Question
माना $\lambda$ के सभी वास्तविक मानों, जिनके लिए समीकरण निकाय $ \lambda x+y+z=1 $ $ x+\lambda y+z=1 $ $ x+y+\lambda z=1$ असंगत है, का समुच्चय $\mathrm{S}$ है, तब $\sum_{\lambda \in S}\left(|\lambda|^2+|\lambda|\right)$ का मान है:
$(\lambda+2)\left|\begin{array}{lll}1 & 1 & 1 \\ 1 & \lambda & 1 \\ 1 & 1 & \lambda\end{array}\right|=0$
$(\lambda+2)\left[1\left(\lambda^2-1\right)-1(\lambda-1)+(1-\lambda)\right]=0$
$(\lambda+2)\left[\left(\lambda^2-2 \lambda+1\right)=0\right.$
$(\lambda+2)(\lambda-1)^2=0 \Rightarrow \lambda=-2, \lambda=1$
at $\lambda=1$ system has infinite solution, for inconsistent $\lambda=-2$
so $\sum\left(|-2|^2+|-2|\right)=6$
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