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If A=[ cosπ...

Question

If $A = [\begin{matrix} cos \frac{π}{3} & sin \frac{π}{3} sin \frac{π}{3} & cos \frac{π}{3} \end{matrix}]$ then $A^{3} = ?$

A

⎡ ⎢ ⎣ \begin{matrix} \frac{5}{4} & \frac{3 \sqrt{3}}{4} \frac{3 \sqrt{3}}{4} & \frac{5}{4} \end{matrix} ⎤ ⎥ ⎦

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B

⎡ ⎢ ⎣ \begin{matrix} \frac{5}{2} & \frac{3 \sqrt{3}}{2} \frac{3 \sqrt{3}}{2} & \frac{5}{2} \end{matrix} ⎤ ⎥ ⎦

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C

- \frac{1}{2} [\begin{matrix} 1 & 0 0 & 1 \end{matrix}]

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D

[\begin{matrix} 0 & 0 1 & 1 \end{matrix}]

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Solution

The correct option is A $⎡ ⎢ ⎣ \begin{matrix} \frac{5}{4} & \frac{3 \sqrt{3}}{4} \frac{3 \sqrt{3}}{4} & \frac{5}{4} \end{matrix} ⎤ ⎥ ⎦$
Given,

$A = ⎡ ⎢ ⎣ \begin{matrix} cos (\frac{π}{3}) & sin (\frac{π}{3}) sin (\frac{π}{3}) & cos (\frac{π}{3}) \end{matrix} ⎤ ⎥ ⎦$

$∴ A^{3} = {⎡ ⎢ ⎣ \begin{matrix} cos (\frac{π}{3}) & sin (\frac{π}{3}) sin (\frac{π}{3}) & cos (\frac{π}{3}) \end{matrix} ⎤ ⎥ ⎦}^{3}$

$A^{2} = A \times A = ⎡ ⎢ ⎣ \begin{matrix} cos (\frac{π}{3}) & sin (\frac{π}{3}) sin (\frac{π}{3}) & cos (\frac{π}{3}) \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} cos (\frac{π}{3}) & sin (\frac{π}{3}) sin (\frac{π}{3}) & cos (\frac{π}{3}) \end{matrix} ⎤ ⎥ ⎦$

$= ⎡ ⎢ ⎣ \begin{matrix} cos (\frac{π}{3}) cos (\frac{π}{3}) + sin (\frac{π}{3}) sin (\frac{π}{3}) & cos (\frac{π}{3}) sin (\frac{π}{3}) + sin (\frac{π}{3}) cos (\frac{π}{3}) sin (\frac{π}{3}) cos (\frac{π}{3}) + cos (\frac{π}{3}) sin (\frac{π}{3}) & sin (\frac{π}{3}) sin (\frac{π}{3}) + cos (\frac{π}{3}) cos (\frac{π}{3}) \end{matrix} ⎤ ⎥ ⎦$

$= ⎡ ⎢ ⎣ \begin{matrix} 1 & \frac{\sqrt{3}}{2} \frac{\sqrt{3}}{2} & 1 \end{matrix} ⎤ ⎥ ⎦$

Now, $A^{3} = ⎡ ⎢ ⎣ \begin{matrix} 1 & \frac{\sqrt{3}}{2} \frac{\sqrt{3}}{2} & 1 \end{matrix} ⎤ ⎥ ⎦ ⎡ ⎢ ⎣ \begin{matrix} cos (\frac{π}{3}) & sin (\frac{π}{3}) sin (\frac{π}{3}) & cos (\frac{π}{3}) \end{matrix} ⎤ ⎥ ⎦$

$= ⎡ ⎢ ⎢ ⎣ \begin{matrix} 1 \cdot cos (\frac{π}{3}) + \frac{\sqrt{3}}{2} sin (\frac{π}{3}) & 1 \cdot sin (\frac{π}{3}) + \frac{\sqrt{3}}{2} cos (\frac{π}{3}) \frac{\sqrt{3}}{2} cos (\frac{π}{3}) + 1 \cdot sin (\frac{π}{3}) & \frac{\sqrt{3}}{2} sin (\frac{π}{3}) + 1 \cdot cos (\frac{π}{3}) \end{matrix} ⎤ ⎥ ⎥ ⎦$

$= ⎡ ⎢ ⎣ \begin{matrix} \frac{5}{4} & \frac{3 \sqrt{3}}{4} \frac{3 \sqrt{3}}{4} & \frac{5}{4} \end{matrix} ⎤ ⎥ ⎦$

$∴ A^{3} = ⎡ ⎢ ⎣ \begin{matrix} \frac{5}{4} & \frac{3 \sqrt{3}}{4} \frac{3 \sqrt{3}}{4} & \frac{5}{4} \end{matrix} ⎤ ⎥ ⎦$

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