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+1 vote
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in Matrices & determinants by (41.3k points)

If x, y, z are not all zero and if ax + by + cz = 0, bx + cy + az = 0, cx + ay + bz = 0, prove that x:y:z = 1:1:1 or 1:ω :ω2 or 1:ω2:ω, where ω is the complex cube roots of unity.

1 Answer

+1 vote
by (41.5k points)
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Best answer

For non-trivial solution,

⇒ (a + b + c) (a2 + b2 + c2 - ab - bc - ca) = 0 (1)

⇒ (a + b + c) (a + ωb + ω2c)(a + ω2b + ωc) = 0 (2)

Now, by using Cramer’s rule

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