3. (a) Given that
$$\mathbf { A } = \left( \begin{array} { c c }
1 & \sqrt { } 2 \\
\sqrt { } 2 & - 1
\end{array} \right)$$
- find \(\mathbf { A } ^ { 2 }\),
- describe fully the geometrical transformation represented by \(\mathbf { A } ^ { 2 }\).
(b) Given that
$$\mathbf { B } = \left( \begin{array} { r r }
0 & - 1 \\
- 1 & 0
\end{array} \right)$$
describe fully the geometrical transformation represented by \(\mathbf { B }\).
(c) Given that
$$\mathbf { C } = \left( \begin{array} { c c }
k + 1 & 12 \\
k & 9
\end{array} \right)$$
where \(k\) is a constant, find the value of \(k\) for which the matrix \(\mathbf { C }\) is singular.