Invertible Solutions of the Yang-Baxter-Like Matrix Equation for Nonzero Singular Matrix
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Assuming the coefficient matrix is a nonzero singular matrix, we demonstrate that the invertible solutions of the Yang-Baxter-like matrix equation must possess at least two elementary divisors. This establishes a necessary condition for an invertible matrix to satisfy the Yang-Baxter-like matrix equation. Building on this finding, we derive several meaningful corollaries. Additionally, we provide some examples to illustrate our results.