Let $A=M_n(F[x_{ij}\mid 1\leq i,j\leq n])$ be the matrix algebra over commutative polynomials in $n^2$ variables $x_{11},\dots,x_{nn}$ over a (nice enough) field $F$. I would like to know what is the centralizer of the matrix $x=(x_{ij})$ in $A$.

If we consider $x$ as an element of $M_n(F(x_{ij}\mid 1\leq i,j\leq n))$, it is not difficult to show that its centralizer consists of polynomials in $x$ with coefficients in $F(x_{ij}\mid 1\leq i,j\leq n)$. Does the centralizer of $x$ in $A$ consists of polynomials in $x$ with coefficients in $F[x_{ij}\mid 1\leq i,j\leq n]$?