On determinant preserver problems (Q1399939)

Early in 1897, Frobenius has proved that let \(F\) be the complex number field and \(\varphi:M_n\to M_n\) be a linear transformation which preserves the determinant, then either \(\varphi(A)=MAN\) or \(\varphi(A)=MA^tN\) where \(M,N\) are matrices such that \(\det(MN)=1\). Recently \textit{G. Dolinar} and \textit{P. Šemrl} [Linear Algebra Appl. 348, 189-192 (2002; Zbl 0998.15011)] proved that if \(\varphi\) is surjective but not necessary linear and \(\det(A+\lambda B)=\det(\varphi(A)+\lambda\varphi(B))\) for all \(A,B\in M_n\) and all \(\lambda\in F\), the same result holds. In this paper, under the weaker restriction that the field \(F\) has more than \(n\) elements, the authors obtain the same results without the surjectivity of \(\varphi\). Furthermore, the map \(\varphi\) can be extended to the following 2 classes: 1. \(\varphi\) is surjective and \(\det(A+\lambda B)=\det(\varphi(A)+\lambda\varphi(B))\) for all \(A,B\in M_n\) and two specific \(\lambda\in F\); 2. \(\varphi\) is additive and preserves determinant. Finally, all these results can be extended to determinant preservers on the vector space \(T_n\) of upper triangular matrices.