Question
The system of linear equations
$$\eqalign{
& x + \lambda y - z = 0 \cr
& \lambda x - y - z = 0 \cr
& x + y - \lambda z = 0 \cr} $$
has a non-trivial solution for:
A.
exactly two values of $$\lambda .$$
B.
exactly three values of $$\lambda .$$
C.
infinitely many values of $$\lambda .$$
D.
exactly one value of $$\lambda .$$
Answer :
exactly three values of $$\lambda .$$
Solution :
For trivial solution,
\[\left| \begin{array}{l}
1\,\,\,\,\,\,\,\,\lambda \,\,\,\,\,\, - 1\\
\lambda \,\,\,\, - 1\,\,\,\,\,\, - 1\\
1\,\,\,\,\,\,\,\,\,1\,\,\,\,\,\, - \lambda
\end{array} \right| = 0\]
$$\eqalign{
& \Rightarrow \, - \lambda \left( {\lambda + 1} \right)\left( {\lambda - 1} \right) = 0 \cr
& \Rightarrow \,\,\lambda = 0, + 1, - 1 \cr} $$