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NDA 2017 Maths · Mathematics · Matrices · medium

If A=beginpmatrixcos θ&sin θ\\-sin θ&cos θendpmatrix with θ=(π)/(9), then the (1,1) entry of A^6 is

Single correct option

If A=(cos⁡θsin⁡θ−sin⁡θcos⁡θ)A=\begin{pmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{pmatrix} with θ=π9\theta=\dfrac\pi9, then the (1,1)(1,1) entry of A6A^6 is
  1. A. 12\dfrac12
  2. B. −32-\dfrac{\sqrt3}2
  3. C. −12-\dfrac12
  4. D. 32\dfrac{\sqrt3}{2}
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Answer: (C) −12-\dfrac12

By induction An=(cos⁡nθsin⁡nθ−sin⁡nθcos⁡nθ)A^n=\begin{pmatrix}\cos n\theta&\sin n\theta\\-\sin n\theta&\cos n\theta\end{pmatrix} (rotation matrices). With nθ=6π9=2π3n\theta=\frac{6\pi}9=\frac{2\pi}3 the entry is cos⁡2π3=−12\cos\frac{2\pi}3=-\frac12.
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Original question written in the NDA 2017 Maths pattern and checked twice for the correct answer. Not an official question paper. How we write and check questions · Corrections policy · Report a mistake