Effective communication strategies with a properly designed source precoding matrix (PM) and a properly designed relay beamforming matrix (BM) can significantly improve the spectral efficiency of multiple-input multiple-output (MIMO) relaying broadcast channels (RBCs). In the present paper, we first propose a general communication scheme with non-regenerative relay that can overcome the half-duplex relay constraint of the general MIMO-RBC. Based on the proposed scheme, the robust source PM and relay BM are designed for imperfect channel state information at the transmitter (CSIT). In contrast to the conventional non-regenerative relaying communication scheme for the MIMO-RBC, in the proposed scheme, the source can send information continuously to the relay and users during two phases. Furthermore, in conjunction with the advanced precoding strategy, the proposed scheme can achieve a full-degree-of-freedom (DoF) MIMO-RBC with that each entry in the related channel matrix is considered to an i.i.d. complex Gaussian variable. The robust source PM and relay BM designs were investigated based on both throughput and fairness criteria with imperfect CSIT. However, solving the problems associated with throughput and fairness criteria for the robust source PM and relay BM designs is computationally intractable because these criteria are non-linear and non-convex. In order to address these difficulties, we first set up equivalent optimization problems based on a tight lower bound of the achievable rate. We then decompose the equivalent throughput problem into several decoupled subproblems with tractable solutions. Finally, we obtain the suboptimal solution for the throughput problem by an alternating optimization approach. We solve the fairness problem by introducing an adjusted algorithm according to the throughput problem. Finally, we demonstrate that, in both cases of throughput and fairness criteria, the proposed relaying communication scheme with precoding algorithms outperforms existing methods.