Hwa Chong Institution
Unit 5.1 Matrices - Matrix Representation
Name: ________________________ (
)
Class: _________
Date: ________
Matrix Multiplication and Word Problems
Objectives:
Worksheet 3
1.
Solve word problems using matrices
2.
Perform matrix multiplication on small order matrices
I
Matrix Multiplication
Visit the following website http://www.math.ncsu.edu/ma107/114/_c1d2a.mov to view the video on matrix
multiplication.
In summary,
1. Two matrices, A and B can be multiplied together iff the number of __________ in A is equal to the
number of __________ in B.
2. The multiplication of two matrices, A and B, will give rise to a matrix AB with the number of
__________ of A and the number of __________of B, i.e.
Amn  Bn p  ABm p .
3. The product of two matrices, A and B, will give rise to a matrix AB, whose element in the ith row and jth
column is the sum of the products formed by multiplying each element in the ith row of A by the
corresponding element in the jth column of B.
Practice
1. Compute the products AB and BA for the following, if possible:
(i)
1 2
 1 4 
A
, B  

3 1
 2 0.5 
 2 3
 2 0.5 1
, B  

 1 2
1 3 4 
(ii) A  
1 1
1 2 2 2


(iii) A  
, B   2 0
0 4 0 3
 3 4


II
Laws of Matrix Multiplication
1.
Given that A  
 3 1
 2 5
1 0
, B  
 and C  
 , find (i) A( BC ) ,
4 0 
 1 3
 2 3
Note:
2.
(ii) ( AB)C .
A( BC )  ( AB)C , this question illustrates that matrix multiplication is associative.
 2 1
 2 4
1 0
, B  
 and C  
 , find
1 0 
 2 3
 0 3
Given that A  
(i)
Note:
A( B  C )
(ii) AB  AC
(iii) ( B  C ) A
(iv) BA  CA
A( B  C )  AB  AC and ( B  C ) A  BA  CA , this illustrates that matrix multiplication is
distributive over addition.
SBGE/MA/SEC1/BEN/MAT/WS3
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3.
 2 1
 4 1
 , find
 and C  
1 3 
0 0 
Given that A  
AB
(ii) BA
AB  BA , this question illustrates that matrix multiplication is not commutative.
(i)
Note:
Summary:
1. Matrix multiplication is associative.
2. Matrix multiplication is distributive over addition and subtraction.
3. Matrix multiplication is not commutative.
III Word Problems
1. An ice-cream stall sells both green tea and mocha ice cream. A small portion of either costs $0.75 and a
large portion costs $1.25. During a short period of time, the number of ice creams sold is own in the table
below.
small
large
Green Tea
3
4
Mocha
6
3
(i)
Write down a column matrix N, representing the cost of each portion of ice cream.
 3 4
 , evaluate MN.
6 3
(ii) Given that M  
(iii) Explain what the numbers given in your answer in (ii) signify.
Answers:
(i)
 0.75 
N 

 1.25 
 3 4  0.75 


 6 3  1.25 
(ii) MN  
 7.25 


 8.25 
(iii) $7.25 is the amount received from the sales of green tea ice cream while $8.25 is the amount received from
the sale of mocha ice cream.
As we can see from the example above, matrices can be used in solving word problems.
SBGE/MA/SEC1/BEN/MAT/WS3
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Practice
The matrix below shows the results of John and Paul in a multiple choice test.
Correct
No attempt
 18
 20
7
John
0
Paul
P= 
Incorrect
The method to award marks is given by:
Marks
 2  Correct
 
Q =  0  No attempt
 
 1 Incorrect
(i) Find the matrix product PQ.
(ii) Explain clearly what your answer to (a) represents.
Answers
Part I 1. columns, rows
2. rows, columns
5 
3
 11 2 
 ; BA  

 1 12.5 
 3.5 4.5 
 7 10 10 
 ; BA( NA)
 4 6.5 7 
Practice 1(i) AB  
1(ii) AB  
1 6 2 5 


1(iii) AB( NA); BA  2 4 4 4


 3 22 6 18 


 5 12 

 8 20 
Part II 1. A( BC )  ( AB)C  
 4 2

 3 4
2. A( B  C )  AB  AC  
10 3 
( B  C ) A  BA  CA  

10 2 
 7 5 
 8 4 
 ; BA  

0 0 
 2 1 
3. AB  
Part III
 31 
 1(ii) John’s score is 31 and Paul’s score is 30.
 30 
Practice 1(i) PQ  
SBGE/MA/SEC1/BEN/MAT/WS3
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Matrices Assignment 3
Do all work on Microsoft Word.
Save your work as MatA3_ClassIndex (e.g. MatA3_3M01)
Send your completed work to (email address) by (date, time)
1.
Evaluate the following:
2
 
(i)  2 3 1 1
 
4
 
2.
3.
 2 3
(ii)  2 3 

1 2
2 3
 2 3 0

(iii) 
  0 2 
 1 1 3   1 3 


c
 2 1 0     9 
Given that 
  1     , find all possible values of c and d.
 0 2 d   d  11
 
The charges for hiring a car from three different companies, based on the number of days for which a car
is hired and/or the number of kilometres for which the car is driven are as follows:
Company A charges $66 per day.
Company B charges 48 cents on per kilometre driven.
Company C charges $30 per day and 25 cents on per kilometre driven.
John needs to hire a car for 4 days to drive 560 kilometres.
(i)
(ii)
4.
Write down two matrices only such that the elements of their product under matrix multiplication
give the charge of hiring a car from the three different companies.
Find this product. Hence state the company that John should hire the car from in order to save cost.
A business makes toy buses and toy lorries. The following table is used in calculating the cost of
manufacturing each toy.
Bus
Lorry
Labour (Hours)
6
3
Wood (Blocks)
4
4
Paint (Tins)
3
2
Labour costs $8 per hour, wood costs $1 per block and paint costs $2 per tin.
8
6 4 3
 
It is given that A  
 , B   1  and C  AB .
3
4
2


 2
 
(a) (i) Evaluate C.
(ii) Explain what the numbers in your answer represent.
(b) In addition, D  100 200  .
(i) Evaluate DC.
(ii) Explain what your answer represents.
(O Level/J98/I/16)
SBGE/MA/SEC1/BEN/MAT/WS3
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