This paper presents a novel approach to solving the one-dimensional asymmetric exclusion model with open boundaries using matrix formulation. The authors represent the weights of configurations in the steady state as a product of non-commuting matrices, reducing the problem to finding two matrices and two vectors that satisfy simple algebraic rules. They derive explicit forms for these matrices, which are infinite-dimensional in general, and use them to obtain exact expressions for current and density profiles. The paper also discusses the phase diagram of the model, which consists of three phases: a low-density phase, a high-density phase, and a maximal current phase. Additionally, the authors explore the case of partially asymmetric exclusion and the mixture of two kinds of particles, showing how the matrix approach can be extended to these more general situations. The results are validated through comparisons with known solutions and provide new insights into the behavior of the asymmetric exclusion model.This paper presents a novel approach to solving the one-dimensional asymmetric exclusion model with open boundaries using matrix formulation. The authors represent the weights of configurations in the steady state as a product of non-commuting matrices, reducing the problem to finding two matrices and two vectors that satisfy simple algebraic rules. They derive explicit forms for these matrices, which are infinite-dimensional in general, and use them to obtain exact expressions for current and density profiles. The paper also discusses the phase diagram of the model, which consists of three phases: a low-density phase, a high-density phase, and a maximal current phase. Additionally, the authors explore the case of partially asymmetric exclusion and the mixture of two kinds of particles, showing how the matrix approach can be extended to these more general situations. The results are validated through comparisons with known solutions and provide new insights into the behavior of the asymmetric exclusion model.