In this paper, we consider a general low-rank matrix optimization problem which is modeled by a general Schatten p-quasi-norm (${\rm 0<p<1}$) regularized matrix optimization. For this nonconvex nonsmooth and non-Lipschitz matrix optimization problem, based on the matrix p-thresholding operator, we first propose a fixed point continuation algorithm with extrapolation (FPCAe) for solving it. Secondly, we prove that any accumulation point of the iterative sequence generated by the proposed algorithm is not only a critical point but also a global stationary point of the problem, where the global stationary point possesses some global optimality which can exclude too many stationary points even some local minimizers of the nonconvex problem. We also prove the rank invariance of the iterative sequence. Thirdly, we prove the global convergence and R-linear convergence rate of the whole iterative sequence generated by the proposed algorithm under some mild conditions. Finally, we conduct a large number of numerical experiments on random square and rectangular matrix completion problem, grayscale image and three-channel image recovery problem. The numerical results illustrate that the proposed FPCAe algorithm is competitive with some state-of-the-art algorithms for low-rank matrix recovery in terms of speed, accuracy, robustness and anti-noise.