Jordan Multiplicative Mappings at Some Points on Matrix Algebras

Let ${\mathcal{M}}_{n}$ be the algebra of all $n\times n$ matrices, and let $\phi: M_{n}\rightarrow M_{n}$ be a linear mapping. We say that $\phi$ is a Jordan multiplicative mapping at $G$ if $\phi(AB+BA)=\phi(A)\phi(B)+\phi(B)\phi(A)$ for any $A, B\in{\mathcal{M}}_{n}$ with AB=G. Fix $G\in{\mathcal{M}}_{n}$, we say that $G$ is a Jordan all-multiplicative point if every Jordan multiplicative linear bijection $\phi$ at $G$ with $\phi({I}_{n})={I}_{n}$ is a multiplicative mapping in ${\mathcal{M}}_{n}$, where ${I}_{n}$ is the unit matrix in ${M}_{n}$. In this paper we mainly show the following result: if $G\in{\mathcal{M}}_{n}$ with det $G$=0, then $G$ is a Jordan all-multiplicative point in ${\mathcal{M}}_{n}$.