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A system of equations is said to be inconsistent if
1. they have one solution 
2. they have no solution 
3. they have one or more solution 
4. none of these

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Correct Answer - Option 2 : they have no solution 

Concept:

The system of linear non-homogeneous equations in matrix form is written as:

A X = B

Let A be a 3 × 3 matrix

\({\rm{A}} = \left[ {\begin{array}{*{20}{c}} {{{\rm{a}}_{11}}}&{{{\rm{a}}_{12}}}&{{{\rm{a}}_{13}}}\\ {{{\rm{a}}_{21}}}&{{{\rm{a}}_{22}}}&{{{\rm{a}}_{23}}}\\ {{{\rm{a}}_{31}}}&{{{\rm{a}}_{32}}}&{{{\rm{a}}_{33}}} \end{array}} \right],{\rm{\;X}} = \left[ {\begin{array}{*{20}{c}} {{{\rm{x}}_1}}\\ {{{\rm{x}}_2}}\\ {{{\rm{x}}_3}} \end{array}} \right],{\rm{\;B}} = \left[ {\begin{array}{*{20}{c}} {{{\rm{b}}_1}}\\ {{{\rm{b}}_2}}\\ {{{\rm{b}}_3}} \end{array}} \right]\)

a11x1 + a12x2 + a13x3 = b1

a21x1 + a22x2 + a23x3 = b2

a31x1 + a32x2 + a33x3 = b3

The set of values x1, x2, x3 which satisfies the above equations are called the solutions of the system.

Consistent and Inconsistent equations/system:

If the equation has one or more than one solution then the equation is called consistent otherwise if there exists no solution then the equation is called inconsistent.

The augmented matrix K is defined as:

\({\rm{K\;}}\left[ {\begin{array}{*{20}{c}} {\rm{A}}&{\rm{B}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{{\rm{a}}_{11}}}&{{{\rm{a}}_{12}}}&{{{\rm{a}}_{13{\rm{\;}}}}}\\ {{{\rm{a}}_{21}}}&{{{\rm{a}}_{22}}}&{{{\rm{a}}_{23}}}\\ {{{\rm{a}}_{31}}}&{{{\rm{a}}_{32}}}&{{{\rm{a}}_{33}}} \end{array}{\rm{\;}}\begin{array}{*{20}{c}} {{{\rm{b}}_1}}\\ {{{\rm{b}}_2}}\\ {{{\rm{b}}_3}} \end{array}} \right]\)

Conditions for consistency and inconsistency:

Case 1 (Consistent equations).

If the rank of A = rank of K then only the system of equations is consistent.

Again, if the rank of A = rank of K = n (n is the number of unknown variables in the system) then the system has a unique solution, and if the rank of A = rank of K < n then the system has infinite solutions.

Case 2 (Inconsistent equations).

If the rank of A ≠ rank of K then the system has no solutions.

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