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

(b) Write $\mathbf { M } ^ { - 1 }$ as the product of two matrices, neither of which is $\mathbf { I }$.\\

(c) Find the equations of the invariant lines, through the origin, of the transformation represented by $\mathbf { M }$.\\

(d) 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 $A B C$.\\

\hfill \mbox{\textit{CAIE Further Paper 1 2024 Q4}}