The matrix \(\mathbf{R}\) represents an anticlockwise rotation through angle \(\varphi\) (\(0^\circ \leqslant \varphi < 360^\circ\)) in two dimensions, and the matrix \(\mathbf{R} + \mathbf{I}\) also represents a rotation in two dimensions. Determine the possible values of \(\varphi\) and deduce that \(\mathbf{R}^3 = \mathbf{I}\).
Let \(\mathbf{S}\) be a real matrix with \(\mathbf{S}^3 = \mathbf{I}\), but \(\mathbf{S} \neq \mathbf{I}\).
Show that \(\det(\mathbf{S}) = 1\).
Given that
\[
\mathbf{S} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
\]
show that \(\mathbf{S}^2 = (a+d)\mathbf{S} - \mathbf{I}\).
Hence prove that \(a + d = -1\).
Let \(\mathbf{S}\) be a real \(2 \times 2\) matrix.
Show that if \(\mathbf{S}^3 = \mathbf{I}\) and \(\mathbf{S} + \mathbf{I}\) represents a rotation, then \(\mathbf{S}\) also represents a rotation.
What are the possible angles of the rotation represented by \(\mathbf{S}\)?