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Let ω be a complex cube root of unity with ω ≠ 1, and let H = [[ω, 0], [0, ω]]. Then the value of H raised to the 70th power is:

1. 0
2. −H
3. H²
4. H

Correct answer: H

Solution

Since H is a diagonal matrix with the same eigenvalue ω on the diagonal, raising H to any power will yield the same matrix scaled by ω raised to that power. Since ω is a cube root of unity, ω³ = 1, and thus ω⁷⁰ = ω^(3*23 + 1) = ω, leading to H⁷⁰ = H.

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