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Let A be a square matrix such that both (A + I/2) and (A - I/2) are orthogonal matrices (where I is the identity matrix). Which of the following must be true?

1. A is symmetric
2. A is skew-symmetric
3. A² = 3I/4
4. A² = -3I/4

Correct answer: A is skew-symmetric

Solution

Let P = A + I/2 and Q = A - I/2, both orthogonal. P*P^T = I => (A+I/2)(A^T+I/2) = I => A*A^T + A/2 + A^T/2 + I/4 = I. Q*Q^T = I => (A-I/2)(A^T-I/2) = I => A*A^T - A/2 - A^T/2 + I/4 = I. Subtracting: (A + A^T) = 0 => A^T = -A => A is skew-symmetric. Adding the two equations: 2*A*A^T + I/2 = 2I => A*A^T = 3I/4. Also since A is skew-symmetric, A^T = -A, so A*(-A) = 3I/4 => -A² = 3I/4 => A² = -3I/4.

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