Write down the \(2 \times 2\) matrix corresponding to each of the following transformations:
a rotation about the origin through \(90 ^ { \circ }\) clockwise;
a rotation about the origin through \(180 ^ { \circ }\).
The matrices \(\mathbf { A }\) and \(\mathbf { B }\) are defined by
$$\mathbf { A } = \left[ \begin{array} { r r }
2 & 4
- 1 & - 3
\end{array} \right] , \quad \mathbf { B } = \left[ \begin{array} { l l }
- 2 & 1
- 4 & 3
\end{array} \right]$$
Calculate the matrix \(\mathbf { A B }\).
Show that \(( \mathbf { A } + \mathbf { B } ) ^ { 2 } = k \mathbf { I }\), where \(\mathbf { I }\) is the identity matrix, for some integer \(k\).
Describe the single geometrical transformation, or combination of two geometrical transformations, represented by each of the following matrices: