Investigate for what values of \(\lambda\) and \(\mu\) the system \(x+y+z=6,\ x+2y+3z=10,\ x+2y+\lambda z=\mu\) has (i) no solution, (ii) a unique solution, (iii) infinitely many solutions. (asked in MU Dec 2024, June 2025 KT and Dec 2025)

Answer: (i) No solution: \(\lambda=3,\ \mu\neq10\). (ii) Unique solution: \(\lambda\neq3\) (any \(\mu\)). (iii) Infinitely many: \(\lambda=3,\ \mu=10\).

Augmented matrix \([A\,|\,B]=\left[\begin{array}{ccc|c}1&1&1&6\\1&2&3&10\\1&2&\lambda&\mu\end{array}\right]\).

\(R_2\to R_2-R_1,\ R_3\to R_3-R_1\): \(\left[\begin{array}{ccc|c}1&1&1&6\\0&1&2&4\\0&1&\lambda-1&\mu-6\end{array}\right]\)

\(R_3\to R_3-R_2\): \(\left[\begin{array}{ccc|c}1&1&1&6\\0&1&2&4\\0&0&\lambda-3&\mu-10\end{array}\right]\)

• (i) No solution when \(\rho(A)\neq\rho(A|B)\): \(\lambda-3=0\) but \(\mu-10\neq0\) gives \(\rho(A)=2,\ \rho(A|B)=3\). So \(\lambda=3,\ \mu\neq10\).
• (ii) Unique solution when \(\rho(A)=\rho(A|B)=3\) (number of unknowns): \(\lambda\neq3\), \(\mu\) any value.
• (iii) Infinitely many solutions when \(\rho(A)=\rho(A|B)=2<3\): \(\lambda=3\) and \(\mu=10\). Then \(y+2z=4,\ x+y+z=6\); with \(z=k\): \(y=4-2k,\ x=2+k\).