In part (i), the elements of each matrix should be expressed in exact numerical form.
(a) Write down the \(2 \times 2\) matrix that represents a rotation of \(210 ^ { \circ }\) anticlockwise about the origin.
(b) Write down the \(2 \times 2\) matrix that represents a stretch parallel to the \(y\)-axis with scale factor 5
The transformation \(T\) is a rotation of \(210 ^ { \circ }\) anticlockwise about the origin followed by a stretch parallel to the \(y\)-axis with scale factor 5
(c) Determine the \(2 \times 2\) matrix that represents \(T\)
$$\mathbf { M } = \left( \begin{array} { r r }
k & k + 3
- 5 & 1 - k
\end{array} \right) \quad \text { where } k \text { is a constant }$$
(a) Find det \(\mathbf { M }\), giving your answer in simplest form in terms of \(k\).
A closed shape \(R\) is transformed to a closed shape \(R ^ { \prime }\) by the transformation represented by the matrix \(\mathbf { M }\).
Given that the area of \(R\) is 2 square units and that the area of \(R ^ { \prime }\) is \(16 k\) square units,
(b) determine the possible values of \(k\).