Question

The set of all values of $\lambda$ for which the system of linear equations:
$2x_1-2x_2+x_3=\lambda x_1$
$2x_1-3x_2+2x_3=\lambda x_2$
$-x_1+2x_2=\lambda x_3$
has a non-trivial solution

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

Answer 1

Answer: (C)

contains two elements

Given set off equation may be written as,
$(2-\lambda )x_1-2x_2+x_3=0$
$2x_1-(3+\lambda )x_2+2x_3=0$
$-x_1+2x_2-\lambda x_3=0$

Now for non-trivial solution determinant of coefficient matrix should be zero.
$\Rightarrow \left|\begin{array}{ccc}2-\lambda & -2& 1\\ 2& -3-\lambda & 2\\ -1& 2& -\lambda \end{array}\right|=0$

Expanding along first column,
$(2-\lambda )({\lambda }^2+3\lambda -4)-2(2\lambda -2)-1(-4+3+\lambda )=0$
$\Rightarrow (\lambda -1{)}^2(\lambda +3)=0\Rightarrow \lambda =1,-3$

Thus set of $\lambda$ will contain two element $-3$ and $1$ for non-trivial solution.