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Proof for updating algorithm for Slater determinant

In this equation, the first line becomes zero for j=i and the second for j \neq i . Therefore, the update of the inverse for the new Slater matrix is given by

\begin{eqnarray} \boxed{d^{-1}_{kj}(\mathbf{x^{new}}) = \left\{ \begin{array}{l l} d^{-1}_{kj}(\mathbf{x^{old}}) - \frac{d^{-1}_{ki}(\mathbf{x^{old}})}{R} \sum_{l=1}^{N} d_{il}(\mathbf{x^{new}}) d^{-1}_{lj}(\mathbf{x^{old}}) & \mbox{if $j \neq i$}\nonumber \\ \\ \frac{d^{-1}_{ki}(\mathbf{x^{old}})}{R} \sum_{l=1}^{N} d_{il}(\mathbf{x^{old}}) d^{-1}_{lj}(\mathbf{x^{old}}) & \mbox{if $j=i$} \end{array} \right.} \end{eqnarray}