4. You are given that \(\boldsymbol { M } = \left( \begin{array} { c c } 1 & - \sqrt { 3 }
\sqrt { 3 } & 1 \end{array} \right)\).
- Show that \(\boldsymbol { M }\) is non-singular.
The hexagon \(R\) is transformed to the hexagon \(S\) by the transformation represented by the matrix \(\boldsymbol { M }\).
Given that the area of hexagon \(R\) is 5 square units,
- find the area of hexagon \(S\).
The matrix \(\boldsymbol { M }\) represents an enlargement, with centre \(( 0,0 )\) and scale factor \(k\), where \(k > 0\), followed by a rotation anti-clockwise through an angle \(\theta\) about \(( 0,0 )\).
- Find the value of \(k\).
- Find the value of \(\theta\).
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