Let $X=(x_{ij})_{mn}$ be a quantum matrix with the commutation relations between entries: \begin{alignat*}{2} & x_{ij} x_{il} = q x_{il} x_{ij}, && j < l, \\ & x_{ij} x_{kj} = q x_{kj} x_{ij}, && i<k, \\ & x_{ij} x_{kl} = x_{kl} x_{ij}, && i<k, j>l, \\ & x_{ij} x_{kl} = x_{kl} x_{ij} + (q - q^{-1})x_{il}x_{kj}, \quad && i < k, j < l. \end{alignat*} For $n \in \mathbb{Z}_{>0}$, denote $[n]=\{1, \ldots, n\}$. For any $I \subset [m]$, $J \subset [n]$, $|I|=|J|=l \in \mathbb{Z}_{>0}$, a quantum minor $\Delta_{I,J}$ is defined as follows \begin{align*} \Delta_{I,J} = \sum_{\sigma \in S_l} (-q)^{\ell(\sigma)} x_{i_1, j_{\sigma(1)}} \cdots x_{i_l, j_{\sigma(l)}}, \end{align*} where $\{i_1 < \cdots < i_l\} = I$, $\{j_1< \cdots < j_l\}=J$, and $\ell(\sigma)$ is the length of the permutation $\sigma$.

When $m=n$ and $I=[n]$, $\Delta_{I,I}=\det_q(X)$ is the quantum determinant of $X$. How to compute $(\det_q(X))^{-1}$. Are there some references about this? Thank you very much.