Inverting the sum of two singular matrices

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