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Let α be a root of the equation x² + x + 1 = 0 and the matrix A = (1/√3) · [[1, 1, 1], [1, α, α²], [1, α², α⁴]], then the matrix A³¹ is equal to:

1. A
2. I₃
3. A²
4. A³

Correct answer: A³

Solution

The matrix A is constructed using the roots of the polynomial, which are related to the cube roots of unity. Since the roots satisfy the relation α³ = 1, raising A to the power of 3 results in a matrix that cycles back to its original form, thus A³ is equivalent to A.

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