SYSTEM OF LINEAR EQUATIONS

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SYSTEM OF LINEAR

EQUATIONS

•

• INTRODUCTION

INTRODUCTION

•

• GAUSS

GAUSS

-

JORDAN ELIMIATION

•

• HOMOGENEOUS LINEAR

HOMOGENEOUS LINEAR

EQUATIONS SYSTEM

•

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INTRODUCTION

• Linear Equations

any straight line in xy-plane can represented by an equations of the form

where a₁,a₂,b : constant and a₁,a₂are not both zero

a linear equation in the variables x and y

More generally

b

x

a

x

a

x

a

₁ ₁

+

₂ ₂

+

...

+

_n _n

=

b

y

a

x

a

₁

+

₂

=

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INTRODUCTION

• System of Linear Equations

1

1 =

−

=

+

y

x

y

x

A sequence : {s₁, s₂, …, s_n} is called solution of system of linear equations if x₁= s₁,x₂= s₂,…,x_n= s_n is a solution of every equation in the system

A finite set of linear equations in the variables x₁,x₂,…,x_n is called a system of linear equations

Example system of linear equations

{

x

=

1 ,

y

=

0 }

Solution of system

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INTRODUCTION

• Solutions of Linear Equations System

0 2 2

0

= −

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

2 0

= +

= −

y x

x-y=0

x+y=2

x-y=0

x-y=-2

2 0

− = −

= −

y x

one solution no solution

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GAUSS

-

JORDAN ELIMIATION

• System of Linear Equations in Matrices form



An arbitrary system of m linear equations and n variables can be written as

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GAUSS

-

JORDAN ELIMIATION

• Augmented Matrices



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)

Augmented matrix [A|b]

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GAUSS

-

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

  

 

  

  =   

 

  

 

  

 

  

 

7 3 8

1 1 2

0 1 1

1 1 3

z y x

  

 

  

  =

7 3 8

| 1 1 2

| 0 1 1

| 1 1 3

] |

[A b

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GAUSS

-

JORDAN ELIMIATION

1 =

x

2 =

y

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

  

 

  

 

3 2 1

| 1 0 0

| 0 1 0

| 0 0 1

3 =

z

or

  

 

  

  =   

 

  

 

3 2 1

z y x

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GAUSS

-

JORDAN ELIMIATION

[ ]

~...~

Example 2



Solution



Reduced row-echelon

Column 3,4 have no leading 1

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GAUSS

-

JORDAN ELIMIATION

1

3

2

₃ ₄

1

+

x

+

x

=

x

2

₄

3

2

−

x

−

x

=

x

x₂ = 2+ s +2t

Example 2 (continued)

  

 

  

 

− −

0 2 1

| 0 0

0 0

| 2 1

1 0

| 3 2

0

1 x₁ =1−2s−3t

The solution is













+

−

=













t

s

t

s

t

s

x

2

3

2

1

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GAUSS

-

JORDAN ELIMIATION

[ ]

~...~

2 1 4

| | |

3 3 3

0 1 2

3 2 1 |

     

   

− =

b A

Example 3

  

 

  

 

− =   

 

  

 

  

 

  

 

2 1 4

3 3 3

0 1 2

3 2 1

z y x

Solution

  

 

  



 −

1 0 0

| 0 0 0

| 2 1 0

| 1 0

1

Reduced row-echelon

0=1 ???

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HOMOGENEOUS LINEAR

EQUATIONS SYSTEM



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

0

Every homogeneous system is consistent

All system has x₁=0,x₂=0,…,x_n=0 as a solution Æ trivial solution

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HOMOGENEOUS LINEAR

EQUATIONS SYSTEM



There are two only possibilities for homogeneous linear system’s 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

Augmented matrix



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HOMOGENEOUS LINEAR

EQUATIONS SYSTEM

  

 

  

 

− −

0 0 0

| 0 0 0 0

| 2 1 1 0

| 3 2 0 1

Example (2)

   

 

   

 

+ −

=    

 

   

 

t s

x x x x

2 3 2

4 3 2 1

The solutions is

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INVERSE MATRICES

   

  =

d c

b a A 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 denoted by A-1

How to find A-1 ?

If A is 2x2 square matrix

A is invertible if ad ≠ bc, then A-1 _{is given by formula}

   

 

−

− −

=

−

a c

b d

bc ad

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INVERSE MATRICES

[ ]

     

    =

1 0 0

0 1 0

0 0 1

| | |

2 2 1

3 2 1

3 1 1

| I A

  

 

  

  =

2 2 1

3 2 1

3 1 1

A

     

   

− −

− 1 0 1 0 1 1

0 0 1

| | |

1 1 0

0 1 0

3 1 1

~

For generally, we can find A-1 _{using Row Operations}

(Gauss-Jordan elimination) at augmented matrix [A|I], the final matrix that we want is [I|A-1_{]. If the left side of final matrix (reduced}

row echelon) is not I, then A is not invertible

Example 1

Find inverse of

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INVERSE MATRICES



Example 1(continued)



Example 2

Showing that A is not invertible

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INVERSE MATRICES

[ ]

~...

1 0 0

0 1 0

0 0 1

| | |

3 2 1

3 1 1

|

     

    = I A

  

 

  

 

− −

−

1 1 0

0 1 1

0 1 2

| | |

0 0 0

0 1 0

3 0 1

~ Example 2 (continued)

We can’t reduced to be Identity matrix A is not invertible

Properties of inverse matrices

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EXERCISES

1. Solve system of linear equations below



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



3. Find inverse matrices of A if A invertible