The evaluation of the right hand side (rhs) term above is carried out by applying the identity \( (A + B)^{-1} = A^{-1} - (A + B)^{-1} B A^{-1} \). In compact notation it yields

$$ \begin{eqnarray*} [\mathbf{D}^{T}(\mathbf{x^{new}})]^{-1} & = & [\mathbf{D}^{T}(\mathbf{x^{old}}) + \Delta^T]^{-1}\\ & = & [\mathbf{D}^{T}(\mathbf{x^{old}})]^{-1} - [\mathbf{D}^{T}(\mathbf{x^{old}}) + \Delta^T]^{-1} \Delta^T [\mathbf{D}^{T}(\mathbf{x^{old}})]^{-1}\\ & = & [\mathbf{D}^{T}(\mathbf{x^{old}})]^{-1} - \underbrace{{[\mathbf{D}^{T}(\mathbf{x^{new}})]^{-1}}}_{\text{By Eq.}{\ref{invDkj}}} \Delta^T [\mathbf{D}^{T}(\mathbf{x^{old}})]^{-1}. \end{eqnarray*} $$