The primary goal of this paper has been the exploration of an analytical formula for the inverse of an invertible \([{\textbf{I}}-\epsilon \textbf{Q}]\) square matrix for two and four energy groups, explicitly in terms of the elements of that matrix. Our work aims to present efficient mathematical techniques to calculate the neutron flux into multi-dimensional homogeneous reactors using the quasi-static method and different types of the Padé approximations, which can be applied to linear and nonlinear kinetics systems. These powerful techniques, which include the analytical inversion of the generated matrix, can be beneficial for cutting down on computing time and speeding up the suggested approximations above the set benchmarks. We have examined the sensitivity of the time step size utilizing different Padé approximations. The results of various Padé approximations are tested and compared with traditional and benchmark techniques for different types of reactivity, including step, ramp, sinusoidal, pulse, and nonlinear reactivity for two and three-dimensional space homogeneous reactors. These practical approximations handle stiffness, are the simplest to execute, and yield results comparable to benchmark techniques.
