Suppose we have a commutative ring $R$ and an $R$-algebra $A$. If $\mathfrak{p}$ is a prime ideal of $R$, then is there any nice representation of $M_n(A)_\mathfrak{p}$, or even $T_n(A)_\mathfrak{p}$, where the latter is the the ring of $n\times n$ triangular matrices. It doesn't seem too much to ask that this should be $M_n(A_\mathfrak{p})$ and $T_n(A_\mathfrak{p})$, respectively, but I am unsure about how to prove this. I am, of course, assuming that I can localize $M_n(-)$ at the prime ideal $\mathfrak{p}$.
My idea was to represent $M_n(A)_\mathfrak{p}$ as $M_n(A)\otimes_AA_\mathfrak{p}$ and perhaps try to show this was isomorphic to $M_n(A\otimes_AA_\mathfrak{p})$.