The paper by Henderson and Searle reviews and derives expressions for the inverse of the sum of two matrices, with a focus on (A + UBV)^{-1}, where A is nonsingular and U, B, V may be rectangular. It discusses generalized inverses of A + UBV and their statistical applications. The paper traces the development of these identities, starting with determinant formulas and partitioned matrix inversion. It highlights the work of Schur, Banachiewicz, and others in deriving the Schur complement formula. The paper also discusses the computational advantages of these formulas, particularly when dealing with large matrices. It presents several identities for inverting a sum of matrices, including (A + UBV)^{-1} = A^{-1} - A^{-1}UB(I + BVA^{-1}U)^{-1}BVA^{-1}, and its variants. The paper also considers symmetric cases and generalized inverses, showing how these formulas can be extended to cases where B is not necessarily square or nonsingular. The paper concludes with a discussion of the historical development and applications of these formulas in statistics and matrix theory.The paper by Henderson and Searle reviews and derives expressions for the inverse of the sum of two matrices, with a focus on (A + UBV)^{-1}, where A is nonsingular and U, B, V may be rectangular. It discusses generalized inverses of A + UBV and their statistical applications. The paper traces the development of these identities, starting with determinant formulas and partitioned matrix inversion. It highlights the work of Schur, Banachiewicz, and others in deriving the Schur complement formula. The paper also discusses the computational advantages of these formulas, particularly when dealing with large matrices. It presents several identities for inverting a sum of matrices, including (A + UBV)^{-1} = A^{-1} - A^{-1}UB(I + BVA^{-1}U)^{-1}BVA^{-1}, and its variants. The paper also considers symmetric cases and generalized inverses, showing how these formulas can be extended to cases where B is not necessarily square or nonsingular. The paper concludes with a discussion of the historical development and applications of these formulas in statistics and matrix theory.