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A new derivation of the three-dimensional (3D) limit equilibrium method for slope stability analyses is presented, which applies linear algebra operations to matrices to achieve a highly efficient algorithm while maintaining the rigorousness of the formulation. It satisfies six equilibrium conditions and includes all of the components of the intercolumnar forces. The generalized 3D Morgenstern-Price's (MP) assumption for the intercolumn forces is recast into incremental forms. The advantage is that the unknowns of the different physical meanings (i.e., factor of safety (FS) and the coefficients in the MP assumption) are determined by balancing the forces and moments independently. Along with the recasting correction coefficients, they are solved with the triple nested algorithm. The numerical instability that is related to the different orders of magnitude for the residuals is avoided. The varication examples show: (i) the insensitivity of the FS to the prescribed functions in the MP assumption, (ii) the reliability of the method to cope with a complicated slope geometry (simplified methods may provide a larger FS for certain cases), and (iii) the superior efficiency of the algorithm (requires several seconds to converge). This study proposes a practical way to track the irregular 3D tension cracking by repeating the analysis and excluding the columns with the negative normal forces at the bottom. This formulation is applied to a valley landfill slope in China. In comparison to the two-dimensional analysis, there are more reasonable/cost-effective measures to increase the system stability through 3D analyses for several conditions (e.g., seismic loading). • A new three-dimensional method is devised by concise linear algebra operations. • It considers 6 equilibrium conditions and all intercolumnar force components. • Triple nested algorithm achieves a high-efficiency and a good convergence. • Factor of safety is insensitive to the intercolumn force distribution function. • A practical way is proposed to track highly irregular tension cracking surface. [ABSTRACT FROM AUTHOR]