| \[x-2y-2z=\lambda x\] |
| \[x+2y+z=\lambda y\] |
| \[-x-y=\lambda z\] |
| has a non-trivial solution |
A) is an empty set
B) contains exactly two elements
C) is a singleton
D) contains more than two elements
Correct Answer: C
Solution :
The given system of linear equations is \[(1-\lambda )x-2y-2z=0\] \[x+(2-\lambda )y+z=0\] \[-x-y\lambda z=0\] Since, it has a non-trivial solution. \[\therefore \]\[\left| \begin{matrix} 1-\lambda & -2 & -2 \\ 1 & 2-\lambda & 1 \\ -1 & -1 & -\lambda \\ \end{matrix} \right|=0\] \[\Rightarrow \]\[(1-\lambda )[(2-\lambda )(-\lambda )+1]+2(-\lambda +1)\]\[-2(-1+2-\lambda )=0\] \[\Rightarrow \]\[(1-\lambda )[-2\lambda +{{\lambda }^{2}}+1+2-2]=0\] \[\Rightarrow \]\[(1-\lambda ){{(1-\lambda )}^{2}}=0\Rightarrow {{(1-\lambda )}^{3}}=0\Rightarrow \lambda =1\]
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