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For what set of values of the parameter lambda does the homogeneous system x + lambda*y - z = 0, lambda*x - y - z = 0, x + y - lambda*z = 0 admit a non-trivial solution?

1. exactly three values of lambda
2. infinitely many values of lambda
3. exactly one value of lambda
4. exactly two values of lambda

Correct answer: exactly one value of lambda

Solution

Setting the coefficient determinant to zero yields a cubic that has only one real root, so there is exactly one value of lambda.

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