Let $\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \ 0 & 1\end{array}\right]$ and $\mathrm{P}=\left[\begin{array}{cc}\cos \theta & -\sin \theta \ \sin \theta & \cos \theta\end{array}\right], \theta0$. If $\mathrm{B}=\mathrm{PAP}^{\mathrm{T}}, \mathrm{C}=\mathrm{P}^{\mathrm{T}} \mathrm{B}^{10} \mathrm{P}$ and the sum of the diagonal elements of $C$ is $\frac{m}{n}$, where $\operatorname{gcd}(m, n)=1$, then $m+n$ is : [JEE Main 2025, 28 Jan (Shift 2)]

Question

Let $\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \ 0 & 1\end{array}\right]$ and $\mathrm{P}=\left[\begin{array}{cc}\cos \theta & -\sin \theta \ \sin \theta & \cos \theta\end{array}\right], \theta>0$. If $\mathrm{B}=\mathrm{PAP}^{\mathrm{T}}, \mathrm{C}=\mathrm{P}^{\mathrm{T}} \mathrm{B}^{10} \mathrm{P}$ and the sum of the diagonal elements of $C$ is $\frac{m}{n}$, where $\operatorname{gcd}(m, n)=1$, then $m+n$ is : [JEE Main 2025, 28 Jan (Shift 2)]

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Let \(\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{array}\right]\) and \(\mathrm{P}=\l…

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Let \(\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} &amp; -2 \\ 0 &amp; 1\end{array}\right]\) and \(\mathrm{P}=\l…

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Let \(\mathrm{A}=\left[\begin{array}{cc}\frac{1}{\sqrt{2}} & -2 \\ 0 & 1\end{array}\right]\) and \(\mathrm{P}=\l…