Inverting the sum of two singular matrices

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