Reducing the adverse effects of noises is one of the key issues for successful signal processing. For decades, there have been numerous studies to achieve this goal. Among them, dual-sensor noise reduction has been widely studied due to its rather high performance with comparatively low cost. In this paper, we propose a novel dual-sensor noise reduction method named a determinant-based generalized sidelobe canceller. As given in the graphical abstract, it is a two-stage noise reduction technique, wherein an input signal is primarily filtered by a determinant-based Wiener filter and then a subsequent noise canceller removes residual noises by utilizing the determinant-based adaptation mode controller. To update the coefficients of the noise canceller for noise-only regions, the output of the more reliable determinant-based signal activity detector is employed. For performance evaluation, five objective measures, the mean opinion score, and spectrograms of the proposed and other competing methods are estimated for both simulation and real data sets. Through the evaluation, we demonstrate that the proposed method outperforms the others in almost all aspects, especially when multiple noises interfere or reverberation exists.