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Sec 3 Integrated Mathematics : Matrices
Name : ______________________ (
) Class (
)
Date : ___________
Definition
A matrix is a rectangular array of numbers and the numbers are known as the elements of the matrix..
Matrices are convenient ways of summarising numerical data.
 2 3 
e.g. matrix A = 
 then 2,  3, 8 and 1 are called the elements of matrix A.
8 1 
Example 1
A company produced 4 types of central heating radiator, known as types A, B, C and D.A builder buys
radiators for all the houses on a new estate. There are 20 small houses, 30 medium-sized houses and 15 large
houses. A small house needs 3 radiators of type A, 2 of type B and 2 of type C.
A medium-sized house needs 2 radiators of type A, 3 of type C and 3 of type D.
A large house needs 1 radiators of type B, 6 of type C and 3 of type D.
GCE “O” N02
 3 2 2 0
We can represent the above information in the form of a matrix such as  2 0 3 3  .
 0 1 6 3


(A) Order of a Matrix
Before we introduce the 3 operations (addition, subtraction & multiplication of matrices) you must
understand the concept of “the order of a matrix”.
The order of a matrix is defined as = number of rows  number of columns.
The order of each of the following matrices are as follow.
6
 
2
8
 
31
1 2


3 4
1 0 2


3 4 0
22
23
5
0 3
13
8
11
(B) Addition and Subtraction of Matrices
Only matrices of the same order can be added or subtracted.
We simply add or minus the corresponding elements. The following illustrate the addition & subtraction of
matrices of order 2  2.
 a b   1 2   a 1 b  2 




 c d  3 4  c  3 d  4
 a b   1 2   a 1 b  2 




 c d  3 4  c  3 d  4
(C) Scalar Multiplication of Matrices (Multiplication of a scalar and a matrix)
 a b   8a 8b 
8


 c d   8c 8d 
Notice that in addition, subtraction and scalar multiplication, the order of the matrices is preserved.
Example 2
Express the following as a single matrix,
 2  1 
   
(a) 2  5   3  0  ,
(b) 2 1 3  5  3 0 ,
 0   2 
   
 0 5  1  4 2 
(c) 3 
 
.
 1 2  2  6 10 
1
 2 14 
 
ANSWERS (a ) 10  (b)  13 6  (c) 

 6 11 
6
 
(D) Multiplication of Two Matrices
To multiply two matrices, the orders of the two matrices NEED NOT be the same. However …
given that the orders of two matrices A and B are p  q and r  s respectively, the product of these two matrices is
defined only when q  r and the order of the resultant product will be p  s .
E.g.
 5 matrix will result in a 1  5 matrix.
4 matrix multiply by a 4  3 matrix will result in a 2  3 matrix.
2 matrix multiply by a 2  6 matrix will result in a 4  6 matrix.
k matrix multiply by a k  p matrix will result in a h  p matrix
The product of a 3  2 matrix and a 3  4 matrix is undefined.
d 
 2
 
e.g. 1 2 3  4    28 
 a b c   e    ad  be  cf 
f
6
 
 

2 
4 
h 
1
3 matrix multiply by a 3
a
 ad ae af 
 


 b   d e f    bd be bf 
c
 cd ce cf 
 


 a b  e f   ae  bg af  bh 




 c d  g h   ce  dg cf  dh 
1
2 4 6 


e.g.  2   2 4 6    4 8 12 
 3
 6 12 18 
 


 1 2  2 4   14 20 
e.g. 



 3 4  6 8   30 44 
Example 3
State which of the following products are not defined and evaluate the others.
1
1
 1 2  0 1 
 
 
(a) 
(b)  2   5 0 1 ,
(c)  2 0 4   3  ,

,
 3 0  3 4 
 3
0
 
 
(d) 1 2 3 5 0 1 ,
1 0

 2
(e)  0 2    ,
 3 1 4


0 2


(f) (1 3 2)  3 1  ,
5 0


 1 2  3 
(g) 
  ,
 3 1  1 
1
 3 2 1  
(h) 
 2
1 0 0  
0
1 2 2 3



(i)  3 4   5 0 
5 61 7



2
 5 0 1
 1 
 7  (i ) undefined
6 9
 


(
d
)
undefined
(
e
)
ANS : (a ) 
(
b
)
10
0
2
(
c
)
2
 

 8  ( f ) 19 5  ( g )  8  ( h)  1 


0
3
 
 


10 
 15 0 3 
 


Example 4
A toothpaste firm supplies tubes of toothpaste to 5 different stores. The number of tubes of toothpaste
supplied per delivery to each store, the sizes and sale prices of the tubes, together with the number of
deliveries made to each store over a 3-month period are shown in the table below.
(i) Write down two matrices only such that the elements of their product under matrix multiplication would
give the volume of toothpaste supplied to each store per delivery.
(ii) Write down two matrices such that the elements of their product under matrix multiplication would
give the number of tubes of toothpaste of each size supplied by the firm over the 3-month period.
Find this product.
(iii) Using the matrix product found in part (ii) and a further matrix, find the total amount of money
which would be obtained from the sale of all the tubes of toothpaste delivered over the 3-month
period.
GCE”O” June 2002
ANS :
 400 300 400 


0 600   50 
 0


(i )  400 0 600   75 



 500 300 0  100 
 600 600 400 


 400 300 400 


0 600 
 0
(ii ) 13 7 10 5 8   400 0 600   16500 10200 18600 


 500 300 0 
 600 600 400 


 2.10 


(iii) 16500 10200 18600   3.00   13500  . Therefore total sales - $13500.
 3.75 


Example 5
A small manufacturing firm produces four types of product, A, B, C and D. Each product requires three
processes–assembly, finishing and packaging. The number of minutes required for each type of product for
each process and the cost, in $ per minute, of each process are given in the following table.
The firm receives an order for 40 of type A, 50 of type B, 50 of type C and 60 of type D.
Write down three matrices such that matrix multiplication will give the total cost of meeting this order.
Hence evaluate this total cost.
GCE”O” Nov 2003
 40 
8 6 6 5 
50
ANS :  0.60 0.20 0.50   5 4 3 2    =…………………..  1111 . Therefore total cost is $1111.00

  50 
3 3 2 2 

 
 60 
Example 6
Given that A, B and C are matrices. State whether the following are true or false.
(a) AB + AC = A(B + C) ?
(b) AB = BA ?
(c) A(BC) = (AB)C ?
Ans : True, False, True
Warning : Must remember that, in general, if A and B are matrices, AB  BA , i.e. matrices are not
commutative. In Arithmetic, multiplication of numbers are commutative e.g. 2  3  3 2 .
(E) Identity Matrices
Before learning how to apply matrices to solve simultaneous equations we must learn the
concept of inverse matrices and identity matrices.
An identity matrix is a square matrix with the digit 1 as the elements along its main diagonal ( sloping
downwards from left to right ) and all the other elements being zeros.
1 0 0 0
1 0 0 

1 0 
 0 1 0 0
E.g 
etc.
, 0 1 0, 
0 0 1 0
0 1 

 0 0 1  

0 0 0 1
The identity matrix is usually denoted by a capital letter I.
Why identity matrix is called identity matrix? Try multiplying any matrix by an identity matrix.
An identity matrix acts something like the number “1” in Arithmetic in that it would not change the identity
of a matrix when multiplied by it.
 1 2  1 0   1 2 
e.g. 


,
 3 4  0 1   3 4 
 1 5  1 0   1 0  1 5   1 5 






 8 7  0 1   0 1  8 7   8 7 
In other words, for any matrix A , AI = IA = A . (must understand and remember this!)
Given that A and B are matrices. State whether the following statement is true or false.
“If AB = BA  A then B must be an identity matrix.”
(F) Determinant of a Matrix (Do not confuse with DISCRIMINANT b 2  4ac )
a b 
The determinant of a matrix M  
 is defined as the value of ad  bc and
c d 
a b
 ad  bc .
we write det M 
c d
2 3
 2 3 
  2  5   3 4   22 .
e.g A  
 , then det A 
4 5
4 5 
(G) Inverse Matrix
The inverse matrix of a matrix A is denoted by A 1 .
1
a b
1  d b 
1  d b 
1  a b 
If A  


 , then A  





det A  c a  ad  bc  c a 
c d
c d
1
2 5
2 5
 4 5  1  4 5 
1
e.g the inverse matrix of 

 is given by  


 

(2)(4)  (5)(3)  3 2  7  3 2 
 3 4 
 3 4 
Note :
(1) A matrix whose determinant is zero has no inverse and hence is known as a singular matrix.
(2) Only square matrices have inverses.
Warning :
Say A, B and C are matrices such that AB = C . And say we know matrices A and C and are asked to find
matrix B. We cannot hope to find B by dividing both sides by matrix A. Do you notice, so far, we have not
introduce “Division of Matrices”. Reason : Such operation is not defined in this branch of mathematics
calls matrices. Then how can we find matrix B? This is where the inverse matrix concept comes in.
You see to find B, we need to eliminate matrix A from the left side of B. How can we eliminate A? Well, by
multiplying by the inverse of A…
AB = C
A1AB = A1C (NOT AA1B = CA1!)
You must multiply matrix A by its inverse A 1 on the left hand side of A ( not right hand side! )
But A 1A = I hence we can simplify to IB = A1C
B = A1C
Bingo!
Example 7
 2 1
 4 2
Given that A = 
 and C  
 . Find the matrix B such that AB = C .
 2 3 
3 1
ANS : From the above we get
B = A1C
Now we have A1 
 3 1 1  3 1
1

 
,
(2)(3)  (1)(2)  2 2  8  2 2 
1  3 1 4 2  1  9 5 
Hence B = A 1C  

 

8  2 2  3 1  8 14 6 
(H) Using Matrices to solve simultaneous equation
We can solve a pair of simultaneous equations
ax  by  m
by expressing it in a matrix form as
cx  dy  n
 a b  x   m 

    
 c d  y   n 
1
1
 a b   a b  x   a b   m 


 
  
  
 c d   c d  y   c d   n 
1
 1 0  x   a b   m 

   
  
 0 1  y   c d   n 
1
 x   a b   m
 
  
 y c d   n 
Example 8 (a)
Use the inverse matrix method to solve the simultaneous equations 6 x  3 y  15 and 4 x  2 y  2 .
What do you notice?
 6 3  x   15 
Ans :
Converting into matrix form, we get 
   
.
 4 2  y   2 
6 3
But det is equal to zero,
  6  2    3 4   0
4 2
Indeed 6 x  3 y  15  y  2 x  5 , 4 x  2 y  2  y  2 x  1 , they are equations of parallel lines. Since
there is no point of intersection between the two lines, there is no solution.
Example 8 (b)
Use the inverse matrix method to solve the simultaneous equations 9 x  6 y  6  0 and 3x  2 y  2  0 .
What do you notice?
 9 6  x   6 
Ans :
Converting into matrix form, we get 
     .
 3 2  y   2 
9 6
But det is also equal to zero,
  9  2    6  3  0
3 2
Both equations could be simplified to y   3 x  1 . They give rise to identical graph. Therefore there are
2
infinite number of solutions.
Example 9
 2005 2006 
1  1 2  as a single matrix.
Given that B  
 , write down BB 

 2007 2008 
3 4
Ans : Do not spend time computing BB 1 , it is equal to identity matrix, hence
 1 2   1 0  1 2   1 2   1 2 
BB1 


  I

.
 3 4   0 1  3 4   3 4   3 4 
End of notes
Sec 3 Integrated Mathematics Enrichment Worksheet
Topic : Application of Matrices to Cryptography
Name : ______________________ (
) Class (
)
Date : ___________
(1) To encode message using matrices, we must first assign numbers to letters of the alphabet.
Let A = 1, B = 2, etc as shown in the table below.
A=1
B=2
C=3
D=4
Space = 27
E=5
F=6
G=7
H=8
I=9
J = 10
K = 11
L = 12
M = 13
N = 14
O = 15
P = 16
Q = 17
R = 18
S = 19
T = 20
U = 21
V = 22
W = 23
X = 24
Y = 25
Z = 26
(2) Say the message you want to encode is “HWA CHONG ”. You need to break the message into groups
of 2 letters (and spaces) each. The message will become
HW
A_
CH
ON
G_
(3) Convert each block of 2 letters into a 2  1 matrix. You will get the following sequence of matrices
 8   1   3  15   7 
         
 23   27   8  14   27 
 2 0
(4) Now use any 2  2 “multiplier” matrix that has an inverse. Say we choose M = 
.
0
1


Find the product of M and each of the preceding matrices gives
 16   2   6   30   14 
         
 23   27   8   14   27 
(5) The recipient of will receive the encoded message as a string of numbers ….
16
23 2
27 6
8
30 14 14 27
which he has to convert them back to sequence of 2  1 matrices.
After multiplying by the inverse, M 1 , the message will be revealed.
Challenge Yourself
You received the following encoded message.
26 21 21 13 31 18 37 19 51 39
You have earlier agreed with the encoder that the multiplier matrix is
What is the encoded message?
 2 1

 .
 1 1