

\text{Total} = 70

[1]

PQ = \begin{pmatrix} 40 \\[8pt] 30 \end{pmatrix}

\text{Total} = 70

Question 3

Answer:

AB = \begin{pmatrix} 7 & 33 \end{pmatrix}

[2]

(1 \times 3)(3 \times 2) \rightarrow 1 \times 2

Column 1:

(-3) \times (-1) + 6 \times (-1) + 2 \times 5 = 7

Column 2:

(-3) \times (-1) + 6 \times 6 + 2 \times (-3) = 33

AB = \begin{pmatrix} 7 & 33 \end{pmatrix}

Question 4

(a)Answer: Order

= 1 \mathbin{\times} 2

[2]

\text{Rows} = 1

\text{Columns} = 2

Order

= 1 \mathbin{\times} 2

(b)Answer:

\text{Element} = 12

[1]

\text{Row } 1: \; 12, \; -4

\text{Element} = 12

Question 5

Answer:

AB = \begin{pmatrix} -17 & 11 \\ -1 & -5 \end{pmatrix}

[3]

(2 \times 2)(2 \times 2) \rightarrow 2 \times 2

Column 1:

(-2) \times 1 + 5 \times (-3) = -17

Column 2:

(-2) \times (-3) + 5 \times 1 = 11

Column 1:

2 \times 1 + 1 \times (-3) = -1

Column 2:

2 \times (-3) + 1 \times 1 = -5

AB = \begin{pmatrix} -17 & 11 \\ -1 & -5 \end{pmatrix}

Question 6

(a)Answer:

\begin{pmatrix} 19 \\ 23.4 \end{pmatrix}

[2]

\text{Row } 1: 2 \times 3.5 + 10 \times 1.2 = 19

\text{Row } 2: 6 \times 3.5 + 2 \times 1.2 = 23.4

PQ = \begin{pmatrix} 19 \\ 23.4 \end{pmatrix}

(b)[1]

The top element is the total amount, in dollars, the bakery collected on Monday.The bottom element is the total amount, in dollars, the bakery collected on Tuesday.

(c)Answer:

42.4

[1]

19 + 23.4 = 42.4

Question 7

(a)Answer:

D = \begin{pmatrix} 0.9 & 0 & 0 \\ 0 & 0.9 & 0 \\ 0 & 0 & 0.8 \end{pmatrix}

[1]

A price reduced by

10\%

0.9

of the price, and so on.

D = \begin{pmatrix} 0.9 & 0 & 0 \\ 0 & 0.9 & 0 \\ 0 & 0 & 0.8 \end{pmatrix}

(b)Answer:

\begin{pmatrix} 18 \\ 9 \\ 7.2 \end{pmatrix}

[1]

DP = \begin{pmatrix} 0.9 & 0 & 0 \\ 0 & 0.9 & 0 \\ 0 & 0 & 0.8 \end{pmatrix}\begin{pmatrix} 20 \\ 10 \\ 9 \end{pmatrix}

= \begin{pmatrix} 18 \\ 9 \\ 7.2 \end{pmatrix}

(c)Answer:

\begin{pmatrix} 817.2 \\ 860.4 \end{pmatrix}

[2]

\text{Row } 1: 23 \times 18 + 28 \times 9 + 21 \times 7.2 = 817.2

\text{Row } 2: 23 \times 18 + 28 \times 9 + 27 \times 7.2 = 860.4

Q(DP) = \begin{pmatrix} 817.2 \\ 860.4 \end{pmatrix}

(d)Answer: The total amount of money taken on Saturday and on Sunday, at the sale prices.[1]

Each row of

Q

is a day's numbers sold;

DP

holds the sale prices, so each element is a day's takings after the discounts.

(e)Answer:

x = 4

[2]

\begin{pmatrix} x & 6 & 12 \end{pmatrix}\begin{pmatrix} 18 \\ 9 \\ 7.2 \end{pmatrix} = 212.4

18x + 54 + 86.4 = 212.4

18x = 72

x = 4

Question 8

(a)Answer:

U = \begin{pmatrix} 44 & 32 & 24 \\ 17 & 29 & 10 \end{pmatrix}

[1]

U = \begin{pmatrix} 44 & 32 & 24 \\ 17 & 29 & 10 \end{pmatrix}

(b)Answer:

\begin{pmatrix} 70 & 40 & 40 \\ 60 & 50 & 40 \end{pmatrix}

[1]

W = \begin{pmatrix} 26 & 8 & 16 \\ 43 & 21 & 30 \end{pmatrix} + \begin{pmatrix} 44 & 32 & 24 \\ 17 & 29 & 10 \end{pmatrix}

= \begin{pmatrix} 70 & 40 & 40 \\ 60 & 50 & 40 \end{pmatrix}

(c)Answer:

\begin{pmatrix} 56 & 32 & 32 \\ 48 & 40 & 32 \end{pmatrix}

[2]

100\% - 20\% = 80\%

0.8W = 0.8\begin{pmatrix} 70 & 40 & 40 \\ 60 & 50 & 40 \end{pmatrix}

= \begin{pmatrix} 56 & 32 & 32 \\ 48 & 40 & 32 \end{pmatrix}

(d)Answer:

\begin{pmatrix} 313 \\ 320 \end{pmatrix}

[2]

P = \begin{pmatrix} 1.5 \\ 2.2 \\ 3 \end{pmatrix}

\text{Row } 1: 70 \times 1.5 + 40 \times 2.2 + 40 \times 3 = 313

\text{Row } 2: 60 \times 1.5 + 50 \times 2.2 + 40 \times 3 = 320

WP = \begin{pmatrix} 313 \\ 320 \end{pmatrix}

(e)Answer: $7[1]

The elements of

WP

are the amounts, in dollars, collected at North and at South.

320 - 313 = 7