Properties of Direction Cosine Matrix
Determinant of DCM is \(\pm 1\)
We know that,
\[\begin{equation} [C][C]^\intercal = [I_{3 \times 3}] \end{equation} \label{eq:c_identity}\]Taking determinant on both sides, we get,
\[\begin{equation} det([C][C]^\intercal) = det([I_{3 \times 3}]) = 1 \end{equation} \label{eq:det_equn}\]For any square matrices \(A\) and \(B\),
\[\begin{equation*} det(AB) = det(A)det(B). \end{equation*}\]Therefore,
\[\begin{equation} det(CC^\intercal) = det(C)det(C^\intercal) = 1. \end{equation}\]Since transposing a square matrix doesn’t change its determinant value, $$det(C) = det(C^\intercal).
Now, equation \eqref{eq:det_equn} can be written as,
\[\begin{equation} det(CC^\intercal) = [\mathrm{det}(C)]^{2} = 1 \end{equation}\]or
\[\begin{equation} \boxed{\mathrm{det}(C) = \pm 1}. \end{equation}\]It can also be shown that \(\mathrm{det}(C) = +1\), if the reference frame base vectors are right-handed.