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					Introduction
• Matrices have many important uses.
• One of the simplest uses of matrices is to
  store data.
• In fact, this is the most common use outside
  mathematics.
• Matrices are used to solve linear systems
  and represent geometric figures and
  transformations. Building on this geometric
  interpretation, matrices can also be applied
  in the study of trigonometry.
•Matrices are a powerful unifying concept,
connecting ideas in mathematics to those in other
content domains as well as connecting various
branches of mathematics. As such, matrices can be
a means of integrated whole rather than a set of
isolated topics.
•Matrices are a rich topic and that many uses of
matrices are minor extensions or applications of
content already in our syllabus. There are several
reasons for introducing matrices into the syllabus:
1
• Matrices introduce a fundamental concept
  in discrete mathematics.
• They model numerous realistic applications.
• They furnish opportunities for doing
  arithmetic and algebraic computation in a
  new context.
• Their properties and their operations lead to
  important theoretical results.
          Why Learn Matrices
•   Storing and organising numerical data
•   Operations with matrices
•   The identity Matrices
•   Inverse Matrices
•   Operations using a Spreadsheet
•   Some interesting Applications of Matrices (
    not in syllabus)
 Storing and organising numerical data




6 rows and 7 columns of data,
we say the matrix has size (order) 6  7
 Example :
Type       Friction Textbooks General Reference

Malay        25       47        22       30

Chinese      40       72        38       40

English      80       85        67       54
                                                    Matrix


Type        Malay   Chinese    English

Friction      25        40       80

Textbook      47        72       85

General       22        38       67

Reference     30        40       54               Matrix
         Text Book Page 198
• Can you name the orders of the following
  matrices?
    (a) 2 by 2        (e) 3 by 5
    (b) 3 by 1        (f) 2 by 1
    (c) 1 by 4        (g) 1 by 1
    (d) 3 by 3
        Operations with Matrices
 • Addition and Subtraction of Matrices




What do you notice about the order of the two
matrices involved in each case?
          Rules for Matrix Addition
           and Matrix Subtraction
• If P and Q are two matrixes of the same order,
  then the sum of P and Q, denoted by ( P + Q ), is
  a matrix with its elements being the sum of the
  corresponding elements of P and Q.
• The difference of P and Q is denoted by
  (P–Q).
• If P and Q are of different orders, then their sum
  and difference are not defined.
  Example :
        Ex 9A Page 201

• Class Work :
• Q1 a to j
Multiplication of a Matrix by a Real Number
• Examples :
             Some Special Matrices
• Square Matrix
  * A matrix has the same number of rows and
  columns.
  Examples :

•Zero matrix or null matrix , denoted by O
 * Every element of a matrix is zero.
 * May be of any order
Examples :
             Ex 9A Page 201
•   Class work :     •   Homework :
•   Q3               •   Q2
•   Q4a, c, e, f     •   Q4b, d
•   Q5a, c, e        •   Q5b, d
•   Q6a              •   Q6b
      Operations With Matrices
Matrix multiplication was invented in the 19th
Century as a useful way of combining two
suitable matrices. It multiplies elements in a row
with elements in a column to produce one single
number.
To make up some matrix multiplication problems
for yourself. And to investigate the condition on
the sizes of two matrices that is necessary before
you can be multiplied.
Example : Maybeline bought 3 apples, 2 oranges and 4
pears.The cost of each item is 25 cents, $1 and 40 cents
respectively. Find the total cost of the purchase.
 Answer : The total cost of the purchase
          = 3  25 cents + 2  100 cents + 4  40 cents
          = 435 cents
          = $ 4.35

Using Matrices :
• Page 202, 203
• For example,
• A 33 matrix multiplied by a 32 matrix
  will result in a 32 matrix.
• A 32 matrix multiplied by a 23 matrix
  will result in a 33 matrix.
• A 32 matrix multiplied by a 33 matrix is
  not possible.
• In general : A m  n matrix can be
  multiplied to a n  p matrix to produce a
  m  p matrix.
               Ex 9B Page 206
•   Class Work :
•   Q1b, d, f         •   Homework :
•   Q2                •   Q1a, c, e
•   Q3a               •   Q3b, c
•   Q4                •   Q5
•   Q6a, c, f         •   Q6b, d, e
•   Q8                •   Q9
•   Q10               •   Q13
•   Q11
•   Q12
•   Q14
                  Matrices Multiplication

                             The prices of the drinks
                             per cup are 55¢ for tea,
                             60¢ for coffee and 75¢ for
                             drinking chocolate.

(a) Form a matrix of prices and use it to find
the total amount taken on each of the 3 days.
(b) What information would be found by pre-
multiplying V by (1 1 1)?
explain why the matrix of prices can be a 31 or 13
matrix in (a) and what information is found if V is
post-multiplied by
Commutative and associative properties


        matrix multiplication is not
       commutative but associative.



               Explain using a
                  problem
             Commutative and associative properties


                                                      Prices




         A                     B                       C
(AB)C is calculated by first finding the number of
doors and information does the matrix AB
   What windows in each of the developments and
  gives? BA?
then finding the total cost of windows and doors for
each development.
A(BC) is calculated by first finding the cost of the
      Calculate (AB)C and A(BC).
doors and windows in each model and then finding
their Explain why they are equal.
      total cost for each development.
 Commutative and associative properties


• In general, matrix multiplication is not
  commutative, i.e. AB  BA. ( Ex 9B Q 2 )
• However, matrix multiplication is
  associative. (AB)C = A(BC) ( Eg of
  previous 2 slides)
• See example 4 on page 205
            Identity Matrices
 • In the addition of numbers, the identity is 0
   since x + 0 = x for every value of x.
 • In the multiplication of numbers, the identity is 1
  since x  1 = x for every value of x.

If you combine the matrix I with any matrix P and
the result is the matrix P, then I is known as the
identity matrix.
Additive Identity :
  Is there a 2  2 identity matrix for matrix
addition. i.e.


  •A + O = O + A = A , where O is the null matrix.
  For example, O =
Multiplicative Identity :
Is there a 2  2 identity matrix for matrix
multiplication. i.e.



 A  I = I  A = A , where I is the identity matrix.
 For example, I =


    When referring to the multiplicative identity, it
 N.B.

 is usually called "the identity matrix".
 Characteristics of Identity matrix :
  •   Is is a square matrix
  •   All elements in the leading diagonal are 1.
  •   All the other elements are 0.
  •   Eg


What do you obtain when A is multiplied by
the identity matrix?

                           AI = A or IA = A
            Inverse Matrices
• The inverse of anything is that which will
  combine with it to give the identity.

Is there an additive inverse for          ?


  The additive inverse of            is


 Reason :

            The additive inverse can be found easily.
         Multiplicative Inverse
  ** When   we say "the inverse of a matrix", it is
referring to the multiplicative inverse




 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.

 IMPT NOTE :                  You are doing Matrices .
 Given A and the inverse of A, denoted by A -1

IMPT NOTE :                 You are doing Matrices .
   To find the inverse of a matrix A =                       .
Step 1 : Find the determinant of the matrix A,
denoted by det A

  Note :
  • If det A = 0, then the inverse of A is not defined.
  •Hence A does not have an inverse.
  •When a matrix does not possess an inverse, it is known as a
  singular matrix.
Step 2 : The inverse of matrix A is
             Ex 9C Page 211
•Class Work:       •Homework:
•Q1a, b, h         •Q1e,i
•Q2a, d,e          •Q2g,h
•Q4                •Q3
•Q5                •Q7
•Q6                •Q8
•Q9
•Q10
           Time to think
• Can a matrix have more than one inverse?
How to prove it?

• Text Book Page 214
• 9.9 Some interesting Properties of
  Matrices
• Q1, 2
       Using Matrices to Solve
       Simultaneous Equations
• 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.
       Using Matrices to Solve
       Simultaneous Equations
• 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
                                   

•Step 2 : Find determinant of

•Step 3 : If              ,
                         then


•Step 3 : If              , the equations have no unique solution.
             Ex 9D Page 214
•   Class work:    •   Homework:
•   Q1, 3, 5,8     •   Q2, 4, 6
•   Q10            •   Q9
•   Q12            •   Q11
•   Q13
•   Q14
                    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?
Operations using a Spreadsheet
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".
   Some Interesting Applications
• 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.
                        Routes Matrices
 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
                                                    B




                                          A                    E
the loop at B gives 2
routes from B to B
but the loop at D gives
                                                        C
only 1 route because
it is one-way only.  R=                             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 1 st row,
  1st 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, 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.
                                         Cryptography
      • 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 =
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.

				
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