    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 _____________.

Open in App
Solution

## 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. $\therefore ∆=\left|\begin{array}{ccc}\lambda & 1& 1\\ -1& \lambda & 1\\ -1& -1& \lambda \end{array}\right|=0$ $⇒\lambda \left({\lambda }^{2}+1\right)-1\left(-\lambda +1\right)+1\left(1+\lambda \right)=0$ $⇒{\lambda }^{3}+\lambda +\lambda -1+1+\lambda =0$ $⇒{\lambda }^{3}+3\lambda =0$ $⇒\lambda \left({\lambda }^{2}+3\right)=0$ $⇒\lambda =0$ (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__.  Suggest Corrections  0      Similar questions  Related Videos   System of Linear Equations
MATHEMATICS
Watch in App  Explore more