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If alpha, beta, gamma are the values of x satisfying det[[x,1,1],[1,x,1],[1,1,x]] = 0, then which of the following holds?

1. alpha + beta + gamma = 0
2. alpha² + beta² + gamma² = 6
3. alpha³ + beta³ + gamma³ = -6
4. alpha⁴ + beta⁴ + gamma⁴ = 18

Correct answer: alpha + beta + gamma = 0

Solution

The determinant equals (x-1)²(x+2), so the roots are 1, 1, -2 with sum 0. (All four listed relations actually hold for these roots.)

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