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The homogeneous system 4x + lambda*y + 2z = 0, 2x - y + z = 0, mu*x + 2y + 3z = 0 (lambda, mu real) has a non-trivial solution. Which statement must hold?

1. mu = 6, lambda is arbitrary
2. lambda = 2, mu is arbitrary
3. mu = 3, lambda is arbitrary
4. mu = -6, lambda is arbitrary

Correct answer: mu = 6, lambda is arbitrary

Solution

Setting the 3x3 determinant to zero makes the lambda terms cancel, leaving a condition that fixes mu = 6 with lambda free.

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