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# System of linear equations. Lecture 4-5

## 1. System of linear equations

SYSTEM OF LINEAR
EQUATIONS
K A R A S H B AY E VA Z H . O .
SENIOR-LECTURER

## 2. overview

OVERVIEW
• Rank of a matrix
• Systems of linear equations
• Matrix representation of SLEs and solution.
• Elementary row and column operations

## 3. Rank of A Matrix

RANK OF A MATRIX
• A matrix of r rows and c columns is said to be of
order r by c. If it is a square matrix, r by r, then the
matrix is of order r.
• The rank of a matrix equals the order of highest-order
nonsingular submatrix.
• Nonsingular matrices have non-zero determinants
• Singular matrices have zero determinants
2- 3

## 5. Computing rank by various methods

COMPUTING RANK BY VARIOUS
METHODS
• By Gauss elimination
• By determinants
• By minors

## 7. Elementary row and column operations

ELEMENTARY ROW AND COLUMN OPERATIONS

## 8. Elementary row and column operations

ELEMENTARY ROW AND COLUMN OPERATIONS

## 12. Example 1: Rank of Matrix

EXAMPLE 1: RANK OF MATRIX
2 3 order matrix,
1 2 4
R
2
4
8
3 square submatrices:
1 2
1 4
R1
, R2
,
2 4
2 8
2 4
R3
4
8
Each of these has a determinant of 0, so the rank is less than 2.
Thus the rank of R is 1.
2- 12

## 13. Example 2: Rank of Matrix

EXAMPLE 2: RANK OF MATRIX
2
A 1
9
4
3
10
6
5
11
Since |A|=0, the rank is not 3. The following
submatrix has a nonzero determinant:
2 4
2(3) 4(1) 2
1 3
Thus, the rank of A is 2.
2- 13

## 14. Systems of linear equations

SYSTEMS OF LINEAR EQUATIONS

## 15.

MATRIX REPRESENTATION OF SLES
• Any SLEs can be formulated in the matrix form:

## 16. Methods of solving SLE

METHODS OF SOLVING SLE

## 17. Methods of solving SLE

METHODS OF SOLVING SLE

## 18. Gauss elimination

GAUSS ELIMINATION
• Two steps
• 1. Forward Elimination
• 2. Back Substitution

## 19. Forward Elimination

FORWARD ELIMINATION
A set of n equations and n unknowns
a11x1 a12 x2 a13 x3 ... a1n xn b1
a21x1 a22 x2 a23 x3 ... a2n xn b2
.
.
.
.
.
.
an1x1 an 2 x2 an3 x3 ... ann xn bn
(n-1) steps of forward elimination

## 20. Forward Elimination

FORWARD ELIMINATION
Step 1
For Equation 2, divide Equation 1 by a11 and
multiply by a21 .
a21
a (a11x1 a12 x2 a13 x3 ... a1n xn b1 )
11
a21
a21
a21
a21x1
a12 x2 ...
a1n xn
b1
a11
a11
a11

## 21. Forward Elimination

FORWARD ELIMINATION
Subtract the result from Equation 2.
a21x1 a22 x2 a23 x3 ... a2n xn b2
a21
a21
a21
− a21x1 a a12 x2 ... a a1n xn a b1
11
11
11
_________________________________________________
a21
a21
a21
a22
a12 x2 ... a2 n
a1n xn b2
b1
a11
a11
a11
or
a x ... a x b
'
22 2
'
2n n
'
2

## 22. Forward Elimination

FORWARD ELIMINATION
Repeat this procedure for the remaining
equations to reduce the set of equations as
a11x1 a12 x2 a13 x3 ... a1n xn b1
'
'
a22
x2 a23
x3 ... a2' n xn b2'
'
'
a32
x2 a33
x3 ... a3' n xn b3'
.
.
.
.
.
.
.
.
.
'
an' 2 x2 an' 3 x3 ... ann
xn bn'
End of Step 1

## 23. Forward Elimination

FORWARD ELIMINATION
Step 2
Repeat the same procedure for the 3rd term of
Equation 3.
a11x1 a12 x2 a13 x3 ... a1n xn b1
'
'
a22
x2 a23
x3 ... a2' n xn b2'
"
a33
x3 ... a3" n xn b3"
.
.
.
.
.
.
"
an" 3 x3 ... ann
xn bn"
End of Step 2

## 24. Forward Elimination

FORWARD ELIMINATION
At the end of (n-1) Forward Elimination steps, the
system of equations will look like
a11 x1 a12 x2 a13 x3 ... a1n xn b1
'
'
a22
x2 a23
x3 ... a2' n xn b2'
"
a33
x3 ... a3" n xn b3"
.
.
.
.
.
.
n 1
n 1
ann
xn bn
End of Step (n-1)

## 25. Matrix Form at End of Forward Elimination

MATRIX FORM AT END OF
FORWARD ELIMINATION
a11 a12
0 a'
22
0
0
0
0
a13
'
a23
a"33
0
a1n x1 b1
'
'
a 2 n x2
b2
a"3n x3 b3"
(n 1 )
xn bn(n-1 )
0 ann

## 26. Back Substitution Starting Eqns

BACK SUBSTITUTION STARTING EQNS
a11 x1 a12 x2 a13 x3 ... a1n xn b1
'
'
a22
x2 a23
x3 ... a2' n xn b2'
"
a33
x3 ... an" xn b3"
.
.
.
n 1
.
.
.
n 1
ann xn bn

## 27. Back Substitution

BACK SUBSTITUTION
( n 1)
n
( n 1)
nn
b
xn
a

## 28. Back Substitution

BACK SUBSTITUTION
( n 1)
n
( n 1)
nn
b
xn
a
xi
bi i 1 ai ,ii 11 xi 1 ai ,ii 12 xi 2 ... ai ,in 1 xn
i 1
aii
i 1
xi
bi
aij i 1 x j
n
j i 1
i 1
ii
a
for i n 1,...,1
for i n 1,...,1

## 32. Existence and Uniqueness of Solutions

EXISTENCE AND UNIQUENESS OF SOLUTIONS

## 33. Existence and Uniqueness of Solutions

EXISTENCE AND UNIQUENESS OF SOLUTIONS

THE END