9 The matrices \(\mathbf { P } = \left( \begin{array} { r r } 0 & 1
- 1 & 0 \end{array} \right)\) and \(\mathbf { Q } = \left( \begin{array} { l l } 2 & 0
0 & 1 \end{array} \right)\) represent transformations \(P\) and \(Q\) respectively.
- Describe fully the transformations P and Q .
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{e449d411-aaa9-4167-aa9c-c28d31446d52-4_625_849_470_648}
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\caption{Fig. 9}
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Fig. 9 shows triangle T with vertices \(\mathrm { A } ( 2,0 ) , \mathrm { B } ( 1,2 )\) and \(\mathrm { C } ( 3,1 )\).
Triangle T is transformed first by transformation P , then by transformation Q . - Find the single matrix that represents this composite transformation.
- This composite transformation maps triangle T onto triangle \(\mathrm { T } ^ { \prime }\), with vertices \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\). Calculate the coordinates of \(\mathrm { A } ^ { \prime } , \mathrm { B } ^ { \prime }\) and \(\mathrm { C } ^ { \prime }\).
T' is reflected in the line \(y = - x\) to give a new triangle, T".
- Find the matrix \(\mathbf { R }\) that represents reflection in the line \(y = - x\).
- A single transformation maps \(\mathrm { T } ^ { \prime \prime }\) onto the original triangle, T . Find the matrix representing this transformation.