You are given two matrices, A and B, where
$$\mathbf { A } = \left( \begin{array} { l l }
1 & 2
2 & 1
\end{array} \right) \text { and } \mathbf { B } = \left( \begin{array} { c c }
- 1 & 2
2 & - 1
\end{array} \right)$$
Show that \(\mathbf { A B } = m \mathbf { I }\), where \(m\) is a constant to be determined.
You are given two matrices, \(\mathbf { C }\) and \(\mathbf { D }\), where
$$\mathbf { C } = \left( \begin{array} { r r r }
2 & 1 & 5
1 & 1 & 3
- 1 & 2 & 2
\end{array} \right) \text { and } \mathbf { D } = \left( \begin{array} { r r r }
- 4 & 8 & - 2
- 5 & 9 & - 1
3 & - 5 & 1
\end{array} \right)$$
Show that \(\mathbf { C } ^ { - 1 } = k \mathbf { D }\) where \(k\) is a constant to be determined.
The matrices \(\mathbf { E }\) and \(\mathbf { F }\) are given by \(\mathbf { E } = \left( \begin{array} { c c } k & k ^ { 2 } 3 & 0 \end{array} \right)\) and \(\mathbf { F } = \binom { 2 } { k }\) where \(k\) is a constant.
Determine any matrix \(\mathbf { F }\) for which \(\mathbf { E F } = \binom { - 2 k } { 6 }\).