Trace and determinant preserving maps of matrices

We show that if a map ϕ on the set of positive definite matrices satisfiesdet⁡(A+B)=det⁡(ϕ(A)+ϕ(B)),ortr(AB−1)=tr(ϕ(A)ϕ(B)−1)with det⁡ϕ(I)=1, then ϕ   is of the form ϕ(A)=M⁎AMϕ(A)=M⁎AM or ϕ(A)=M⁎AtMϕ(A)=M⁎AtM for some invertible matrix M   with det⁡(M⁎M)=1det⁡(M⁎M)=1. We also characterize the map ϕ:S→Sϕ:S→S preserving the similar trace equality or the determinant equalitydet⁡(tA+(1−t)B)=det⁡(tϕ(A)+(1−t)ϕ(B)),t∈[0,1],in SS, where SS denotes the set of complex matrices, symmetric matrices, or upper triangular matrices, respectively.