Question

The set of all values of $\lambda$ for which the system of linear equations
$x-2y-2z=\lambda x$
$x+2y+z=\lambda y$
$-x-y=\lambda z$
has a non-trivial solution :

• A. is a singleton
• B. is an empty set
• C. contains exactly two elements
• D. contains more than two elements

Answer 1

The correct option is A is a singleton

For non-trivial solution, $\Delta =0$

$\left|\begin{array}{ccc}1-\lambda & -2& -2\\ 1& 2-\lambda & 1\\ -1& -1& -\lambda \end{array}\right|=0$

$R_2\to R_2+R_3$
$\left|\begin{array}{ccc}1-\lambda & -2& -2\\ 0& 1-\lambda & 1-\lambda \\ -1& -1& -\lambda \end{array}\right|=0$

$\Rightarrow (1-\lambda )\left|\begin{array}{ccc}1-\lambda & -2& -2\\ 0& 1& 1\\ -1& -1& -\lambda \end{array}\right|=0$

Expanding along $R_2$, we get
$(1-\lambda )({\lambda }^2-2\lambda +1)=0$
$\Rightarrow (1-\lambda )(\lambda -1{)}^2=0$
$\therefore \lambda =1$