$$\mathbf { A } = \left( \begin{array} { l l }
3 & 0
0 & 1
\end{array} \right) \quad \mathbf { B } = \left( \begin{array} { r r }
- \frac { \sqrt { 3 } } { 2 } & \frac { 1 } { 2 }
- \frac { 1 } { 2 } & - \frac { \sqrt { 3 } } { 2 }
\end{array} \right)$$
(a) Describe fully the single transformation represented by the matrix \(\mathbf { A }\).
(b) Describe fully the single transformation represented by the matrix \(\mathbf { B }\).
The transformation represented by \(\mathbf { A }\) followed by the transformation represented by \(\mathbf { B }\) is equivalent to the transformation represented by the matrix \(\mathbf { C }\).
(c) Find \(\mathbf { C }\).
\(\mathbf { M } = \left( \begin{array} { c c } 2 k + 5 & - 4 1 & k \end{array} \right)\), where \(k\) is a real number.
Show that \(\operatorname { det } \mathbf { M } \neq 0\) for all values of \(k\).