Let $ A = \begin{bmatrix} 1 & \sqrt{2} \ -2 & 0 & 1 \end{bmatrix} $ and $ P = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} $, $ \theta 0 $. If $ B = P A P^T $, $ C = P^T B P $, 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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Let $ A = \begin{bmatrix} 1 & \sqrt{2} \ -2 & 0 & 1 \end{bmatrix} $ and $ P = \begin{bmatrix} \cos \theta & -\sin \theta \ \sin \theta & \cos \theta \end{bmatrix} $, $ \theta > 0 $. If $ B = P A P^T $, $ C = P^T B P $, 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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Step 1: Calculate $ B $ We are given matrices $ A $, $ P $, and $ B = P A P^T $. First, multiply $ P $ and $ A $: $$ B = P A P^T $$ Since $ P^T P = I $, we know from matrix multiplication rules that: $$ B = P A P^T = P \left( P^T B P \right) = C $$ Step 2: Use the Formula to Calculate the Diagonal Sum Through matrix computations, the sum of the diagonal elements of $ C $ is: $$ \frac{1}{32} + 1 = \frac{33}{32} $$ Thus, $$ m + n = 65 $$

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The Correct Option is B

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