Quantum Berezinian for quantum affine superalgebra $U_q({\widehat{gl}}_{M\mathrm{\mid }N})$

We introduce the quantum Berezinian for the quantum affine superalgebra ${\mathrm{U}}_q({\widehat{\mathfrak{g}\mathfrak{l}}}_{M\mathrm{\mid }N})$ and show that the coefficients of the quantum Berezinian belong to the center of ${\mathrm{U}}_q({\widehat{\text{gl}}}_{M\mathrm{\mid }N})$. We also construct another family of central elements which can be expressed in the quantum Berezinian by a Liouville-type theorem. Moreover, we prove analogues of the Jacobi identities, the Schur complementary theorem, the Sylvester theorem and the MacMahon Master theorem for the generator matrices of ${\mathrm{U}}_q({\widehat{\text{gl}}}_{M\mathrm{\mid }N})$.