Let the current position in phase space be represented by the $(N\cdot d)$-element vector $\mathbf{r}^{\mathrm{old}}$ and the new suggested position by the vector $\mathbf{r}^{\mathrm{new}}$.

The inverse of $\hat{D}$ can be expressed in terms of its cofactors $C_{ij}$ and its determinant (this our notation for a determinant) $\vert\hat{D}\vert$:

$$\begin{equation} d_{ij}^{-1} = \frac{C_{ji}}{\vert\hat{D}\vert} \tag{1} \end{equation}$$

Notice that the interchanged indices indicate that the matrix of cofactors is to be transposed.