Square matrices of the form à = A + eDf* are considered. An explicit expression for the inverse is given, provided à and D are invertible with rank(Ã) = 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