4 The transformations \(T _ { A }\) and \(T _ { B }\) are represented by the matrices \(\mathbf { A }\) and \(\mathbf { B }\) respectively, where \(\mathbf { A } = \left( \begin{array} { r r } 0 & - 1
1 & 0 \end{array} \right)\) and \(\mathbf { B } = \left( \begin{array} { l l } 0 & 1
1 & 0 \end{array} \right)\)

1. Describe geometrically the single transformation consisting of \(T _ { A }\) followed by \(T _ { B }\).
2. By considering the transformation \(\mathrm { T } _ { \mathrm { A } }\), determine the matrix \(\mathrm { A } ^ { 423 }\). The transformation \(\mathrm { T } _ { \mathrm { C } }\) is represented by the matrix \(\mathbf { C }\), where \(\mathbf { C } = \left( \begin{array} { l l } \frac { 1 } { 2 } & 0
   0 & \frac { 1 } { 3 } \end{array} \right)\). The region \(R\) is defined by the set of points \(( x , y )\) satisfying the inequality \(x ^ { 2 } + y ^ { 2 } \leqslant 36\). The region \(R ^ { \prime }\) is defined as the image of \(R\) under \(\mathrm { T } _ { \mathrm { C } }\).
3. 1. Find the exact area of the region \(R ^ { \prime }\).
   2. Sketch the region \(R ^ { \prime }\), specifying all the points where the boundary of \(R ^ { \prime }\) intersects the coordinate axes.