The real value

The system of homogeneous equations $\lambda x+y+z=0,-x+\lambda y+z=0$ and $-x-y+\lambda z=0$ has a non-zero solution or an infinite many solutions.

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

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

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

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

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

$\Rightarrow \lambda=0 \quad\left(\lambda^{2}+3 \neq 0\right.$ for any real value of $\left.\lambda\right)$

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 $\lambda$ for which the system of equations $\lambda x+y+z=0,-x+\lambda y+z=0,-x-y+\lambda z=0$ has a non-zero solution, is 0