Abstract: Let A be a 3×3 singular or diagonalizable matrix, all solutions to the Yang-Baxter-like matrix equation have been determined. However, finding all solutions for full rank, non-diagonalizable matrices remains challenging. By utilizing classification techniques, we establish all solutions of the Yang-Baxter-like matrix equation in this paper when the coefficient matrix A is similar to non-diagonalizable matrix diag(λ,J₂(λ)) with λ= 0. More specifically, we divide the non-diagonal elements of the solution into 10 different cases. By discussing each situation, we establish all solutions of the YangBaxter-like matrix equation. The results of this work enrich the existing ones.

Key words: Yang–Baxter–like matrix equation, Yang–Baxter equation, Commuting solutions, Non–commuting solutions