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Let a, b, c be real numbers with a + b + c not equal to 0. If the homogeneous system ax + by + cz = 0, bx + cy + az = 0, cx + ay + bz = 0 has a non-trivial solution, which of the following must be true?

1. a + c - b = 0
2. a = b = c
3. a + b - c = 0
4. None of these

Correct answer: a = b = c

Solution

The determinant of the circulant matrix is det = a³+b³+c³-3abc = (a+b+c)(a²+b²+c²-ab-bc-ca). For det=0 and a+b+c≠0, we need a²+b²+c²-ab-bc-ca = 0. This equals (1/2)[(a-b)²+(b-c)²+(c-a)²] = 0, which requires a = b = c.

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