Let α be a root of the equation x 2 + x + 1 = 0 and the matrix A = 1 3 1 1 1 1 α α 2 1 α 2 α 4 , then the matrix A 31 is equal to

Question

Let α be a root of the equation x 2 + x + 1 = 0 and the matrix A = 1 3 1 1 1 1 α α 2 1 α 2 α 4 , then the matrix A 31 is equal to, A 3, I 3, A 2, A

MARKS App · Accepted Answer

Here, α = ω , ω 2 where ω is complex cube root of unity. A 2 = 1 3 1 1 1 1 ω ω 2 1 ω 2 ω 1 1 1 1 ω ω 2 1 ω 2 ω = 1 0 0 0 0 1 0 1 0 ⇒ A 4 = 1 0 0 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 = 1 0 0 0 1 0 0 0 1 ⇒ A 31 = A 28 × A 3 = I × A 3 = A 3

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