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Question
Answer the following question:
If A = `[(1, omega),(omega^2, 1)]`, B = `[(omega^2, 1),(1, omega)]`, where ω is a complex cube root of unity, then show that AB + BA + A −2B is a null matrix
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Solution
ω is a complex cube root of unity
∴ ω3 = 1 and ω4 = ω3·ω = ω ...(1)
Also 1 + ω + ω2 = 0 ...(2)
AB = `[(1, omega),(omega^2, 1)] [(omega^2, 1),(1, omega)]`
= `[(omega^2 + omega,1 + omega^2),(omega^4 + 1, omega^2 + omega)]`
BA = `[(omega^2, 1),(1, omega)] [(1, omega),(omega^2, 1)]`
= `[(omega^2 + omega^2, omega^3 + 1),(1 + omega^3, omega + omega)]`
= `[(2omega^2, 2),(2, 2omega)]` ...[∵ ω3 = 1]
∴ AB + BA + A – 2B
= `[(omega^2 + omega, 1 + omega^2),(omega^4 + 1, omega^2 + omega)] + [(2omega^2, 2),(2, 2omega)] + [(1, omega),(omega^2, 1)] -2[(omega^2, 1),(1, omega)]`
= `[(omega^2 + omega, 1 + omega^2),(omega^4 + 1, omega^2 + omega)] + [(2omega^2, 2),(2, 2omega)] + [(1, omega),(omega^2, 1)] - [(2omega^2, 2),(2, 2omega)]`
= `[(omega^2 + omega + 2omega^2 + 1 - 2omega^2, 1 + omega^2 + 2 + omega - 2),(omega^4 + 1 + 2 + omega^2 - 2,omega^2 + omega + 2omega + 1 - 2omega)]`
= `[(1 + omega + omega^2, 1 + omega + omega^2),(1 + omega + omega^2, 1 + omega + omega^2)]` ...[∵ ω4 = ω]
= `[(0, 0),(0, 0)]` ...[By (2)]
which is a null matrix.
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