Question

The equations $\lambda x-y=2,2x-3y=-\lambda ,3x-2y=-1$ are consistent for

• A. $\lambda =-4$
• B. $\lambda =-1,4$
• C. $\lambda =-1$
• D. $\lambda =1,-4$

Answer 1

The correct option is B $\lambda =-1,4$

$\left|\begin{array}{ccc}\lambda & -1& -2\\ 2& -3& \lambda \\ 3& -2& 1\end{array}\right|=0$
$\Rightarrow \lambda (-3+2\lambda )+1(2-3\lambda )-2(-4+9)=0\Rightarrow 2{\lambda }^2-6\lambda -8=0\Rightarrow {\lambda }^2-3\lambda -4=0\Rightarrow (\lambda -4)(\lambda +1)=0\therefore \lambda =-1,4$