4 The matrix \(\mathbf { M }\) is given by \(\mathbf { M } = \left( \begin{array} { c c } \frac { 1 } { 2 } & - \frac { 1 } { 2 } \sqrt { 3 }
\frac { 1 } { 2 } \sqrt { 3 } & \frac { 1 } { 2 } \end{array} \right) \left( \begin{array} { c c } 14 & 0
0 & 1 \end{array} \right)\).

1. The matrix \(\mathbf { M }\) represents a sequence of two geometrical transformations in the \(x - y\) plane. Give full details of each transformation, and make clear the order in which they are applied.
2. Write \(\mathbf { M } ^ { - 1 }\) as the product of two matrices, neither of which is \(\mathbf { I }\).
3. Find the equations of the invariant lines, through the origin, of the transformation represented by \(\mathbf { M }\).
4. The triangle \(A B C\) in the \(x - y\) plane is transformed by \(\mathbf { M }\) onto triangle \(D E F\). Given that the area of triangle \(D E F\) is \(28 \mathrm {~cm} ^ { 2 }\), find the area of triangle \(A B C\).