On Deriving the Inverse of a Sum of Matrices

Available expressions are reviewed and new ones derived for the inverse of the sum of two matrices, one of them being nonsingular. Particular attention is given to $({\text{{\bf A}}} + {\text{{\bf UBV}}})^{ - 1} $, where ${\textbf{A}}$ is nonsingular and ${\textbf{U}}$, B and ${\textbf{V}}$ may be rectangular; generalized inverses of ${\textbf{A}} + {\textbf{UBV}}$ are also considered. Several statistical applications are discussed.