Mathematics Question from JEE Main 2025 · 28 January, Shift 2

Let $A = \left[ \begin{array}{cc} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{array} \right]$ and $P = \left[ \begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array} \right], \theta > 0.$ If $B = P A P^T$, $C = P^T B^{10} P$, and the sum of the diagonal elements of $C$ is $\frac{m}{n}$, where $\gcd(m, n) = 1$, then $m + n$ is: