Preserving Inverses Of Matrices Problems Over Fields

Let F be a field, F₂ be a field with only two elements and F^* be a set includes all elements of F except zero and one. Let M_n(F) denote the n×nfull matrix algebra over F.We say that a linear map f from M_n(F) to M_n(F) preserves inversesof matrices if f(A) is also invertible and f(A)^-1 = f[A^-1) for every invertiblematrix A∈M_n (F). Chongguang Cao determines the forms of linear maps over symmetric matrix space preserving inverses of matrices when chF≠2,3 (see Linear operators preserving inverses of matrices over division rings). And also, there are many results concerning additive maps preserving inverses of matrices over the additive group of all upper triangular matrices when chF≠2,3 (seeAdditive operators preserving inverses of matrices). But we have never found anyresult about linear maps preserving inverses of matrices over full matrix space M_n(F) which is over any field F. In this paper, we relax the limitation ofchF≠2,3 to that F is any field with at least four elements. Furthermore, wedetermine the forms of linear maps preserving inverses of matrices over full matrix space M_n(F).