Square matrices of the form à = 𝐀 + 𝐞𝐷𝐟* are considered. An explicit expression for the inverseis given, provided à and 𝐷 are invertible with rank(Ã) = rank(𝐀) + rank(𝐞𝐷𝐟*). The inverse ispresented in two ways, one that uses singular value decomposition and another that dependsdirectly on the components 𝐀, 𝐞, 𝐟 and 𝐷. Additionally, a matrix determinant lemma forsingular matrices follows from the derivations.