Christmas Blast!!!

Algebra Level 5

If

\(\displaystyle X_r = \begin{pmatrix} cos\left(\dfrac{\pi}{2^{r}} \right) & sin\left(\dfrac{\pi}{2^{r}} \right) \\ -sin\left(\dfrac{\pi}{2^{r}} \right) & cos\left( \dfrac{\pi}{2^{r}} \right) \end{pmatrix}\)

And

\(\displaystyle K = \lim_{r \to \infty } X_1.X_2.X_3 \ldots X_r\)

Then find \(100K\).

Clarification

  • \(X_r\) is a \(2 \times 2\) ordered matrix

  • K(matrix) can be represented as a real number


Merry Christmas to all


This question is flagged because it is not clear how "A matrix can be represented as a real number".

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