  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.
