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

翻译：考虑形如$\widetilde{\mathbf{A}} =\mathbf{A} + \mathbf{e}D \mathbf{f}^*$的方阵。在$\widetilde{\mathbf{A}}$与$D$可逆且满足$\text{rank}(\widetilde{\mathbf{A}}) =\text{rank}(\mathbf{A})+\text{rank}(\mathbf{e}D \mathbf{f}^*)$的条件下，给出了其逆矩阵的显式表达式。该逆矩阵以两种方式呈现：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$D$。此外，从推导过程中得出了针对奇异矩阵的矩阵行列式引理。