Exact Solutions and Stability for First-Order Linear Discrete Matrix Equations with Multiple Delays and Non-Permutable Matrices

This study formulates closed-form solution expressions for linear discrete matrix equations that involve several time delays, without requiring the coefficient matrices or the non-homogeneous term to commute. Using a generalized multinomial series and exponential matrix functions adapted to multiple delays, we establish fundamental solutions in a setting where matrix multiplication is not assumed to be commutative. These explicit representations are subsequently utilized to analyze the stability properties of the system, specifically establishing Hyers–Ulam stability. The analysis elucidates the influence of both delay structure and noncommutativity on solution behavior and robustness. A representative example is provided to illustrate the practical applicability of the proposed method and to highlight the significant qualitative effects induced by delays and noncommutative matrix interactions. Notably, the results extend classical theories by addressing noncommutative settings and yield novel contributions that remain significant even in the absence of delays.