Non Homogeneous Equations
If there is n equation & n unknown you will not find the unique solution always.
x+y=10 there is no solution for these equation.
Working rule for finding the solution of linear non homogeneous equation (AX=B)
- Suppose the coefficient matrix 'A' is of type M*N
- Write the augmented matrix [AB] and reduce to echlon form by applying only row transformations.
- This echelon form will enable us to know the ranks of the augmented matrix [AB] and the coefficient matrix A. Then the following different cases arise
Case1: Rank A < Rank [AB] ⇒AX=B is inconsistant ⇒ No solution
Case2: Rank A = Rank [AB]=r(say)
a) r=n ⇒ unique solution
b) r<n ⇒ n-r linearly independnt solution
⇒n-r variable will be assigned random values
⇒infinite no of solutions
therefor ,unique solution
z=5 , y=3, x=1