Question

34. The system of linear equations x+λy−z=0 λx−y−z=0 x+y−λz=0 has a non-trivial solution for :
(1) exactly one value of λ. (2) exactly two values of λ. (3) exactly three values of λ. (4) infinitely many values of λ.

Answer 1

dear student

$$\begin{gathered} thesystemwillhavenontrivialsolutionsif \\ ∆=\left|\begin{array}{ccc}1& \lambda & -1\\ \lambda & -1& -1\\ 1& 1& -\lambda \end{array}\right|=0 \\ (\lambda +1)-\lambda (-{\lambda }^2+1)-1(\lambda +1)=0 \\ {\lambda }^3-\lambda =0 \\ \lambda (\lambda +1)(\lambda -1)=0 \\ \lambda =0,\pm 1 \\ optionC \end{gathered}$$

regards