4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { l l } \mathrm { a } & \mathrm { b } ^ { 2 }
\mathrm { c } ^ { 2 } & \mathrm { a } \end{array} \right)\), where \(a , b , c\) are real constants and \(b \neq 0\).

1. Show that \(\mathbf { M }\) does not represent a rotation about the origin.
2. Find the equations of the invariant lines, through the origin, of the transformation in the \(x - y\) plane represented by \(\mathbf { M }\).
   It is given that \(\mathbf { M }\) represents the sequence of two transformations in the \(x - y\) plane given by an enlargement, centre the origin, scale factor 5 followed by a shear, \(x\)-axis fixed, with \(( 0,1 )\) mapped to \(( 5,1 )\).
3. Find \(\mathbf { M }\).
4. The triangle \(D E F\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(P Q R\). Given that the area of triangle \(D E F\) is \(12 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(P Q R\).