The transformation represented by the \(2 \times 2\) matrix \(\mathbf { P }\) is an anticlockwise rotation about the origin through 45 degrees.
Write down the matrix \(\mathbf { P }\), giving the exact numerical value of each element.
$$\mathbf { Q } = \left( \begin{array} { c c }
k \sqrt { 2 } & 0
0 & k \sqrt { 2 }
\end{array} \right) \text {, where } k \text { is a constant and } k > 0$$
Describe fully the single geometrical transformation represented by the matrix \(\mathbf { Q }\).
The combined transformation represented by the matrix \(\mathbf { P Q }\) transforms the rhombus \(R _ { 1 }\) onto the rhombus \(R _ { 2 }\).
The area of the rhombus \(R _ { 1 }\) is 6 and the area of the rhombus \(R _ { 2 }\) is 147