Mixed-type Reverse Order Laws For Generalized Inverses

Generalized inverse of the matrix theory is a subject that was studied by many scholars in the last several decades, they have been made remarkable development. Generalized inverse of the matrix theory originally by I.Fredholm proposed in1903,he first gave a generalized inverse of integral operators.The article from the most simple generalization(AB)-1=B-1A-1to any number of products{1,3}-and{1,4}一inverse mixed-type, using known conditions or some equations, research the following any number of hybrid reverse order law for general-ized inverses of matrices.According to the conclusion of maximal and minimal ranks of matrix of generalized Schur complement rank, we can obtain reverse order law nec-essary and sufficient condition for the establishment: An{1,i}An-1{l,i}...A1{1,j}(?)(A1A2…An){1,2}, An{l,i}An-l{1,i}...A1{l,i}(?)(A1A2...An){l,2,i}, An{1,2,i}An-l{1,2,i)…A1{l,2,i}(?)(AlA2…An){1}å'ŒAn{1,2,i}An-1{1,2,i}...A1{l,2,i}(?)(A1A2…An){l,2}, where i=3,4.The necessary and sufficient conditions are composed by the rank of a matrix equation, and it is easy to verify.