"the identity matrix".

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21 Νοε 2013 (πριν από 3 χρόνια και 6 μήνες)

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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)

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.


For matrices, the number
1
is

1 0 0
1 0
0 1 0
0 1
0 0 1
or
 
 
 
 
 
 
 
 
Multiplicative Identity :

Is there a
2



楤敮i楴i慴物⁦潲慴物
浵汴楰i楣慴i潮⸠椮.⸠

1 0
0 1
 
 
 
A



I

=
I



A

=
A ,
where

I
is the identity matrix.


For example,

I =


















d
b
c
a
d
b
c
a
(?)
N.B.
When referring to the multiplicative identity, it
is usually called "
the identity matrix
".



Is is a square matrix

`
All elements in the leading diagonal are
1
.

`
All the other elements are
0
.

`
Eg




1 0 0
1 0
,0 1 0,
0 1
0 0 1
etc
 
 
 
 
 
 
 
 
What do you obtain when A is multiplied by


the identity matrix?

1 0
0 1
a b a b
c d 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

IMPT NOTE : if two matrices are inverses and you multiply them,
then the result is the IDENTITY MATRIX.

-1
-1
AA I
A A I


Step
1
:

Find the determinant of the matrix
A
,
denoted by det
A

det A =
a b
ad bc
c d
 
To find the inverse of a matrix A = .

a b
c d
 
 
 
Step
2

: The inverse of matrix A is

Note :



If det
A
=
0
, then the inverse of
A

is not defined.


Hence
A

does not have an inverse.

1
det
d b
c a
A

 
 

 
Find the inverse if it exists:





Find the inverse if it exists:


3 1
4 1
 
 

 
1 1
1
4 3
31 1( 4)

 

 
 
 
1 1
1
4 3
7

 

 
 
1 1
7 7
3
4
7 7

 
 
 
6 3
8 4
 
 
 
6 3
1
8 4
6 4 3 4
 

 

 
6 3
1
8 4
0
 

 
 
Determine whether each pair of matrices are inverses





3
2
2 1
2 2
1
3 4
and

 
 
 
 

 
 
If
2
matrices are inverses, when you
multiply them you get the identity
matrix.

1 0
0 1
 
 
 
3
2
2 1
2 2
1
3 4

 
 

 
 

 
 
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
    

    
    



Step
2
: Find determinant of

a b
c d
 
 
 

Step
3
: If ,

0
a b
c d

then

1
x d b h
y c a k
ad bc

    

    


    

Step
3
: If ,

0
a b
c d

the equations have no unique solution.


Class work:


Q
1
,
3
,
5
,
8


Q
10


Q
12


Q
13


Q
14


Homework:


Q
2
,
4
,
6


Q
9


Q
11

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
three
-
bedroom 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.


A

E

D

C

B

To
A
B
C
D
E
A
0
1
2
0
1
From
B
1
2
0
0
1
C
1
0
0
1
2
D
1
0
1
1
1
E
0
1
1
1
0
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.













0
1
1
1
0
1
1
1
0
1
2
1
0
0
1
1
0
0
2
1
1
0
2
1
0
R =

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



Multiplying this matrix by itself gives
R
2

which gives the number
of possible two
-
stage routes from place to place. E.g. the
number in the
1
st

row,
1
st

column is
3
showing there are
3
two
-
stage 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,
R
3

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






Choose an arbitrary
2


2
matrix
A

which
has an inverse
A
-
1
. Say
A

= and


A
-
1

=














26
14
A
M













19
7
H
T

















18
8
I
S













21
8
F
S
U 6
N 13
   

   
   






2
1
3
2








2
1
3
2
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.