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Problem 19: For what values of the $\lambda$ will the following equations fail to have unique solution

Problem 19: For what values of the $\lambda$ will the following equations fail to have unique solution

\begin{align*} 3x-y+\lambda z =1\\ 2x+y+z=2\\ x+2y-\lambda z = -1? \end{align*} will the equations have any solution for these values of $\lambda$

Solution:

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

\[\left[\begin{array}{c|c} A & B \end{array} \right] = \begin{align*} \left[\begin{array}{rrr|r} 3 & -1 & \lambda & 1 \\ 2 & 1 & 1& 2 \\ 1 & 2 & -\lambda & -1 \\ \end{array}\right] \end{align*}\] \[\xrightarrow{R_1\leftrightarrow R_3 } \left[\begin{array}{rrr|r} 1 & 2 & -\lambda & -1 \\ 2 & 1 & 1& 2 \\ 3 & -1 & \lambda & 1 \\ \end{array}\right]\] \[\xrightarrow{R2\rightarrow R_2-2R_1,R3\rightarrow R_3-3R_1} \left[\begin{array}{rrr|r} 1 & 2 & -\lambda & -1 \\ 0 & -3 & 1+2\lambda & 4 \\ 0 & -7 & 4\lambda & 4 \\ \end{array}\right]\]

Therefore the coefficient matrix will be non-singular if and only if, \begin{align*} (-3)4\lambda - (1+2\lambda)(-7) \ne 0\\ -12\lambda +7 + 14 \lambda \ne 0\\ \lambda \ne -\frac{7}{2} \end{align*} Thus the given system will have a uniuqe solution if $\lambda \ne -\frac{7}{2}$.

In case If $\lambda = -\frac{7}{2}$, \begin{align*} \left[\begin{array}{rrr|r} 1 & 2 & \frac{7}{2} & -1 \\ 0 & -3 & -6 & 4 \\ 0 & -7 & -14 & 4 \\ \end{array}\right] \end{align*} \[\xrightarrow{R3\rightarrow R_3-\frac{7}{3}R2} \left[\begin{array}{rrr|r} 1 & 2 & \frac{7}{2} & -1 \\ 0 & -3 & -6 & 4 \\ 0 & 0 & 0 & -frac{16}{3} \\ \end{array}\right]\]

showing that given equations are inconsistent in this case. Thus if $\lambda = -\frac{7}{2}$ , no solution exists.

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