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)

exactly three values of $\lambda$

Answer 2

$x + \lambda y -z = 0$
$\lambda x - y - z = 0$
$x +y - \lambda z = 0$
Let's consider that, system of equation has a non trivial solution;
$\Rightarrow \ \begin{vmatrix}1&\lambda&-1\\\ \lambda&-1&-1\\\ 1&1&-\lambda\end{vmatrix}=0$
$\lambda + 1 - \lambda \\{ - \lambda^2 + 1 \\} - (\lambda + 1 ) = 0$
$\lambda (\lambda^2 - 1 ) = 0$
$\lambda = 0 , 1 ,-1$
Hence, the system of equation has non-trivial solution for exactly three values of $\lambda$.