Matrix equations are crucial in solving many problems in various fields such as mechanics computational physics geology structural design molecular spectroscopy electrical engineering parameter identification automatic control business intelligence linear system theory big data analysis and dynamic analysis. This study investigates the solution of the matrix equation AX = B providing new criteria for the existence of solutions and their general expressions. It extends the conditions for solvability and the general forms of the solutions of the matrix equation AX = B. Examples demonstrate that this approach simplifies the solution process of the matrix equation AX = B as well as the calculation of linear representations of vector sets and the transition matrix between bases. This contributes to enriching the solution theory of matrix equations and simplifying computations.