Determinants Preserving Maps on the Spaces of Symmetric Matrices and Skew-Symmetric Matrices

Denote by Σn and Qn the set of all n × symmetric skew-symmetric matrices over a field $\mathbb {F}$ , respectively, where $\text {char}(\mathbb {F})\neq 2$ $|\mathbb {F}| \geq n^{2}+1$ . A characterization $\phi ,\psi :{\varSigma }_{n} \rightarrow {\varSigma }_{n}$ for which at least one them is surjective, satisfying $ \det (\phi (x)+\psi (y))=\det (x+y)\qquad (x,y\in }_{n}) given. Furthermore, if even :Q_{n} Q_{n}$ ψ surjective ψ(0) = 0, satisfy $\det Q_{n}), then ϕ must be bijective linear map preserving determinant.