In this paper, we propose a novel baseband (BB) phase shifter (PS) using a fixed-gain-amplifier (FGA) matrix. The proposed BB PS consists of 5 stages of a vector synthesis type FGA matrix with in-phase/quadrature-phase (I/Q) input/output interfaces. In order to achieve low gain variation between phase shift states, 3rd to 5th stages are designed to have a phase shift of +φ_i and -φ_i (i=3,4,5). To change between +φ_i and -φ_i phase shift states, two FGAs with DC bias in-phase/out-phase switches are used. The two FGAs have the same gain, therefore ideally no gain variation can be achieved. Using this configuration, phase shift error and gain variation caused by process mismatch and temperature variation can be reduced. Fabricated 5-bit BB PS has 3-dB bandwidth of 1.05GHz, root-mean-square (rms) phase errors lower than 2.2°, rms gain variations lower than 0.42dB. Power consumption of the PS core and output buffer are 4.9mW and 14.3mW, respectively. 1-dB compression output power is -12.5dBm. The fabricated PS shows that the total phase shift error and gain variation are within the required accuracy of a 5-bit PS with no requirement of calibration.