Question

Number of values of ${}^{\prime }{\lambda }^{\prime }$ for which the system of linear equations $\lambda x+y+z=1$, $x+\lambda y+z=0$, $x+y+\lambda z=0$ has infinitely many solutions are

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Answer 1

Answer: (D)

$3$

$Givenequationarex+\lambda y-z=0\lambda x-y-z=0x+y-\lambda z=0$

$Thegivensystemwillhaveanon-trivialsolutiononlyiftheerminatsofthecoefficientmatrixiszero\left|\begin{array}{ccc}1& \lambda & -1\\ \lambda & -1& -1\\ 1& 1& -\lambda \end{array}\right|=0\Rightarrow 1\left(\lambda +1\right)+\lambda \left(-{\lambda }^2+1\right)-1\left(\lambda +1\right)=0\Rightarrow {\lambda }^3-\lambda =0\Rightarrow \lambda \left({\lambda }^2+1\right)=0\therefore \lambda =0and\lambda =\pm i$

$So,givensystemoflinearequationhasanon-trivialsolutionforexcatlythreesolutionof\lambda .$

Hence, this is the answer.