SYSTEM OF EQUATION & CRITERIAN FOR CONSISTENCY
(A) GAUSS - JORDAN METHOD
Q1. x + y + z = 6
x – y + z = 2
2x + y – z = 1
Q2. x + 2y + 3z = 2
2x + 4y + 5z = 3
3x + 5y + 6z = 4
A X = B Þ A–1 A X = A–1 B
X = A–1 B = {(Adj A ) /|A|} .B
Note :
(1) If |A| ¹ 0 ,system is consistent having unique solution
(2) If |A| ¹ 0 & (adj A) . B ¹ Null matrix,
system is consistent having unique non-trivial solution.
(3) If |A| ¹ 0 & (adj A) . B = 0 (Null matrix) ,
system is consistent having trivial solution.
(4) If | A | = 0 ,
(B) GAUSS - ELIMINATION METHOD
(SOLVING SYSTEM OF EQUATION BY ELEMENTRY ROW OPERATION
Q1 x + y + z = 4
3x + 2y – 2z = 1
5x + y – z = 4
The coefficient of this system is
Augmented matrix is
Elementry row operations that can be performed are
(i) Interchanging two rows
(ii) Multiplying any row by a non-zero scalar
(iii) Subtracting (adding) elements of any row by equimultiples of corresponding elements of any other row.
These operations are performed on Augmented matrix
If the system has infinite solutions the equivalent augmented matrix will be of the form
If the system is inconsistent the equivalent augmented matrix will be of the form
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