FP1 January 2009 Q10

10. \[\mathbf{A} = \begin{pmatrix} 3\sqrt{2} & 0 \\ 0 & 3\sqrt{2} \end{pmatrix},\quad \mathbf{B} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},\quad \mathbf{C} = \begin{pmatrix} \frac{1}{\sqrt{2}} & -\frac{1}{\sqrt{2}} \\[4pt] \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{pmatrix}\]

It is given that the matrix \(\mathbf{D} = \mathbf{CA}\), and that the matrix \(\mathbf{E} = \mathbf{DB}\).

The triangle \(ORS\) has vertices at the points with coordinates \((0,\ 0)\), \((-15,\ 15)\) and \((4,\ 21)\). This triangle is transformed onto the triangle \(OR'S'\) by the transformation described by \(\mathbf{E}\).