Problem 18: Investigate for what values of $\lambda, \mu$ the equations

Problem 18: Investigate for what values of $\lambda, \mu$ the equations

\begin{align*} x+y+z=6\\ x+2y+3z=10\\ x+2y+\lambda z = \mu \end{align*} have i. no solution, ii. a unique solution. iii. an infinite number of solutions.

Solution:

Consider the augmented matrix of this system and apply row operations.

\[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 1 & 1 & 1& 6 \\ 1 & 2 & 3& 10 \\ 1 & 2 & \lambda& \mu \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R2\rightarrow R_2-R_1,R3\rightarrow R_3-R_1 } \left[\begin{array}{rrr|r} 1 & 1 & 1& 6 \\ 0 & 1 & 2& 4 \\ 0 & 1 & \lambda-1& \mu-6 \\ \end{array}\right]\] \[\xrightarrow{R3\rightarrow R_3-R_2} \left[\begin{array}{rrr|r} 1 & 1 & 1& 6 \\ 0 & 1 & 2& 4 \\ 0 & 0 & \lambda-3& \mu-10 \\ \end{array}\right]\]

Case i: If $\lambda = 3$ and $\mu \ne 10$, \begin{align*} \left[\begin{array}{rrr|r} 1 & 1 & 1& 6 \\ 0 & 1 & 2& 4 \\ 0 & 0 & 0& \mu-10 \\ \end{array}\right] \end{align*} rank of coefficient matrix = 2,rank of augmented matrix = 3. Since rank of coefficient matrix $\ne$ rank of augmented matrix, no solution possible.

Case ii: If $\lambda \ne 3$, \begin{align*} \left[\begin{array}{rrr|r} 1 & 1 & 1& 6 \\ 0 & 1 & 2& 4 \\ 0 & 0 & \lambda-3& \mu-10 \\ \end{array}\right] \end{align*}

the rank of the coefficient matrix = rank of augmented matrix = 3 = number of unknowns. This implies that the system is consistent and the solution is unique for any value of $\mu$.

Case iii: If $\lambda = 3$ and $\mu = 10$,\begin{align*} \left[\begin{array}{rrr|r} 1 & 1 & 1& 6 \\ 0 & 1 & 2& 4 \\ 0 & 0 & 0& 0 \\ \end{array}\right] \end{align*}

rank of coefficient matrix = rank of augmented matrix = 2 $<$ number of unknowns. The system is consistent and it has infinitely many solutions.