The answer is b. A quick way to see this is to observe that the product of this number with \(q\) is \(1\).

Note that the multiplicative inverse of \(q\) can be computed directly as follows: \begin{eqnarray*} q^{-1} & = & \frac{1}{q} \\ & = & \frac{1}{\frac{1}{2}+\frac{1}{2}\sqrt{5}} \\ & = & \frac{2}{1 + \sqrt{5}} \\ & = & \frac{2}{1+\sqrt{5}}\cdot \frac{1-\sqrt{5}}{1- \sqrt{5}} \\ & = & \frac{2 - 2\sqrt{5}}{ 1^2 - (\sqrt{5})^2} \\ & = & \frac{2 - 2\sqrt{5}}{-4} \\ & = & -\frac{1}{2} + \frac{1}{2}\sqrt{5}. \end{eqnarray*}