Write down the matrix \(\mathbf { C }\) which represents a stretch, scale factor 2 , in the \(x\)-direction.
The matrix \(\mathbf { D }\) is given by \(\mathbf { D } = \left( \begin{array} { l l } 1 & 3 0 & 1 \end{array} \right)\). Describe fully the geometrical transformation represented by \(\mathbf { D }\).
The matrix \(\mathbf { M }\) represents the combined effect of the transformation represented by \(\mathbf { C }\) followed by the transformation represented by \(\mathbf { D }\). Show that
$$\mathbf { M } = \left( \begin{array} { l l }
2 & 3
0 & 1
\end{array} \right)$$
Prove by induction that \(\mathbf { M } ^ { n } = \left( \begin{array} { c c } 2 ^ { n } & 3 \left( 2 ^ { n } - 1 \right) 0 & 1 \end{array} \right)\), for all positive integers \(n\).