Trace, product of matricesHard

Matrix Power and Diagonal Trace

Let $A = \left[ {\begin{array}{*{20}{c}} {\frac{1}{{\sqrt 2 }}}&{ - 2}\\[0.9em] 0&1 \end{array}} \right]$ and $P = \left[ {\begin{array}{*{20}{c}} {\cos \theta }&{-\sin \theta }\\[0.9em] {\sin \theta }&{\cos \theta } \end{array}} \right]$ , θ > 0. If B = PAP^T, C = P^TB¹⁰P and the sum of the diagonal elements of C is $\frac{m}{n}$ , where gcd(m, n) = 1, then m + n is :

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