2 The matrix \(\mathbf { A }\) is given by \(\mathbf { A } = \left( \begin{array} { r r } 2 & - 2
1 & 3 \end{array} \right)\).
- Calculate \(\operatorname { det } \mathbf { A }\).
- Write down \(\mathbf { A } ^ { - 1 }\).
- Hence solve the equation \(\mathbf { A } \binom { \mathrm { x } } { \mathrm { y } } = \binom { - 1 } { 2 }\).
- Write down the matrix \(\mathbf { B }\) such that \(\mathbf { A B } = 4 \mathbf { I }\).
Matrices \(\mathbf { C }\) and \(\mathbf { D }\) are given by \(\mathbf { C } = \left( \begin{array} { l } 2
0
1 \end{array} \right)\) and \(\mathbf { D } = \left( \begin{array} { l l l } 0 & 2 & p \end{array} \right)\) where \(p\) is a constant. - Find, in terms of \(p\),
- the matrix CD
- the matrix DC.
It is observed that \(\mathbf { C D } \neq \mathbf { D C }\). - The result that \(\mathbf { C D } \neq \mathbf { D C }\) is a counter example to the claim that matrix multiplication has a particular property. Name this property.