SYSTEM OF LINEAR EQUATIONS

  INTRODUCTION

  

SYSTEM OF LINEAR

SYSTEM OF LINEAR

EQUATIONS EQUATIONS

  INTRODUCTION

  GAUSS GAUSS - - JORDAN ELIMIATION JORDAN ELIMIATION

HOMOGENEOUS LINEAR HOMOGENEOUS LINEAR EQUATIONS SYSTEM EQUATIONS SYSTEM

  INVERSE

  INVERSE INTRODUCTION

  INTRODUCTION

• Linear Equations

  any straight line in xy-plane can represented by an equations of the form where a ₁ ,a ₂ ,b : constant and a ₁ ,a ₂ are not both zero a linear equation in the variables x and y

  More generally b x a x a x a n n = + + + ...

  2

  2

  1

  1

  = b y a x a +

  2

  1 a linear equation in the variables x

  1 ,x

  2 ,…,x n INTRODUCTION

  INTRODUCTION

• System of Linear Equations

  

A finite set of linear equations in the variables x ,x ,…,x is

  1 2 n called a system of linear equations

A sequence : {s , s , …, s } is called solution of system of

  1 2 n

linear equations if x = s ,x = s ,…,x = s is a solution of

  1

  1

  2 2 n n every equation in the system Example system of linear equations

• x y

  1 = Solution of system x = y

  1 , =

  { } x y

  1 − = 1+ 0 = 1 1 – 0 = 1 INTRODUCTION

  INTRODUCTION

• Solutions of Linear Equations System

  2 2 = − = −

  y x y x

In the xy-plane, solutions system of linear equations can be

represented as

  

A system of linear equations that has no solutions is said to be

inconsistent x-y=0 2x-2y=0

  y x y x x-y=0 x+y=2 x-y=0 x-y=-2

  = 2 − − = −

  y x y x one solution no solution

• + 2 =

  = −

  Infinitely many solutions GAUSS GAUSS

  JORDAN ELIMIATION JORDAN ELIMIATION

• System of Linear Equations in Matrices form

  1

  12

  2

  1

  1

11 M M M M

  n n mn m m ^{n n} b b b x x x a a a a a a a a a

  22

  Denoted by b x A = or Ax = b

       

       

  =      

       

       

  21

  1

  2

  M M K M M M K K _{2 1 2 1}

  2

  2

  1

  1

       

  2

  ... ...

  = + + + = + + + = + + + ...

  2 ^{1 2 22 21 1 12 11} An arbitrary system of m linear equations and n variables can be written as n n mn m m n n n n b x a x a x a b x a x a x a b x a x a x a

  2

• Augmented Matrices

  M M K M M M K K _{2 1 2 1}

  2 ^{1 2 22 21 1 12 11}

  M K M M M K K _{2 1}

  n mn m m ^{n n} b b b a a a a a a a a a

       

       

  Augmented matrix [A|b]

  2 ^{1 2 22 21 1 12 11} Augmented matrices of system Ax=b is matrices which the values is join of entries of A (left side) and entries of b (right side)

  n n mn m m ^{n n} b b b x x x a a a a a a a a a

  GAUSS GAUSS

       

       

       

  =      

       

       

  JORDAN ELIMIATION JORDAN ELIMIATION

  | | | |

  3

  1

  1

  3 z y x

       

     

  =

  7

  8 |

  1

  1

  1

  2 |

  1

  1 |

  1

  1

  1

  2

  GAUSS GAUSS

       

  JORDAN ELIMIATION JORDAN ELIMIATION Definition

A systematic procedure for solving system of linear equations

by reducing augmented matrix to be reduced echelon form Example 1

  Solve this system of linear equations

  7

  2

  3

  8

  3 = + + = + = + + z y x y x z y x

     

  1

  

=

    

       

     

      

  7

  3

  8

  1

  3 ] | [ b A Augmented matrix [A|b]

  1

  1 z y x

  2

  3

      

  =     

     

       

  3 = z

or

  1 |

  GAUSS GAUSS

  1 |

  1 |

  2

  3

      

      

  Example 1 (continued) We have eliminated this matrix to be reduced row echelon before (see matrices ppt page 17) The reduced row echelon form of the augmented matrix is

  2 = y

  JORDAN ELIMIATION JORDAN ELIMIATION 1 = x

  Solution of linear equations system

  4 ^{3 2 1} x x x x

  1

  4

  3

  1

  1

  1

  3

  1

  2

  2

  1

  Solution      

  4

     

  − −

  2

  1 | |

  2

  1

  1 |

  3

  2

  Column 3,4 have no leading 1 x

  3 =s , x

  4 =t

  1

  3

  GAUSS GAUSS

      

      

       

  =      

      

      

  Example 2 Solve this system of linear equations

  JORDAN ELIMIATION JORDAN ELIMIATION [ ]

1 Reduced row-echelon

      

  4

  = b A

  ~ ... ~

  3

  4

  1 | | |

  1

  3

  1 |

  1

  1

  1

  3

      

  2

  2

  1

       

  1 |

  3

  2

  1 t s x

  3

  2

  1 ₁ − − =

  The solution is

       

  2

  − − =

       

       

  t s t s

t s

x x x x

  2

  2

  3

  2

  1

  1

  1 | |

  2

  GAUSS GAUSS

  JORDAN ELIMIATION JORDAN ELIMIATION

  1

  

3

  2

  4

  3

  1 = + + x x x

• =

  2

  2

  4

  3

  2 = − − x x x t s x

  2

  2 ₂

  Example 2 (continued)      

     

  − −

  4 ^{3 2 1}

  3

  2

  1

  4

  3

  3

  3

  1

  2

  2

     

  1 z y x

  Solution      

     

  −

1 |

|

  2

  1 |

  1

  0=1 ??? No solution

      

       

  GAUSS GAUSS

  − =     

     

  Example 3 Solve this system of linear equations      

  JORDAN ELIMIATION JORDAN ELIMIATION [ ]

1 Reduced row-echelon

      

  1 |

  2

  3

      

  1

  3

  3

  3

  4 | | |

  1

  2

  ~ ... ~

  − = b A

  2 HOMOGENEOUS LINEAR HOMOGENEOUS LINEAR EQUATIONS SYSTEM EQUATIONS SYSTEM

       

  22

  2 =0,…,x n=

  1 =0,x

  = x A Every homogeneous system is consistent All system has x

  M M M M Denoted by

  = + + + n mn m m n n n n x a x a x a x a x a x a x a x a x a

  11 = + + + = + + +

  1

  12

  2

  1

  21

  1

  2

       

  2

  1

  1

  2

  2

  A system of linear equations is said to be homogeneous if the constant term are all zero; that is, the system has the form ... ... ...

  n mn m m ^{n n} x x x a a a a a a a a a

  K M M M K K

  2 ^{1 2 22 21 1 12 11} M M

        _{2 1}

       

       

  =      

  0 as a solution Æ trivial solution If there are other solutions, they are called nontrivial solutions

• The system has only the trivial solution
• The system has infinitely many solutions in addition to the trivial solution Example Solve the homogeneous system of linear equations

  2

  1 Reduced row-echelon

  2

  3

  solutions

  2

  

3

  HOMOGENEOUS LINEAR HOMOGENEOUS LINEAR EQUATIONS SYSTEM EQUATIONS SYSTEM      

  3

  1

1 There are two only possibilities for homogeneous linear system’s

  1

  3

     

  =       

       

  1 |

  1

  2

  − − | |

     

  Augmented matrix      

     

      

  2

  1

  1

  3

  1

  1

  1

  

4

  

1

  4

     

       

  4 ^{3 2 1} x x x x

  1

  1

  2

  1

  | | |

  4 ^{3 2 1} The solutions is infinitely many solutions / nontrivial solutions

  1 Example (2)

  2

  3

  2

       t s t s t s x x x x

     

       

       

  2

  HOMOGENEOUS LINEAR HOMOGENEOUS LINEAR EQUATIONS SYSTEM EQUATIONS SYSTEM

  3

  1 |

  1

  2

  − − | |

• − =   

      

      

INVERSE MATRICES

  Definition If A,B is n x n square matrix, if B can be found such that AB=BA=I, then A is said to be invertible and B is called an inverse of A and

• 1

  denoted by A

• 1 ?

  How to find A a b

   

  A

  =

  If A is 2x2 square matrix

   

  c d

   

• 1

  A is invertible if ad ≠ bc, then A is given by formula d − b

  1   − ₁ A

  =   ad − bc − c a

   

INVERSE MATRICES

• 1
• 1

      

  1

  1

  1 A     

      

  − −

  −

  1

  1

  1

  1 | | |

  1

  1

  1

  1

  3

  1

  1 ~

  For generally, we can find A

  using Row Operations (Gauss- Jordan elimination) at augmented matrix [A|I], the final matrix that we want is [I|A

  ]. If the left side of final matrix (reduced row echelon) is not I, then A is not invertible Example 1 Find inverse of Solution

  3

  2

      

  2

  =

  1

  1

  1 | | |

  2

  2

  1

  3

  1

  [ ]

  3

  1

  1 | I A

      

      

  =

  2

  2

  1

  3

INVERSE MATRICES

      

  1 ~

  1

  1

  1

  1

  3

  4

  2 | | |

  1

  1

     

  

I

Example 2 Showing that A is not invertible      

  − −

  =

  3

  2

  1

  3

  2

  1

  3

  1

  −

      

      

  3

  − −

  − −

  1

  1

  1

  1

  1

  2 | | |

  1

  1

  1 ~

      

       

     

  − − − = ⁻

  1

  1

  1

  1

  3

  4

  2 ₁ A Example 1(continued)

1 A

INVERSE MATRICES

  2 | | |

  −

  1

  1

  1

  1

  1

  1

      

  3

  1 ~

  Example 2 (continued) We can’t reduced to be Identity matrix A is not invertible

  Properties of inverse matrices (AB)

  = B

  A

  − −

      

  [ ] ... ~

  3

  1

  1

  1 | | |

  3

  2

  1

  2

  I A

  1

  3

  1

  1 |

      

      

  =

• 1
• 1
• 1

  EXERCISES EXERCISES

1. Solve system of linear equations below

  x   ₁

  4

  3 5 x

  1      

  3

  2

  2

  1       x

        ₂

        a .

  2

  2 3 y =

  2 b .

  2

  1

  3 4 =

  2

       

       x  ₃

  

  3

  3 5   z   3 

   1 − 1 −

1 −

2   − 1 

         

      x ₄

   

2. Find solutions of homogeneous linear system Ax = 0 where

  1

  1

  1  

  1

  1

  2

  2  

     

  a . A =

  2

  2

  2

  b . A =

  2

  1

  3

  3  

    

  1 1   

  1

  1

  1  

   

3. Find inverse matrices of A if A invertible

  − 1 −

  1

  2

  3

  2 − 3 − 3 − 7    

       

   

  b . A =

  2

  1 2 c . A =

  2

  1

  3

  a . A =

  2

  6

  7    

    

  3

  1 1  

  2 1  

  1

  2 3  − −

       