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The two-level quantum system (qubit) in a precessing magnetic field and incontact with a heat bath is investigated. The exact reduced dynamics for thequbit in question is obtained. We apply the approach based on the blockoperator matrices theory, in contrast with the standard methods provided by thetheory of the open quantum systems. We also present the solution of the Riccatioperator equation associated with the Hamiltonian of the system. Next, we studythe adiabatic approximation for the system under consideration using quantumfidelity as a way to measure validity of the adiabatic theory. We find that inthe weak coupling domain the standard condition that ensures adiabatic behaviorof the spin in the magnetic field also guarantees its adiabatic evolution inthe open system variant of this model. Therefore, we provide the explicitexample of the open quantum system that satisfies the adiabatic approximationfirstly formulated for the closed quantum systems.

Exact reduced dynamics for a qubit in a precessing magnetic field and in the contact with a heat-bath