Topic 4: Matrices Test A
Name: ___________________________
Multiple choice
Consider the following matrix sum
1
5 2   2  1 7 b 
1 a     1  3   0  1 .

 
 

The values of a and b respectively are:
A a = −4, b = −1
B a = −4, b = 1
C a = −2, b = 1
D a = 2, b = −1
E a = 2, b = 1
2
3
1 2 4 0 
The order of the matrix 0 7 1 1 


5 3 6 2 
is:
A34
B43
C2×4
D7
E 12
1
 5
Consider the matrix B  
9

 2
The element b21 is:
A −5
B2
C4
D7
E9
4
2 
.
0

7
The following information relates to
Questions 4 and 5.
The following matrix shows the three
different ticket prices (in dollars) for two
rides, Terror and Stopper, at a major theme
park.
T erro r S t o p p er
1 5 . 0 0 1 7 . 5 0 
C o n ces s i o n  8 . 5 0 1 2 . 0 0 
C h i ld ren  7 . 5 0 1 0 . 5 0 
Ad u lt s
4
The cost of a concession ticket on the
Terror ride is:
A $7.50
B $8.50
C $10.50
D $12.00
E $15.00
5
A family buys one or more of each
type of ticket to ride the Stopper. They
pay $50.50. The number of each type
of ticket the family purchased is:
A One adult, one concession and one
child
B One adult, two concessions and one
child
C One adult, one concession and two
children
D Two adults, one concession and
one child
E Two adults, two concessions and
two children
© John Wiley & Sons Australia, Ltd
Page 1 of 9
6
The table below shows the number of
spectators (in thousands) attending the
top five sports in Australian for
between 2005–6 and 2009–10.
Sport
AFL
Horse racing
Rugby league
Motor sports
Soccer
2005–6
2526.7
2003.7
1486.4
1485.2
560.7
2003.7 1486.4 1485.2 560.7
C
 2 5 2 6 .7
 2 0 0 3 .7

1 4 8 6 .4

1 4 8 5 .2
 5 6 0 .7
2 8 3 1 .8
1 9 4 0 .3 
1 5 6 3 .8 

1 4 2 3 .0 
9 3 8 .8 
The product matrix RTS has an order
of 2  4 . Matrix R has an order of
2  m , Matrix S has order of n  p and
matrix T has order of 3  2 . The values
of m, n and p respectively are:
A 2, 3 and 4
B 2, 4 and 3
C 3, 2 and 4
D 3, 4 and 2
E 4, 3 and 2
2009–10
2831.8
1940.3
1563.8
1423.0
938.8
A 1 × 5 matrix that could be used to
represent the attendance (‘000s) in
2009–10 is:
A
 2526.7 
 2003.7 


1486.4 


1485.2 
 560.7 
B
2526.7
7
8
Which of the following matrices do
not have inverses?
1  1
I 

3 3 
 2  2
II 

 1 1 
6
3
III 

  4  8
 4  3
IV 

12 9 
A I and II
B II and III
C I, II and III
D I, II and IV
E I, II, III and IV
D
2831.8 1940.3 1563.8 1423.0 938.8
E
 2 8 3 1 . 8
1 9 4 0 . 3 


1 5 6 3 . 8 


1 4 2 3 . 0 
 9 3 8 . 8 
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Page 2 of 9
9
Which one of the following will find
the solution of the matrix equation
shown?
2 3   x   7 
 3  6  y     2

   
 x   7    6  3
A     

 y    2   4 2 
1 2 3   7 
 x
B  

 
 y   24  4  6   2
2 3   x   7 
C 
    
 4  6  y    2
1   6  3  7 
 x
D  

 
 y   24   4 2    2
1  7    6  3
 x
E  
 

 y   24   2   4 2 
10
The diagram shows the number of
roads connecting five towns: A, B, C,
D and E.
A
A 1

B 1
B 
C 1

D 2
E
0
A
A 0

B 1
C 
C 1

D 2
E
0
A
A 0

B 1
D 
C 1

D 2
E
0
A
A 0
Which one of the following adjacency
matrices represents the number of
roads connecting the five towns?
A
A 1

B 2
A 
C 1

D 1
E
0
B
C
D

B 1
E 
C 1

D 2
E
0
B
C
D
E
1 1 2 0
1 1 0 1 
1 1 1 0

0 1 1 0
1 0 0 1 
B
C
D
E
1 1 2 0
0 1 0 0 
1 0 1 0

0 1 0 1
0 0 1 0 
B
C
D
E
1 1 2 0
0 1 0 0 
1 0 1 1

0 1 0 0
0 1 0 0 
B
C
D
E
1 1 2 0
0 1 0 1 
1 0 1 0

0 1 0 0
1 0 0 0 
E
2 1 1 0
1 1 1 1 
1 1 1 1

1 1 1 1
1 0 1 1 
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Page 3 of 9
Short answer
1
Evaluate the following.
 5    1  7 
(a)        
  2    3   4 
6 1  0  2 
(b) 2 
  3

5  4   3 5 
1
2
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Page 4 of 9
2
(a) By stating the order of each matrix,
determine if the following product
matrices exist.
3
(i)   2 6  
2
4
 3 2
(ii)  2  
 1 5 

 1 
1
8  1  3 
(iii) 
 
3 2   4 
11 3   1 6 
(iv) 


 2 5  4 7 
(b) Find the product matrices for those that
exist.
1
1
1
2
2
2
3
Find the determinant of each matrix and
hence state whether its inverse matrix exists.
5 3
(a) 

4 2
  2  1
(b) 
3 
6
© John Wiley & Sons Australia, Ltd
1
1
Page 5 of 9
4
5
Solve the following matrix equations by first
finding the inverse of the 2 × 2 matrix.
3 1   x   9 
(a) 
    
1 2  y  8
 4  2  x   1 
(b) 
   

 3 4   y  14.5
4
4
4
Evaluate the following.
1 4 
(a) 

0 3
3
1 0 
(b) 

0 1 
6
2
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Page 6 of 9
Extended response
1
Five hundred people attended the premier of the
blockbuster Galaxy Wars. Two types of tickets
were available for the premier viewing: gold
class and platinum class. Gold class tickets cost
$45.95 and platinum class tickets cost $52.75.
The total ticket sales for the premier were
$23879.40. Two simultaneous equations were
constructed to represent this information, where
g and c represent the number of each type of
ticket sold.
2
2
2
(a) Complete the simultaneous equations by
finding the values that are missing.
(b) Construct a matrix equation to represent the
simultaneous equations.
(c) By referring to the value of the determinant,
explain why the pair of simultaneous
equations will have a solution.
(d) Using the inverse matrix, solve the matrix
equation and hence state the number of each
type of ticket sold for the movie premier.
© John Wiley & Sons Australia, Ltd
4
Page 7 of 9
2
Three major supermarkets — Alcosts, Coolies
and Waldos — stock the same brand of a
leading cereal, SuperFlakes. The price in
dollars of a 500-g packet of Superflakes at each
supermarket is shown in matrix P below.
 2 . 9 5  Alco s t s
P   4 . 6 0  C o o li es
 4 . 5 0  W ald o s
1
1
1
3
(a) How much in dollars would you pay for
two 500-g packets of Superflakes at
Coolies?
The table shows the number of 500-g packets
of Superflakes sold in the first week of April at
each supermarket.
Number of 500g packets of
Supermarket
Superflakes sold
Alcosts
357
Coolies
432
Waldos
495
2
A row matrix, N, is to be constructed to
represent the number of 500-g packets of
Superflakes sold in the first week of April.
(a) (i) Construct the row matrix, N, to
represent this information.
(ii) State the order of N.
2
2
A matrix multiplication will be performed to
determine the total sales amount, in dollars, for
the first week of April.
(c) By referring to the order of matrices N and
P, explain why the product matrix is NP
and not PN.
(d) By performing the matrix multiplication
determine the total sales amount, in dollars,
for Superflakes in the first week of April.
(continued)
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Page 8 of 9
Due to an increase in the price of wheat, the
selling price of each 500-g packet of
Superflakes is increased by 5%. A new price
 2.95
matrix, k  4.60  , is constructed to reflect this


 4.50 
increase as shown.
(e) Write the value of k.
Increased fuel costs adds additional costs to
each packet of Superflakes as shown in the
matrix sum:
 2 . 9 5  0 . 6 5
k  4 . 6 0    0 . 3 5 
 4 . 5 0   0 . 4 0 
(f) Using the value of k from part (e) and the
additional transportation costs from the
matrix sum, construct a new price matrix
for each packet of Superflakes at the three
different supermarkets. Give your answers
correct to the nearest cent.
© John Wiley & Sons Australia, Ltd
Page 9 of 9