Question

Let \( A = \begin{bmatrix} \frac{1}{\sqrt{2}} & -2 \\ 0 & 1 \end{bmatrix} \) 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:

• A. 258
• B. 65
• C. 127
• D. 2049

Hint:

When dealing with matrix transformations and diagonal sums, use matrix multiplication and properties of orthogonal matrices to simplify calculations.

Answer 1

The Correct Option is B

Given matrices \( A \), \( P \), and \( B = P A P^T \), the objective is to determine the sum of the diagonal elements of matrix \( C \).
Step 1: Compute \( B \). The initial operation involves multiplying \( P \) by \( A \): \[ B = P A P^T \] With the condition \( P^T P = I \) and applying matrix multiplication principles, we derive: \[ B = P A P^T = P \left( P^T B P \right) = C \] 
Step 2: Apply the formula for diagonal sum calculation. Based on matrix operations, the sum of the diagonal elements of \( C \) is determined to be: \[ \frac{1}{32} + 1 = \frac{33}{32} \] Consequently, \( m + n = 65 \).