Discussion
What is the maximum number of free independent real parameters specifying an \(n\)-dimensional orthogonal matrix?
Answer: C
The number is \(\frac{n(n - 1)}{2}\).
An \(n \times n\) real matrix has \(n^{2}\) entries. Orthogonality, \(O^{T}O = I\), imposes one condition for each entry of the symmetric matrix \(O^{T}O\), that is \(\frac{n(n + 1)}{2}\) independent conditions. The free parameters are
\[ n^{2} - \frac{n(n + 1)}{2} = \frac{n(n - 1)}{2} \]
For \(n = 2\) this gives 1 (a rotation angle) and for \(n = 3\) it gives 3 (the Euler angles), as expected. \(\frac{n(n + 1)}{2}\) is the number of constraints, not of free parameters, and \((n - 1)^{2}\) and \(n(n - 2)\) fail the \(n = 2\) and \(n = 3\) checks.
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