6 The set \(S\) is given by \(S = \{ \mathbf { A } , \mathbf { B } , \mathbf { C } , \mathbf { D } \}\) where
\(\mathbf { A } = \left[ \begin{array} { l l } 1 & 0
0 & 0 \end{array} \right]\)
\(\mathbf { B } = \left[ \begin{array} { l l } 1 & 0
0 & 1 \end{array} \right]\)
\(\mathbf { C } = \left[ \begin{array} { l l } 0 & 0
0 & 1 \end{array} \right]\)
\(\mathbf { D } = \left[ \begin{array} { l l } 0 & 0
0 & 0 \end{array} \right]\) 6

1. Complete the Cayley table for \(S\) under matrix multiplication.

   	A	B	C	D
   A	A		D
   B		B
   C			C
   D				D

   6
2. Using the Cayley table above, explain why \(\mathbf { B }\) is the identity element of \(S\) under matrix multiplication.
   [0pt] [1 mark] 6
3. Sam states that the Cayley table in part (a) shows that matrix multiplication is commutative. Comment on the validity of Sam's statement.