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The number of real values of lambda for ...

The number of real values of `lambda` for which the system of equations `lambdax + y + z = 0, x-lambday-z = 0, x + y - lambdaz=0` will have nontrivial solution is

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The correct Answer is:
`lambda =0`
Given system `lambda x +y +z =0, -x +lambday +z =0`
`"and" -x-y +lambdaz =0`
will have non-zero solution, if
`[{:(lambda,1, 1),(-1,lambda, 1), (-1, -1,lambda):}] = 0`
`rArr (lambda^(2) +1)-1(-lambda +1) + 1(1+lambda) =0`
`rArr lambda^(3) + lambda + lambda -1+1 +lambda =0`
`rArr lambda^(3) + 3lambda =0`
`rArr lambda(lambda^(2) +3)=0 rArr lambda =0`
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