We have studied the muon neutrino and antineutrino quasi-elastic (QEL) scattering reactions (v(mu)n -> mu(-)p and (v) over bar mu p -> mu(+)n) using a set of experimental data collected by the NOMAD Collaboration. We have performed measurements of the cross-section of these processes on a nuclear target (mainly carbon) normalizing it to the total v(mu) ((v) over bar mu) charged-current cross section. The results for the flux-averaged QEL cross sections in the (anti) neutrino energy interval 3-100 GeV are v(mu) = (0.92 +/- 0.02(stat) +/- 0.06(syst)) x 10(-38) cm(2) and (v) over bar (mu) = (0.81 +/- 0.05(stat) +/- 0.09(syst)) x 10(-38) cm(2) for neutrino and antineutrino, respectively. The axial mass parameter M(A) was extracted from the measured quasi-elastic neutrino cross section. The corresponding result is M(A) = 1.05 +/- 0.02(stat)+/- 0.06(syst) GeV. It is consistent with the axial mass values recalculated from the antineutrino cross section and extracted from the pure Q(2) shape analysis of the high purity sample of v(mu) quasi-elastic 2-track events, but has smaller systematic error and should be quoted as the main result of this work. Our measured MA is found to be in good agreement with the world average value obtained in previous deuterium filled bubble chamber experiments. The NOMAD measurement of MA is lower than those recently published by K2K and Mini-BooNE Collaborations. However, within the large errors quoted by these experiments on MA, these results are compatible with the more precise NOMAD value.