Matrix Reloaded vs 2018

Geometry Level 3

If A = ( 3 2 0 − 1 2 0 − 1 0 1 2 0 3 2 ) A = \begin{pmatrix} \frac{\sqrt{3}}{2} & 0 & \frac{-1}{2} \\ 0 & -1 & 0 \\ \frac{1}{2} & 0 & \frac{\sqrt{3}}{2} \end{pmatrix}

then, what is the product of the diagonal elements of A 2018 A^{2018} ?

Bonus.- Enter A 2018 A^{2018}


The answer is 0.25.

Guillermo Templado by Guillermo Templado , 46, Spain

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1 solution

By induction, ∀ n ∈ N , A n = ( cos ⁡ ( n π 6 ) 0 − sin ⁡ ( n π 6 ) 0 ( − 1 ) n 0 sin ⁡ ( n π 6 ) 0 cos ⁡ ( n π 6 ) ) ⇒ A 2018 = ( 1 / 2 0 − 3 / 2 0 1 0 3 / 2 0 1 / 2 ) . . . \displaystyle \forall \space n \in \mathbb{N}, \space A^{n} = \begin{pmatrix} \cos (\frac{n \pi}{6}) & 0 & - \sin (\frac{n \pi}{6}) \\ 0 & (-1)^n & 0 \\ \sin (\frac{n \pi}{6}) & 0 & \cos (\frac{n \pi}{6}) \end{pmatrix} \Rightarrow A^{2018} = \begin{pmatrix} 1/2 & 0 & -\sqrt{3}/2 \\ 0 & 1 & 0 \\ \sqrt{3}/2 & 0 & 1/2 \end{pmatrix} ...

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