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The system of linear equation $x+\lambda y-z=\alpha,  hx-y-z=\alpha, x+y-\lambda z=0$

has a non-trivial solution for 

Answer:

Explanation:

Given a system of linear equation is $x+\lambda y-z=\alpha,  hx-y-z=\alpha, x+y-\lambda z=0$

  Note that given system will have a non trivial solution only if determined  of coefficient matris is zero, i.e,

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

$\Rightarrow \lambda+1+\lambda^{2}-\lambda -\lambda-1=0$

$\Rightarrow \lambda+1+\lambda^{2}-\lambda -\lambda-1=0$

$\Rightarrow \lambda^{3}-\lambda=0=\lambda(\lambda^{2}-1)=0$

$\Rightarrow \lambda=0 or \lambda=\pm 1$

Hence , given system of linear equation has a non trivial solution for exactly three values of λ