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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 two values of λ.

Solution

The system has a non-trivial solution when the determinant of the coefficient matrix is zero, which leads to a characteristic polynomial in λ. This polynomial can yield exactly two distinct values of λ where the system has non-trivial solutions.

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