The algebra \(M_n(R)\) of the \(n\times n\) matrices over \(R\) is considered, where \(R\) is a unital commutative ring with invertible 2. The main result shows that the algebra \(M_n(R)\) is idempotent elements determined, i.e. for any \(R\)-module \(X\) and for any symmetric bilinear map \(\{\cdot,\cdot\}:M_n(R)\times M_n(R)\rightarrow X\), the following two conditions are equivalent:{\parindent7mm \begin{itemize}\item[(i)]\(\{x,x\}=\{e,x\}\) whenever \(x^2=x\); \item[(ii)] there exists a linear map \(f:M_n(R)\rightarrow X\) such that \(\{x,y\}=f(x\circ y)\) for all \(x,y\in M_n(R)\), \end{itemize}}where \(e\) denotes the unit matrix and the Jordan product \(\circ\) is defined by \(x\circ y=2^{-1}(xy+yx)\). The main result is applied to show that an invertible linear transformation \(\varphi\) on \(M_n(R)\) fixing the identity preserves the idempotence if and only if \(\varphi\) is a Jordan automorphism, i.e. \(\varphi(x)\circ\varphi(y)=\varphi(x\circ y)\) for all \(x,y\in M_n(R)\). If \(A\) is an associative algebra over the ring \(R\), a linear transformation \(\varphi\) on \(A\) is said to be Jordan derivable at a point \(p=x\circ y\in A\) if \(\varphi(p)=\varphi(x)\circ y+x\circ\varphi(y)\); \(\varphi\) is called a Jordan derivation if it is Jordan derivable at any point. Again by using the main result, it is shown that a linear transformation in \(M_n(R)\) is a Jordan derivation if and only if it is Jordan derivable at all idempotent points.