公理化定义矩阵行列式的性质推导

Axiomatic Definition of Determinant of Matrix and Derivation of its Properties

行列式的理论和应用是代数中的一个经典问题,矩阵的行列式是赋予矩阵的一个数,将矩阵的列向量视为变量,行列式可以视为列向量组的一个函数,其公理化定义为矩阵的性质推导和分析带来方便。基于行列式的公理定义,从公理出发,给出相关重要性质的详细推导,直接由性质研究线性方程组解的存在性和解的表达式。推导过程的简洁与性质间的关联,体现了公理化定义的优点。

The theory and application of determinant is a classical problem in algebra,the determinant of a matrix is a number assigned to the matrix.The column vectors of the matrix are viewed as variables,and then the determinant is a function of the column vectors.Its axiomatic definition has many advantages for the derivation and analysis of the properties of matrices.Based on the axiomatic conditions in the definition,we give the detail derivation of some important properties directly from the axioms,and investigate the existence and the expression of the solution of the system of linear equations.The derivation processes and the relations between the properties show the advantages of the axiomatic definition.