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The system of linear equations x + λy − z = 0 λx − y − z = 0 x + y − λz = 0 has a non-trivial solution for:

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

Correct answer: exactly three values of λ.

Solution

Non-trivial solutions need det=0. Expanding the coefficient determinant gives lambda^3 - lambda = lambda(lambda-1)(lambda+1) = 0, so lambda = 0, 1, -1 -> exactly three values.

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