Consider the system of linear equations \(x+y+z=5\), \(x+2 y+\lambda^2 z=9, x+3 y+\lambda z=\mu\), where \(\lambda, \mu …

\(\left|\begin{matrix}1 & 1 & 1 \\ 1 & 2 & {\lambda }^{2} \\ 1 & 3 & \lambda \end{matrix}\right|=0\\ \begin{matrix} & \Rightarrow 2{\lambda }^{2}−\lambda −1=0 \\ & \lambda =1,−\frac{1}{2}\end{matrix}\\ \left|\begin{matrix}1 & 1 & 5 \\ 2 & {\lambda }^{2} & 9 \\ 3 & \lambda & \mu \end{matrix}\right|=0\Rightarrow \mu =13\)

Infinite solution \(\lambda =1&\mu =13\)
For unique solution \(\lambda \neq 1\), \(\mu \neq 13\)
For no solution\(\lambda =1&\mu \neq 13\)
Considering the case when\(\lambda =-\frac{1}{2}\) and \(\mu \neq 13\) this will generate no solution case.