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

Homogeneous and Non-homogeneous Systems of Linear Equations

Category : JEE Main & Advanced

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.

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