  We explore the invariant theory of quantum symmetric spaces of orthogonal and
symplectic types by employing R-matrix techniques. Our focus involves
establishing connections among the quantum determinant, Sklyanin determinants
associated with the orthogonal and symplectic cases, and the quantum Pfaffians
over the symplectic quantum space. Drawing inspiration from twisted Yangians,
we not only demonstrate but also extend the applicability of q-Jacobi
identities, q-Cayley's complementary identities, q-Sylvester identities, and
Muir's theorem to Sklyanin minors in both orthogonal and symplectic types,
along with q-Pfaffian analogs in the symplectic scenario. Furthermore, we
present expressions for Sklyanin determinants and quantum Pfaffians in terms of
quasideterminants.
