9 The triangle ABC has vertices at \(\mathrm { A } ( 0,0 ) , \mathrm { B } ( 0,2 )\) and \(\mathrm { C } ( 4,1 )\). The matrix \(\left( \begin{array} { r r } 1 & - 2
3 & 0 \end{array} \right)\) represents a transformation T .
- The transformation \(T\) maps triangle \(A B C\) onto triangle \(A ^ { \prime } B ^ { \prime } C ^ { \prime }\). Find the coordinates of \(A ^ { \prime } , B ^ { \prime }\) and \(C ^ { \prime }\).
Triangle \(A ^ { \prime } B ^ { \prime } C ^ { \prime }\) is now mapped onto triangle \(A ^ { \prime \prime } B ^ { \prime \prime } C ^ { \prime \prime }\) using the matrix \(\mathbf { M } = \left( \begin{array} { l l } 4 & 0
0 & 2 \end{array} \right)\). - Describe fully the transformation represented by \(\mathbf { M }\).
- Triangle \(\mathrm { A } ^ { \prime \prime } \mathrm { B } ^ { \prime \prime } \mathrm { C } ^ { \prime \prime }\) is now mapped back onto ABC by a single transformation. Find the matrix representing this transformation.
- Calculate the area of \(A ^ { \prime \prime } B ^ { \prime \prime } C ^ { \prime \prime }\).