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

Answer 1

Answer: (B)

is a singleton

$x-2y-2z=\lambda x$

$(1-\lambda )x-2y-2z=0.....\left(1\right)$

$x+2y+z=\lambda y$

$x+(2-\lambda )y+z=0$

$-x-y=\lambda z$

$x+y+\lambda z=0$

Since the system of given linear equations has a non-trivial solution.

Therefore,

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

$\left(\right(1-\lambda \left)\right(\lambda (2-\lambda )-1)-2(\lambda -1)-2(1-(2-\lambda ))=0$

${(1-\lambda )}^3=0$

$\Rightarrow \lambda =1$

Thus, the set of values of $\lambda$ contains exactly one element.

Hence the correct answer is $\left(B\right)$.