Question

If the system of homogeneous equations $ 2x-y+z=0,\,\,x-2y+z=0,\,\,\lambda \,\,x-y+2z=0 $ has infinitely many solutions, then

• A. $ \lambda =5 $
• B. $ \lambda =-5 $
• C. $ \lambda \ne \pm 5 $
• D. None of these

Answer 1

Answer: (A)

If given system of equations have infinitely many solutions, then $ \left| \begin{matrix} 2 & -1 & 1 \\ 1 & -2 & 1 \\ \lambda & -1 & 2 \\ \end{matrix} \right|=0 $
$ \Rightarrow $ $ 2(-4+1)+1(2-\lambda )+1(-1+2\lambda )=0 $
$ \Rightarrow $ $ -6+2-\lambda -1+2\lambda =0 $
$ \Rightarrow $ $ \lambda -5=0 $
$ \Rightarrow $ $ \lambda =5 $