4.
$$\mathbf { A } = \left( \begin{array} { c c }
- \frac { 1 } { \sqrt { 2 } } & \frac { 1 } { \sqrt { 2 } }
- \frac { 1 } { \sqrt { 2 } } & - \frac { 1 } { \sqrt { 2 } }
\end{array} \right)$$
- Describe fully the single geometrical transformation represented by the matrix \(\mathbf { A }\).
- Hence find the smallest positive integer value of \(n\) for which
$$\mathbf { A } ^ { n } = \mathbf { I }$$
where \(\mathbf { I }\) is the \(2 \times 2\) identity matrix.
The transformation represented by the matrix \(\mathbf { A }\) followed by the transformation represented by the matrix \(\mathbf { B }\) is equivalent to the transformation represented by the matrix \(\mathbf { C }\).
Given that \(\mathbf { C } = \left( \begin{array} { r r } 2 & 4
- 3 & - 5 \end{array} \right)\), - find the matrix \(\mathbf { B }\).