The matrices
\[\begin{pmatrix} 2 & a \\ -3 & -1 \end{pmatrix} \quad \text{and} \quad \begin{pmatrix} -\frac{1}{16} & b \\ \frac{3}{16} & \frac{1}{8} \end{pmatrix}\]are inverses.  Enter the ordered pair $(a,b).$
The product of the matrices is
\[\begin{pmatrix} 2 & a \\ -3 & -1 \end{pmatrix} \begin{pmatrix} -\frac{1}{16} & b \\ \frac{3}{16} & \frac{1}{8} \end{pmatrix} = \begin{pmatrix} \frac{3a}{16} - \frac{1}{8} & \frac{a}{8} + 2b \\ 0 & -3b - \frac{1}{8} \end{pmatrix}.\]We want this to be the identity matrix, so $\frac{3a}{16} - \frac{1}{8} = 1,$ $\frac{a}{8} + 2b = 0,$ and $-3b - \frac{1}{8} = 1.$  Solving, we find $(a,b) = \boxed{\left( 6, -\frac{3}{8} \right)}.$