Several researchers had worked on the fixed point of some iteration schemes. For instance, in Mann constructed an iteration scheme in a convex Banach space E that converged to the fixed-point of the transformation T defined on E while in, Dotson applied the iterative process introduced by Mann in 1953 to the approximation of fixed-point of quasi non-expansive mapping in Hilbert space and in uniformly convex and strictly convex Banach spaces. Furthermore, Rhoades in used an infinite matrix of weighted mean to determine the convergence of an iteration scheme to the fixed-point of the self-mapping defined on the iteration scheme in Banach space. This paper presents the fixed-point properties of three infinite matrices acting on some sequence spaces by classifying them to be either a source or sink fixed points. In addition, the stability of these fixed points are rigorously analysed, with each point being identified as either stable or unstable. Furthermore, we determine the fixed set or Fix(A) of the convergent domains of the operators in the sequence spaces, as well as their fixed subsets alongside with the conditions for stability. A significant topological property of these domains is established, underscoring the structural interplay between infinite matrix transformations and related fixed-points.