The authors calculate the primitive spectrum of \({\mathcal O}_q(M_2(k))\) (the quantized coordinate ring of quantum \(2\times 2\) matrices) over an algebraically closed field \(k\) of characteristic \(0\) when \(q\) is not a root of unity, as well as the symplectic ideals of \({\mathcal O}(M_2(k))\) relative to the associated Poisson structure, and exhibit a bijection between \(\text{Prim }{\mathcal O}_q(M_2(k))\) and \(\text{Symp }{\mathcal O}(M_2(k))\). The second author has proved that such a bijection also exists when \(k\) is not necessarily algebraically closed, as well as a bijection between \(\text{Spec }{\mathcal O}_q(M_2(k))\) and the Poisson spectrum of \({\mathcal O}(M_2(k))\) [Symplectic ideals of Poisson algebras and the Poisson structure associated to quantum matrices, Commun. Algebra (to appear)].