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.