Question
The system of equations
$$\eqalign{
& 2x - y + z = 0 \cr
& x - 2y + z = 0 \cr
& \lambda x - y + 2z = 0 \cr} $$
has infinite number of nontrivial solutions for
A.
$$\lambda = 1$$
B.
$$\lambda = 5$$
C.
$$\lambda = - 5$$
D.
no real value of $$\lambda $$
Answer :
$$\lambda = 5$$
Solution :
If given system of equations have infinitely many solutions, then
\[\begin{array}{l}
\left| \begin{array}{l}
2\,\,\,\, - 1\,\,\,\,\,1\\
1\,\,\,\, - 2\,\,\,\,\,1\\
\lambda \,\,\,\, - 1\,\,\,\,\,2
\end{array} \right| = 0\\
\Rightarrow 2\left( { - 4 + 1} \right) + 1\left( {2 - \lambda } \right) + 1\left( { - 1 + 2\lambda } \right) = 0\\
\Rightarrow - 6 + 2 - \lambda - 1 + 2\lambda = 0\\
\Rightarrow \lambda - 5 = 0\\
\Rightarrow \lambda = 5
\end{array}\]