Square matrices of the form $\widetilde{\mathbf{A}} =\mathbf{A} + \mathbf{e}D \mathbf{f}^\mathsf{T}$ are considered. An explicit expression for the inverse is given, provided $\widetilde{\mathbf{A}}$ and $D$ are invertible with $\text{rank}(\widetilde{\mathbf{A}}) =\text{rank}(\mathbf{A})+\text{rank}(\mathbf{e}D \mathbf{f}^\mathsf{T})$. The inverse is presented in two alternative ways, one that uses singular value decomposition and another that depends directly on the components $\mathbf{A}$, $\mathbf{e}$, $\mathbf{f}$ and $D$. As a consequence of the derivations follows a rank-deficient matrix determinant lemma.

翻译：考虑形如$\widetilde{\mathbf{A}} =\mathbf{A} + \mathbf{e}D \mathbf{f}^\mathsf{T}$的方阵。在假设$\widetilde{\mathbf{A}}$和$D$可逆且满足$\text{rank}(\widetilde{\mathbf{A}}) =\text{rank}(\mathbf{A})+\text{rank}(\mathbf{e}D \mathbf{f}^\mathsf{T})$的条件下，给出了该矩阵逆的显式表达式。该逆以两种等价形式呈现：一种利用奇异值分解，另一种直接依赖于分量$\mathbf{A}$、$\mathbf{e}$、$\mathbf{f}$和$D$。作为推导结果，得到了一个秩亏矩阵的行列式引理。