6 Three matrices, \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\), are given by \(\mathbf { A } = \left( \begin{array} { c c } 1 & 2
a & - 1 \end{array} \right) , \mathbf { B } = \left( \begin{array} { c c } 2 & - 1
4 & 1 \end{array} \right)\) and \(\mathbf { C } = \left( \begin{array} { c c } 5 & 0
- 2 & 2 \end{array} \right)\) where \(a\) is a

1. Using \(\mathbf { A } , \mathbf { B }\) and \(\mathbf { C }\) in that order demonstrate explicitly the associativity property of matrix multiplication.
2. Use \(\mathbf { A }\) and \(\mathbf { C }\) to disprove by counterexample the proposition 'Matrix multiplication is commutative'. For a certain value of \(a , \mathbf { A } \binom { x } { y } = 3 \binom { x } { y }\).
3. Find
   • \(y\) in terms of \(x\),
   • the value of \(a\).
     \(7 C\) is the locus of numbers, \(z\), for which \(\operatorname { Im } \left( \frac { z + 7 \mathrm { i } } { z - 24 } \right) = \frac { 1 } { 4 }\).
     By writing \(z = x + \mathrm { i } y\) give a complete description of the shape of \(C\) on an Argand diagram.