Correct Answer - Option 2 : λ = 6, μ ≠ 20
Concept:
The number of solutions can be determined by finding out the rank of the Augmented matrix and the rank of the Coefficient matrix.
- If rank(Augmented matrix) = rank(Coefficient matrix) = no. of variables then no of solutions = 1.
- If rank(Augmented matrix) ≠ rank(Coefficient matrix) then no of solutions = 0.
- If rank(Augmented matrix) = rank(Coefficient matrix) < no. of variables, no of solutions = infinite.
Calculation:
The augmented matrix for the system of equations is
\(\left[ {A{\rm{|}}B} \right] = \left[ {\left. {\begin{array}{*{20}{c}} 1&1&1\\ 1&4&6\\ 1&4&\lambda \end{array}} \right|\begin{array}{*{20}{c}} 6\\ {20}\\ \mu \end{array}} \right]\)
Performing: R3 → R3 – R2
\(\left[ {A{\rm{|}}B} \right] = \left[ {\left. {\begin{array}{*{20}{c}} 1&1&1\\ 1&4&6\\ 0&0&{\lambda - 6} \end{array}} \right|\begin{array}{*{20}{c}} 6\\ {20}\\ {\mu - 20} \end{array}} \right]\) …
If λ = 6 and μ ≠ 20 then
Rank (A | B) = 3 and Rank (A) = 2
∵ Rank (A | B) ≠ Rank (A)
∴ Given the system of equations has no solution for λ = 6 and μ ≠ 20