Transcript "the identity matrix".

Section 4-7
In the multiplication of numbers, the identity element is the number 1
since x ·1 = x for every value of x.
(it gives the original number its identity back)
For matrices, the number 1 is
1 0 0 
1 0  

or
0
1
0
0 1  


 0 0 1 


If you multiply the matrix I with any matrix P and the result is the matrix P, then
I is known as the identity matrix.
Multiplicative Identity :
Is there a 2  2 identity matrix for matrix
multiplication. i.e.  a c 
a c 

(?)  

b d 
b d 
A  I = I  A = A , where I is the identity matrix.
For example, I =  1 0 


0 1
When referring to the multiplicative identity, it
is usually called "the identity matrix".
N.B.




Is is a square matrix
All elements in the leading diagonal are 1.
All the other elements are 0.
Eg
1

0
1
0 
,0
1 
0
0
1
0
0

0  , etc
1

What do you obtain when A is multiplied by
the identity matrix?
 a b  1 0   a b 




c
d
0
1
c
d


 

AI = A or IA = A
** When
we say "the inverse of a matrix", it is
referring to the multiplicative inverse
 2 3
 3 3 
If A = 
 , and B = 
,
1 4
 5 2 
 2 3  3 3   1 0 
then AB = 


  I and BA= I
 1 4  5 2   0 1 
If A and B are two matrices and AB = BA = I,
then A is said to be the inverse of B, denoted by B-1;
B is said to be the inverse of A, denoted by A-1.
Given A and the inverse of A, denoted by A-1
AA  I
-1
A A  I
-1
IMPT NOTE : if two matrices are inverses and you multiply them,
then the result is the IDENTITY MATRIX.
To find the inverse of a matrix A =
a b 

.
c d
Step 1 : Find the determinant of the matrix A,
denoted by det A
a b
det A =
 ad  bc
c d
Note :
• If det A = 0, then the inverse of A is not defined.
•Hence A does not have an inverse.
Step 2 : The inverse of matrix A is
1  d b 


det A  c a 

Find the inverse if it exists: 


4
1


3
1
1 1 1 1 1  17
1
 
 



3 1  1(4)  4 3  7  4 3   74
Find the inverse if it exists:  6
3
8 4 


6 3 
1
1 6 3 
 




6 4  3 4 8 4 
0 8 4 
 17 
3 
7 
Determine whether each pair of matrices are inverses
2
 2 2
 3 4 and   3


 2
 2 2  2
 3 4   3

 2
1
1 
If 2 matrices are inverses, when you
multiply them you get the identity
matrix.
1 1 0



1  0 1 
1 0 
0 1 




To solve simultaneous equations by using
simple algebra, if there is no solution or
infinite solutions, what will you say about
the two equations?
The simultaneous equations will represent
either two parallel lines or the same
straight line.


When the simultaneous equations is
expressed in the matrix form, and if the
determinant of the 22 matrix is zero, then
the two simultaneous equations will represent
either two parallel lines or the same straight
line.
The equations have no unique solution.
Using Matrices to Solve
Simultaneous Equations
•Step 1 : Given ax + by = h
and
cx + dy = k

 a b  x   h 

    
 c d  y   k 
a b 
•Step 2 : Find determinant of 

c d

a b
•Step 3 : If
0 ,
 x
1  d b  h 
then   

 
y

c
a
ad

bc
 

 k 
c d
•Step 3 : If
a
b
c
d
 0 , the equations have no unique solution.






Class work:
Q1, 3, 5,8
Q10
Q12
Q13
Q14
•
•
•
•
Homework:
Q2, 4, 6
Q9
Q11
Why learn Matrices ?
The interior design company is given the job of putting up
the curtains for the windows, sliding doors and the living
room of the entire new apartment block of the NTUC
executive condominium. There are a total of 156 threebedroom units and each unit has 5 windows, 3 sliding
doors and 2 living rooms. Each window requires 6 m of
fabric, each sliding door requires 14 m of fabric and each
living room requires 22 m of fabric. Given that each metre
of the fabric for the window cost $12.30, the fabric for the
sliding door costs $14.50 per metre and each metre of the
fabric for the living room is $16.50.
We can write down three matrices whose product shows
the total amount of needed to put up the curtains for each
unit of the executive condominium.
NE Message:
The property market in Singapore went up very rapidly in
the 1990’s. Many Singaporeans dream of owning a private
property were dashed and many call for some form of help
from the government to realise their dream. NTUC Choice
Home was set up to go into property business as a way of
stabilising the market and to help Singaporeans achieve
their dream of owning private properties. With the onset of
the Asian economic crises, the property market went under
and the public start to question the need for NTUC Choice
Home and urged NTUC to dissolve NTUC Choice Homes.
Do you think this is a good request? How long do you
think it will take to set up a company to run the property
business?
The Microsoft Excel matrix functions are:
 MDETERM(array)
Returns the matrix
determinant
of an array
 MINVERSE(array)
Returns the inverse of the
matrix of an array
 MMULT(array A, array B)
Returns the matrix
product
 TRANSPOSE(array)
Returns the transpose of an
array. The first row of
the input
becomes the first column
of the
output array, etc.
 *Except for MDETERM(), these are array functions
and must be completed with "Crtl+shift+Enter".



Routes matrices or Matrices for Graphs
Matrices can be used to store data about graphs.
The graph here is a geometric figure consisting of
points (vertices) and edges connecting some of
these points. If the edges are assigned a direction,
the graph is called directed.
Cryptography
Matrices are also used in cryptography, the art of
writing or deciphering secret codes.
Example If 5 places A, B, C, D, E are connected by a road
system shown in the graph. The arrows denote one-way roads,
then this can be listed as
To
From
A
B
C
D
E
A
0
1
1
1
0
B
1
2
0
0
1
C
2
0
0
1
1
the loop at B gives 2
routes from B to B
but the loop at D gives
only 1 route because
it is one-way only.
R=
D
0
0
1
1
1
0
1
1
1
0

B
E
1
1
2
1
0
1
2
0
0
1
A
2
0
0
1
1
0
0
1
1
1
1
1
2
1
0 
E
C
D


Multiplying this matrix by itself gives R2 which gives the number
of possible two-stage routes from place to place. E.g. the
number in the 1st row, 1st column is 3 showing there are 3 twostage routes from A back to A (One is ABA, another is ACA using
the two-way road and the third is ACA out along the one-way
road and back along the two-way road.)
Similarly, R3 gives the number of possible three-stage routes
from place to place and vice versa.
A spreadsheet can be used for the tedious matrix
operations as shown below.

One way of encoding is associating numbers
with the letters of the alphabet as show
below. This association is a one-to-one
correspondence so that no possible
ambiguities can arise.
A
B
C
D
E
F
G
H
I
J
K
L
M
N
O
P
Q
R
S
T
U
V
W
X
Y
Z


























26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
In this code, the word PEACE looks like 11 22 26 24 22.
Suppose we want to encode the message: MATHS IS
FUN
If we decide to divide the message into pairs of
letters, the message becomes MA TH IS SF UN.

(If there is a letter left over, we arbitrarily
assign Z to the last position). Using the
correspondence of letters to numbers
given above, and writing each pair of
letters as a column vector, we obtain
 M    14   T    7   S    8   S    8   U    6 
 H arbitrary
     2 matrix
A   26  an
 Choose
N  13 
  19   I 2 18
  F   21A which
has an inverse A-1. Say A =
A-1 =
 2  3 
 1 2 
 2 3 
 1 2
and
Now transform the column vectors by multiplying each
of them on the left by A:
The encoded message is 106 66 71 45 70 44 79 50 51 32.
To decode, multiple by A-1 and reassigning letters to the
numbers.