Question
The set of all values of $$\lambda $$ for which the system of linear equations:
$$\eqalign{
& 2{x_1} - 2{x_2} + {x_3} = \lambda {x_1} \cr
& 2{x_1} - 3{x_2} + 2{x_3} = \lambda {x_2} \cr
& - {x_1} + 2{x_2} = \lambda {x_3} \cr} $$
has a non-trivial solution
A.
contains two elements
B.
contains more than two elements
C.
is an empty set
D.
is a singleton
Answer :
contains two elements
Solution :
\[\left. \begin{array}{l}
\,\,\,2{x_1} - 2{x_2} + {x_3} = \lambda {x_1}\\
2{x_1} - 3{x_2} + 2{x_3} = \lambda {x_2}\\
\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - {x_1} + 2{x_2} = \lambda {x_3}
\end{array} \right\}\]
$$\eqalign{
& \Rightarrow \,\,\left( {2 - \lambda } \right){x_1} - 2{x_2} + {x_3} = 0 \cr
& \,\,\,\,\,\,\,2{x_1} - \left( {3 + \lambda } \right){x_2} + 2{x_3} = 0 \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - {x_1} + 2{x_2} - \lambda {x_3} = 0 \cr} $$
For non - trivial solution, $$\Delta = 0$$
\[{\rm{i}}{\rm{.e}}{\rm{.,}}\,\,\left| \begin{array}{l}
2 - \lambda \,\,\,\,\,\,\,\,\,\,\, - 2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,1\\
\,\,\,\,2\,\,\,\,\,\,\,\,\,\, - \left( {3 + \lambda } \right)\,\,\,\,\,\,\,\,\,\,\,\,2\\
\,\, - 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,2\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - \lambda
\end{array} \right| = 0\]
$$\eqalign{
& \Rightarrow \,\,\left( {2 - \lambda } \right)\left[ {\lambda \left( {3 + \lambda } \right) - 4} \right] + 2\left[ { - 2\lambda + 2} \right] + 1\left[ {4 - \left( {3 + \lambda } \right)} \right] = 0 \cr
& \Rightarrow \,\,{\lambda ^3} + {\lambda ^2} - 5\lambda + 3 = 0 \cr
& \Rightarrow \,\,\lambda = 1,1,3 \cr} $$
Hence, $$\lambda $$ has 2 values.