浅析教学实践中引入伴随矩阵的原因

Analysis on Why Introduce Adjoint Matrix in the Teaching Practice

伴随矩阵是线性代数课程矩阵理论教学中的一个关键点,在教学实践过程中许多学生迷惑于伴随矩阵的众多性质,而忽略对伴随矩阵出现的原因及其作用的探究,抓不住学习的重点。本文从教学实践出发,分析了矩阵理论中引入伴随矩阵的原因,指出伴随矩阵的主要作用是由其基本性质可得到方阵可逆的三个充分必要条件,从而使得可逆矩阵的判定可以从方阵的行列式、方阵的秩的角度来考虑,且判别方法更加简单便捷。本文利用伴随矩阵的性质对可逆矩阵的三个充分必要条件进行了详细论证,并举例说明利用伴随矩阵来求一个具体方阵的逆矩阵的可行性不高,指出在矩阵论中求逆矩阵的主要方法是初等变换法。

Adjoint matrix is a key point in the teaching of the matrix theory in linear algebra.Many students are puzzled by the properties of adjoint matrix so that they lose sight of the reason of introducing adjoint matrix and the application of adjoint matrix.This manuscript analyses the reason of introducing the adjoint matrix in the theory of matrix.As a result,it points out that the key role of the adjoint matrix is that it brings out three sufficient prerequisites of the invertible matrix.Therefore,people can say whether a matrix is invertible from its determinant or its rank,which are easier and simpler than from the definition of the invertible matrix.This paper also gives the proof of the three sufficient prerequisites of the invertible matrix.And it points out it is impracticable to get the inverse of a matrix by its adjoint matrix.Therefore,the elementary transformations are used to get the inverse matrix.