The paper by Henderson and Searle reviews and derives new expressions for the inverse of the sum of two matrices, one of which is nonsingular. The focus is on $(A + UBV)^{-1}$, where $A$ is nonsingular and $U$, $B$, and $V$ can be rectangular. Generalized inverses of $A + UBV$ are also considered. The authors discuss several statistical applications, including inverting partitioned matrices, modifying a matrix by adding another matrix, and estimating variance components in the general linear model. They review the development of these identities, starting with determinants and partitioned matrices, and provide new forms of the inverse that do not require symmetry or squareness of the matrices involved. The paper also explores generalized inverses of $A + UBV$ under certain conditions.The paper by Henderson and Searle reviews and derives new expressions for the inverse of the sum of two matrices, one of which is nonsingular. The focus is on $(A + UBV)^{-1}$, where $A$ is nonsingular and $U$, $B$, and $V$ can be rectangular. Generalized inverses of $A + UBV$ are also considered. The authors discuss several statistical applications, including inverting partitioned matrices, modifying a matrix by adding another matrix, and estimating variance components in the general linear model. They review the development of these identities, starting with determinants and partitioned matrices, and provide new forms of the inverse that do not require symmetry or squareness of the matrices involved. The paper also explores generalized inverses of $A + UBV$ under certain conditions.