Question

The real value of λ for which the system of equations λx + y + z = 0, -x + λy + z = 0, - x - y + λz = 0 has a non-zero solution, is _____________.

Answer 1

The system of homogeneous equations λx + y + z = 0, −x + λy + z = 0 and −x − y + λz = 0 has a non-zero solution or an infinite many solutions.

$\therefore ∆=\left|\begin{array}{ccc}\lambda & 1& 1\\ -1& \lambda & 1\\ -1& -1& \lambda \end{array}\right|=0$

$\Rightarrow \lambda \left({\lambda }^2+1\right)-1\left(-\lambda +1\right)+1\left(1+\lambda \right)=0$

$\Rightarrow {\lambda }^3+\lambda +\lambda -1+1+\lambda =0$

$\Rightarrow {\lambda }^3+3\lambda =0$

$\Rightarrow \lambda \left({\lambda }^2+3\right)=0$

$\Rightarrow \lambda =0$ (λ² + 3 ≠ 0 for any real value of λ)

Thus, the real value of λ for which the given system of homogeneous equations has a non-zero solution is 0.

The real value of λ for which the system of equations λx + y + z = 0, −x + λy + z = 0, −x − y + λz = 0 has a non-zero solution, is __0__.