Consider the system of linear equations $x+y+z=5, x+2 y+\lambda^2 z=9, x+3 y+\lambda z=\mu$, where $\lambda, \mu \in \mathbb{R}$. Then, which of the following statement is NOT correct?

<p>$$\begin{aligned} & \left|\begin{array}{ccc} 1 & 1 & 1 \\ 1 & 2 & \lambda^2 \\ 1 & 3 & \lambda \end{array}\right|=0 \\ & \Rightarrow 2 \lambda^2-\lambda-1=0 \\ & \lambda=1,-\frac{1}{2} \\ & \left|\begin{array}{ccc} 1 & 1 & 5 \\ 2 & \lambda^2 & 9 \\ 3 & \lambda & \mu \end{array}\right|=0 \Rightarrow \mu=13 \end{aligned}$$</p> <p>Infinite solution $\lambda=1 \& \mu=13$</p> <p>For unique $\operatorname{sol}^{\mathrm{n}} \lambda \neq 1$</p> <p>For no $\operatorname{sol}^{\mathrm{n}} \lambda=1 \& \mu \neq 13$</p> <p>If $\lambda \neq 1$ and $\mu \neq 13$</p> <p>Considering the case when $\lambda=-\frac{1}{2}$ and $\mu \neq 13$ this will generate no solution case</p>