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?
Solutions · 1
Official solution
\[\begin{aligned}
& \left|\[\begin{array}{|ccccccc|}1 & 1 & 1 \\ 1 & 2 & \lambda^2 \\ 1 & 3 & \lambda\end{array}\end{aligned}\]\right|=0 \\
\& \Rightarrow 2 \lambda\^2-\lambda-1=0 \\
\& \lambda=1,-\frac{1}{2} \\
\& \left|\[1231λ2λ59μ\]\right|=0 \Rightarrow \mu=13
\]
Infinite solution \[\lambda=1 \& \mu=13\]
For unique \[\operatorname{sol}^{\mathrm{n}} \lambda \neq 1\]
For no \[\operatorname{sol}^{\mathrm{n}} \lambda=1 \& \mu \neq 13\]
If \[\lambda \neq 1\] and \[\mu \neq 13\]
Considering the case when \[\lambda=-\frac{1}{2}\] and \[\mu \neq 13\] this will generate no solution case