Question

The system of linear equations
$x + \lambda y - z = 0$
$\lambda x -y - z = 0$
$x + y - \lambda z = 0$
has a non-trivial solution for :

• A. infinitely many values of $\lambda$
• B. exactly one value of $\lambda$
• C. exactly two values of $\lambda$
• D. exactly three values of $\lambda$

Answer 1

Answer: (D)

$x + \lambda y -z = 0 $
$\lambda x - y - z = 0 $
$x +y - \lambda z = 0$
For non-trivial solution $\Rightarrow \Delta = 0$
$\Rightarrow \ \begin{vmatrix}1&\lambda&-1\\ \lambda&-1&-1\\ 1&1&-\lambda\end{vmatrix}=0$
$\lambda + 1 - \lambda \{ - \lambda^2 + 1 \} - (\lambda + 1 ) = 0$
$\lambda (\lambda^2 - 1 ) = 0 $
$\lambda = 0 , \pm 1 $
Exactly $3$ values of $\lambda$.