9 Matrix \(\mathbf { R }\) is given by \(\mathbf { R } = \left( \begin{array} { c c c } a & 0 & - b
0 & 1 & 0
b & 0 & a \end{array} \right)\) where \(a\) and \(b\) are constants.
- Find \(\mathbf { R } ^ { 2 }\) in terms of \(a\) and \(b\).
The constants \(a\) and \(b\) are given by \(a = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } + 1 )\) and \(b = \frac { \sqrt { 2 } } { 4 } ( \sqrt { 3 } - 1 )\).
- By determining exact expressions for \(a b\) and \(a ^ { 2 } - b ^ { 2 }\) and using the result from part (a), show that \(\mathbf { R } ^ { 2 } = k \left( \begin{array} { c c c } \sqrt { 3 } & 0 & - 1
0 & 2 & 0
1 & 0 & \sqrt { 3 } \end{array} \right)\) where \(k\) is a real number whose value is to be determined. - Find \(\mathbf { R } ^ { 6 } , \mathbf { R } ^ { 12 }\) and \(\mathbf { R } ^ { 24 }\).
- Describe fully the transformation represented by \(\mathbf { R }\).
\section*{END OF QUESTION PAPER}