Consider a non-homogeneous system of linear equations an over-determined system [CE: GATE – 2005]

Consider a non-homogeneous system of linear equations representing mathematically an over-determined system. Such a system will be [CE: GATE – 2005]
(a) consistent having a unique solution (b) consistent having many solutions
(c) inconsistent having a unique solution (d) Inconsistent having no solution


System of Linear Equations - Consistent, Inconsistent, Number of Solutions and Number of Linearly Independent Solutions ( Rank of [A] and [A | b], Rank of coefficient matrix and augmented matrix)

In this lecture we consider system of linear equations ( nonhomogeneous and homogeneous ) where number or equations and number of unknown are different. In matrix form such system of equations can be written as

A x = b

where
A is coefficient matrix of size m x n. Here m is number of equations and n is number of unknowns
x is n x 1 is vector of unknown variables
b is n x 1 vector of constants

if b is not equal to zero then such system of equations is called non homogeneous system of equations.

If b is equal to zero then such system of equation is called homogeneous system of equations.


Nonhomogeneous system of equations ( b is not equal to zero )

(i) Number of equations is equal to number of unkowns ( m = n )
This case be analysed using the determinant method. We can also use the rank method that we will learn in this lecture.
(ii) Number of equations is less than number of unknowns ( m is less than n )
This is called underdetermined system. Such system is always consistent and has infinite number of solutions.
(iii) Number of equations is more than number of unknowns ( m is greater then n )
Such system is called overdetermined system. If all the equations are linearly independent then such system of equations is inconsistent. In general, if the number of linearly independent equations is more than number of unknown then the system is inconsistent.

In the general case where number of equations is different from number of unknowns, the coefficient matrix A will not be a square matrix so we cannot use determinant concept to check whether a given system of equation is consistent or inconsistent ( nonconsistent ). So here learn about the rank method to check whether the given system of equations is consistent or not.

A system of equations will be inconsistent (nonconsistent) if rank( [A] ) is not equal to rank( [A | b] ) .

A system of equations will be consistent if rank( [A] ) = rank( [A | b] ) .

Conceptually rank [A] denotes number of constrained variables and rank [A | b] denotes number of linearly independent equations. So if number of constrained variables is equal to number of linearly independent equations then system of equations will be consistent.


If rank( [A] ) = rank( [A | b] ) = n then there will be unique solution.

If rank( [A] ) = rank( [A | b] ) = r less than n then there will be infinitely many solutions.

If rank( [A] ) = rank( [A | b] ) = r then there will be (n-r) free variables, i.e., variables that can take arbitrary values and still satisfy given system of equations, and r constrained variables, i.e., variables whose values need to be determined by solving r linearly independent equations. Free variables will be solved in terms of constants and free variables.


Homogenous system of equation ( b is equal to zero )

As b is equal to zero the augmented matrix will be same as coefficient matrix so rank( [A] ) = rank( [A | b] ) always.

Here rank( [A] ) denotes number of constrained variable, i.e., variables whose values need to be determined by solving the r linearly independent equations.

rank( [A | b] ) denotes number of linearly independent equations.

So if number of constrained variables is equal to number of linearly independent equations then system of equations will be consistent.

If rank( [A] ) = rank( [A | b] ) = n then there will be unique solution. This solution will be the trivial solutions, i.e. solution vector will be a zero vectors. In other words all the variables will be zero.

If rank( [A] ) = rank( [A | b] ) = r less than n then there will be infinitely many solutions. In this case there will be nontrivial (nonzero ) solutions also.

If rank( [A] ) = rank( [A | b] ) = r then there will be (n-r) free variables, i.e., variables that can take arbitrary values and still satisfy given system of equations, and r constrained variables, i.e., variables whose values need to be determined by solving r linearly independent equations.

In this case there will be (n-r) linearly independent solutions and all other solutions ( infinitely many ) can be written as linear combination of these (n-r) solutions.


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Alpha Academy, Udaipur

Minakshi Porwal (9460189461)

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