JEE Main & Advanced Mathematics Determinants & Matrices Homogeneous and Non-homogeneous Systems of Linear Equations

A system of equations \[AX=B\] is called a homogeneous system if \[B=O\]. If \[B\ne O\], it is called a non-homogeneous system of equations.

e.g., \[2x+5y=0\]

\[3x-2y=0\]

is a homogeneous system of linear equations whereas the system of equations given by

e.g., \[2x+3y=5\]

\[x+y=2\]

is a non-homogeneous system of linear equations.

(1) Solution of Non-homogeneous system of linear equations

(i) Matrix method : If \[AX=B\], then \[X={{A}^{-1}}B\] gives a unique solution, provided A is non-singular.

But if A is a singular matrix i.e.,  if \[|A|=0\], then the system of equation \[AX=B\] may be consistent with infinitely many solutions or it may be inconsistent.

(ii) Rank method for solution of Non-Homogeneous system \[AX=B\]

(a) Write down A, B

(b) Write the augmented matrix \[[A:B]\]

(c) Reduce the augmented matrix to Echelon form by using elementary row operations.

(d) Find the number of non-zero rows in A and \[[A:B]\] to find the ranks of A and \[[A:B]\] respectively.

(e) If \[\rho (A)\ne \rho (A:B),\] then the system is inconsistent.

(f) \[\rho (A)=\rho (A:B)=\] the number of unknowns, then the system has a unique solution.

If \[\rho (A)=\rho (A:B)<\] number of unknowns, then the system has an infinite number of solutions.

(2) Solutions of a homogeneous system of linear equations : Let \[AX=O\] be a homogeneous system of 3 linear equations in 3 unknowns.

(a) Write the given system of equations in the form \[AX=O\] and write A.

(b) Find \[|A|\].

(c) If \[|A|\ne 0\], then the system is consistent and \[x=y=z=0\] is the unique solution.

(d)  If \[|A|=0\], then the systems of equations has infinitely many solutions. In order to find that put \[z=K\] (any real number) and solve any two equations for \[x\] and \[y\] so obtained with \[z=K\] give a solution of the given system of equations.