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MATH / Let ω be a nonreal root of x³=1, and let

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem

Let ω\omega be a nonreal root of x3=1,x^3 = 1, and let \[\mathbf{M} = \begin{pmatrix} -\omega^2 & - \omega \\ 1 & 0 \end{pmatrix}.\]Find the sum of the entries of M+M2+M3+…+M2009.\mathbf{M} + \mathbf{M}^2 + \mathbf{M}^3 + \dots + \mathbf{M}^{2009}.
Plain-text mathematical notation (without MathML)
Let ω be a nonreal root of x³=1, and let
\[\mathbf{M} = \begin{pmatrix} -\omega^2 & - \omega \\ 1 & 0 \end{pmatrix}.\]Find the sum of the entries of M+M²+M³+…+M²⁰⁰⁹.
Original LaTeX notation
Let $\omega$ be a nonreal root of $x^3 = 1,$ and let
\[\mathbf{M} = \begin{pmatrix} -\omega^2 & - \omega \\ 1 & 0 \end{pmatrix}.\]Find the sum of the entries of $\mathbf{M} + \mathbf{M}^2 + \mathbf{M}^3 + \dots + \mathbf{M}^{2009}.$

level

Level 5

type

Precalculus

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Source and history

task_id
ab189ca5-9ede-5222-a36a-722f5ff72783
task_key
precalculus--test--ab189ca5-9ede-5222-a36a-722f5ff72783
task_revision_id
1
config / split
precalculus / test

Official source

initial import