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.