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$S$ denote the set of all real values of $\lambda$ such that the system of equations $\lambda x + y + z =1$ ; $x +\lambda y + z =1$ ; $x + y +\lambda z =1$ is inconsistent, then $\sum_{\lambda \in S}\left(|\lambda|^2+|\lambda|\right)$ is equal to
$2$
$12$
$4$
$6$
Solution
$\left|\begin{array}{lll}\lambda & 1 & 1 \\ 1 & \lambda & 1 \\ 1 & 1 & \lambda\end{array}\right|=0$
$(\lambda+2)\left|\begin{array}{lll}1 & 1 & 1 \\ 1 & \lambda & 1 \\ 1 & 1 & \lambda\end{array}\right|=0$
$(\lambda+2)\left[1\left(\lambda^2-1\right)-1(\lambda-1)+(1-\lambda)\right]=0$
$(\lambda+2)\left[\left(\lambda^2-2 \lambda+1\right)=0\right.$
$(\lambda+2)(\lambda-1)^2=0 \Rightarrow \lambda=-2, \lambda=1$
at $\lambda=1$ system has infinite solution, for inconsistent $\lambda=-2$
so $\sum\left(|-2|^2+|-2|\right)=6$
