Trace and determinant preserving maps of matrices

Abstract: Suppose a map $\phi$ on the set of positive definite matrices satisfies $\det(A+B)=\det(\phi(A)+\phi(B))$. Then we have $${\rm tr}(AB^{-1}) = {\rm tr}(\phi(A){\phi(B)}^{-1}).$$ Through this viewpoint, we show that $\phi$ is of the form $\phi(A)= M^*AM$ or $\phi(A)= M^*A^tM$ for some invertible matrix $M$ with $\det (M^*M)=1$. We also characterize the map $\phi: \mathcal{S} \rightarrow \mathcal{S}$ preserving the determinant of convex combinations in $\mathcal{S}$ by using similar method. Here $\mathcal{S}$ can be the set of complex matrices, positive definite matrices, symmetric matrices, and upper triangular matrices.