The system of linear equations [pmatrix x + y - z = 0 x - y - z = 0 x + y - z = 0 pmatrix …

Question

The system of linear equations

$$\begin{pmatrix} {x + \lambda y - z = 0}  \
   {\lambda x - y - z = 0}  \
   {x + y - \lambda z = 0}  \ \end{pmatrix} $$
has a non-trivial solution for :

Accepted Answer

For non-trivial solution, we have

$$\left| {\begin{pmatrix} 1 & \lambda  & { - 1}  \
   \lambda  & { - 1} & { - 1}  \
   1 & 1 & { - \lambda }  \

\end{pmatrix} } \right| = 0$$

$$ \Rightarrow 1(\lambda  + 1) - \lambda ( - {\lambda ^2} + 1) - 1(\lambda  + 1) = 0$$

$$ \Rightarrow \lambda ({\lambda ^2} - 1) = 0$$

$$ \Rightarrow \lambda  =  - 1,0,1$$

Solutions · 1