The values of λ for which the following system of equations has uniqu…

$\lambda x+3 y-z=1$      ......(1)

$x+2 y+z=2$      ......(2)

$-\lambda x+y+2 z=-1$      ......(3)

so  $A=\left[\begin{array}{ccc}\lambda & 3 & -1 \\ 1 & 2 & 1 \\ -\lambda & 1 & 2\end{array}\right]$

for unique solution

$|A| \neq 0$

so $|A|=\left|\begin{array}{ccc}\lambda & 3 & -1 \\ 1 & 2 & 1 \\ -\lambda & 1 & 2\end{array}\right|$

so $|A|=\lambda\left|\begin{array}{ll}2 & 1 \\ 1 & 2\end{array}\right|-3\left|\begin{array}{cc}1 & 1 \\ -\lambda & 2\end{array}\right|-1\left|\begin{array}{cc}1 & 2 \\ -\lambda & 1\end{array}\right|$

upon solving

$|A|=-2 \lambda-7 \neq 0$

$\Rightarrow 2 \lambda \neq-7$

$\lambda \neq \frac{-7}{2}$