  We study the invariant theory for the quantum symmetric spaces of orthogonal
and symplectic types using techniques of the R-matrix. We explicitly realize
the quantum symmetric spaces as subrings of the quantum coordinate ring
$M_q(N)$ and study the relations among the quantum determinant, the Sklyanin
determinant of orthogonal and symplectic types, and their associated quantum
Pfaffians. Using similar ideas as $q$-Yangians, we prove and generalize the
$q$-Jacobi identities, $q$-Cayley's complementary identities, the $q$-Sylvester
identities, the $q$-minor identities in both orthogonal and symplectic types as
well as their $q$-Pfaffian analogues. We also formulate the Sklyanin
determinants and the quantum Pfaffians in terms of quasideterminants. Finally
we obtain Muir's theorem for the Sklyanin determinant.

Quantum Algebra;Rings and Algebras