2x2 matrix

Algebra Level 4

Calculate A 2009 A^{2009} when a 2x2 matrix A is given as A = ( cos ⁡ 3 6 ∘ − sin ⁡ 3 6 ∘ sin ⁡ 3 6 ∘ cos ⁡ 3 6 ∘ ) . A=\begin{pmatrix} \cos36^{\circ} & -\sin36^{\circ} \\ \sin36^{\circ} & \cos36^{\circ} \end{pmatrix}.

A 2 A^{2} A A A − 1 A^{-1} I I

Krishna Singh by Krishna Singh , 22, India

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1 solution

Varsha Dani
Sep 27, 2018

A A is the matrix for rotation by 3 6 ∘ 36^{\circ} , so A 10 = I A^{10} = I . It follows that A 2009 = ( A 10 ) 200 A 9 = A 9 = A − 1 A^{2009} = (A^{10})^{200} A^9 = A^9 = A^{-1} .

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