PPT: System of Linear Equations | Engineering Mathematics - Civil Engineering (CE) PDF Download

 Page 1


Linear System
? Linear equation: ax
1
+ bx
2
= c
where, a, b, c = constants                 x
1, 
x
2
= variables
? A linear system consists of more than one linear equations.
? For example:- a
1
x
1
+ b
1
x
2 
+ c
1
x
3
= k
1
……………….(1)
a
2
x
1
+ b
2
x
2
+ c
2
x
2
= k
2
……………….(2)
a
3
x
1
+ b
3
x
2
+ c
3
x
3
= k
3
……………….(3)
? To find a solution of this system means to find the value/values of x
1, 
x
2
, x
3
which satisfies all 
the equations in the linear system.
? A linear system can have a unique solution, more than one solutions or no solution.
Page 2


Linear System
? Linear equation: ax
1
+ bx
2
= c
where, a, b, c = constants                 x
1, 
x
2
= variables
? A linear system consists of more than one linear equations.
? For example:- a
1
x
1
+ b
1
x
2 
+ c
1
x
3
= k
1
……………….(1)
a
2
x
1
+ b
2
x
2
+ c
2
x
2
= k
2
……………….(2)
a
3
x
1
+ b
3
x
2
+ c
3
x
3
= k
3
……………….(3)
? To find a solution of this system means to find the value/values of x
1, 
x
2
, x
3
which satisfies all 
the equations in the linear system.
? A linear system can have a unique solution, more than one solutions or no solution.
Linear System
? Rewriting the above equation in form of matrix
AX = B
A = 
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
?? 3
?? 3
?? 3
, X = 
?? 1
?? 2
?? 3
, B = 
?? 1
?? 2
?? 3
? The matrix A is called the coefficient matrix and the block matrix [A B] , is the augmented 
matrix of the linear system. Type eq uati on h er e.
Augmented matrix : 
?? 1
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
?? 2
?? 3
?? 3
?? 3
?? 3
? If all the elements of B are zero then linear system is called Homogeneous otherwise Non-
Homogeneous.
Page 3


Linear System
? Linear equation: ax
1
+ bx
2
= c
where, a, b, c = constants                 x
1, 
x
2
= variables
? A linear system consists of more than one linear equations.
? For example:- a
1
x
1
+ b
1
x
2 
+ c
1
x
3
= k
1
……………….(1)
a
2
x
1
+ b
2
x
2
+ c
2
x
2
= k
2
……………….(2)
a
3
x
1
+ b
3
x
2
+ c
3
x
3
= k
3
……………….(3)
? To find a solution of this system means to find the value/values of x
1, 
x
2
, x
3
which satisfies all 
the equations in the linear system.
? A linear system can have a unique solution, more than one solutions or no solution.
Linear System
? Rewriting the above equation in form of matrix
AX = B
A = 
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
?? 3
?? 3
?? 3
, X = 
?? 1
?? 2
?? 3
, B = 
?? 1
?? 2
?? 3
? The matrix A is called the coefficient matrix and the block matrix [A B] , is the augmented 
matrix of the linear system. Type eq uati on h er e.
Augmented matrix : 
?? 1
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
?? 2
?? 3
?? 3
?? 3
?? 3
? If all the elements of B are zero then linear system is called Homogeneous otherwise Non-
Homogeneous.
Row Reduced Echelon Form of a 
Matrix
? A matrix C is said to be in the row reduced form if
1. The first non-zero entry in each row of C is 1
2. The column containing this 1 has all its other entries zero
e.g. A = 
1 1 3
0 3 2
0 0 1
B = 
1 0 1 - 2
0 1 2 3
0 0 4 1
0 0 - 1 - 2
C = 
1 - 1
0 2
Page 4


Linear System
? Linear equation: ax
1
+ bx
2
= c
where, a, b, c = constants                 x
1, 
x
2
= variables
? A linear system consists of more than one linear equations.
? For example:- a
1
x
1
+ b
1
x
2 
+ c
1
x
3
= k
1
……………….(1)
a
2
x
1
+ b
2
x
2
+ c
2
x
2
= k
2
……………….(2)
a
3
x
1
+ b
3
x
2
+ c
3
x
3
= k
3
……………….(3)
? To find a solution of this system means to find the value/values of x
1, 
x
2
, x
3
which satisfies all 
the equations in the linear system.
? A linear system can have a unique solution, more than one solutions or no solution.
Linear System
? Rewriting the above equation in form of matrix
AX = B
A = 
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
?? 3
?? 3
?? 3
, X = 
?? 1
?? 2
?? 3
, B = 
?? 1
?? 2
?? 3
? The matrix A is called the coefficient matrix and the block matrix [A B] , is the augmented 
matrix of the linear system. Type eq uati on h er e.
Augmented matrix : 
?? 1
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
?? 2
?? 3
?? 3
?? 3
?? 3
? If all the elements of B are zero then linear system is called Homogeneous otherwise Non-
Homogeneous.
Row Reduced Echelon Form of a 
Matrix
? A matrix C is said to be in the row reduced form if
1. The first non-zero entry in each row of C is 1
2. The column containing this 1 has all its other entries zero
e.g. A = 
1 1 3
0 3 2
0 0 1
B = 
1 0 1 - 2
0 1 2 3
0 0 4 1
0 0 - 1 - 2
C = 
1 - 1
0 2
Gauss Elimination Method
? Gaussian elimination is a method of solving a linear system AX = B (consisting of m 
equations in n unknowns) by bringing the augmented matrix
[A B] = 
?? 11 ??1 2 ? ??1 ?? ??1
??2 1 ??2 2 ? ??2 ?? ??2
? ? ? ? ?
????1 ????2 ? ?????? ????
? to an upper triangular form (or reduced row echelon form)
?? 11 ?? 12 ? ?? 1?? ?? 1
0 ?? 22 ? ?? 2?? ?? 2
? ? ? ? ?
0 0 ? ?? ???? ?? ?? ? This elimination process is also called the forward elimination method.
Page 5


Linear System
? Linear equation: ax
1
+ bx
2
= c
where, a, b, c = constants                 x
1, 
x
2
= variables
? A linear system consists of more than one linear equations.
? For example:- a
1
x
1
+ b
1
x
2 
+ c
1
x
3
= k
1
……………….(1)
a
2
x
1
+ b
2
x
2
+ c
2
x
2
= k
2
……………….(2)
a
3
x
1
+ b
3
x
2
+ c
3
x
3
= k
3
……………….(3)
? To find a solution of this system means to find the value/values of x
1, 
x
2
, x
3
which satisfies all 
the equations in the linear system.
? A linear system can have a unique solution, more than one solutions or no solution.
Linear System
? Rewriting the above equation in form of matrix
AX = B
A = 
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
?? 3
?? 3
?? 3
, X = 
?? 1
?? 2
?? 3
, B = 
?? 1
?? 2
?? 3
? The matrix A is called the coefficient matrix and the block matrix [A B] , is the augmented 
matrix of the linear system. Type eq uati on h er e.
Augmented matrix : 
?? 1
?? 1
?? 1
?? 1
?? 2
?? 2
?? 2
?? 2
?? 3
?? 3
?? 3
?? 3
? If all the elements of B are zero then linear system is called Homogeneous otherwise Non-
Homogeneous.
Row Reduced Echelon Form of a 
Matrix
? A matrix C is said to be in the row reduced form if
1. The first non-zero entry in each row of C is 1
2. The column containing this 1 has all its other entries zero
e.g. A = 
1 1 3
0 3 2
0 0 1
B = 
1 0 1 - 2
0 1 2 3
0 0 4 1
0 0 - 1 - 2
C = 
1 - 1
0 2
Gauss Elimination Method
? Gaussian elimination is a method of solving a linear system AX = B (consisting of m 
equations in n unknowns) by bringing the augmented matrix
[A B] = 
?? 11 ??1 2 ? ??1 ?? ??1
??2 1 ??2 2 ? ??2 ?? ??2
? ? ? ? ?
????1 ????2 ? ?????? ????
? to an upper triangular form (or reduced row echelon form)
?? 11 ?? 12 ? ?? 1?? ?? 1
0 ?? 22 ? ?? 2?? ?? 2
? ? ? ? ?
0 0 ? ?? ???? ?? ?? ? This elimination process is also called the forward elimination method.
Gauss Elimination Method
E.X. 1       y + z = 2
2x + 3z = 5
x + y + z = 5
Solution:- Augmented matrix =  
0 1 1 2
2 0 3 5
1 1 1 3
? Interchange 1
st
and 2
nd
equation
2 0 3 5
0 1 1 2
1 1 1 3