Correct Answer - Option 2 :
Concept:
Consider the system of 'm' linear equations with 'n' unknown.
a11 x1 + a12 x2 + … + a1n xn = b1
a21 x1 + a22 x2 + … + a2n xn = b2
…
am1 x1 + am2 x2 + … + amn xn = bm
To determine whether the above system of equations is consistent or not, we need to find the rank of the following matrix.
\(C = \left[ {\begin{array}{*{20}{c}} {{a_{11}}}&{{a_{12}}}& \ldots &{{a_{1n}}}&{b_1}\\ {{a_{21}}}&{{a_{22}}}& \ldots &{{a_{2n}}}&{b_2}\\ \ldots & \ldots & \ldots & \ldots \\ {{a_{m1}}}&{{a_{m2}}}& \ldots &{{a_{mn}}}&{b_m} \end{array}} \right]\)
C is the augmented matrix of the given system of equations.
Rank of A = Rank of C = no of unknown (consistent and unique solution)
Rank of A = Rank of C < no of unknown (inconsistent and infinite solution)
Rank of A < Rank of C (inconsistent and no solution)
Calculation:
Given:
x + y = 2
2x + 2y = 5
\(C=\begin{bmatrix} 1 & 1 &2 \\ 2& 2 &5 \end{bmatrix}\)
R2' → R2 - 2R1
\(C=\begin{bmatrix} 1 & 1 &2 \\ 0& 0 &1\end{bmatrix}\)
rank of C = 2 and rank of A = 1
i.e. Rank of A < Rank of C
∴ the given matrix is inconsistent and has no solution.