Center of a quantum matrix algebra Let $p \in k^\times$ be a nonroot of unity. It seems to be a well-known fact that the center of the quantum matrix algebra $\mathcal{O}_p(M_n(k))$ is generated by the quantum determinant $D_p$. It is not difficult to check that $D_p$ is indeed central but showing that it generates the whole center seems considerably more difficult. I'm sure this has been proved somewhere but I've been unable to track it down.

You can find a proof in Section 1.5 of the paper "Finite dimensional representations of the quantum group $GL_q(n; C)$ and the zonal spherical functions on $U_q(n-1)\setminus U_q(n)$" by Noumi, Yamada and Mimachi, Japanese Journal of Mathematics (New Series), Vol 19 (1993) no 1. p 31-80.