5 In this question you must show detailed reasoning.
You are given that the matrix \(\mathbf { M } = \left( \begin{array} { c c c } \frac { 1 } { 2 } & - \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { 2 }
\frac { 1 } { \sqrt { 2 } } & 0 & - \frac { 1 } { \sqrt { 2 } }
\frac { 1 } { 2 } & \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { 2 } \end{array} \right)\) represents a rotation in 3-D space.
- Explain why it follows that \(\mathbf { M }\) has 1 as an eigenvalue.
- Find a vector equation for the axis of the rotation.
- Show that the characteristic equation of \(\mathbf { M }\) can be written as
$$\lambda ^ { 3 } - \lambda ^ { 2 } + \lambda - 1 = 0 .$$
- Find the smallest positive integer \(n\) such that \(\mathbf { M } ^ { n } = \mathbf { I }\).
- Find the magnitude of the angle of the rotation which \(\mathbf { M }\) represents. Give your reasoning.
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