1、 1 矩阵求逆 摘 要 本文在借鉴参考文献的基础上，对高等代数学这门课程中的一 些有关矩阵求逆的内容简要地进行了分析、研究和总结。笔者在参考的各种 不同版本的教材中发现，大多教材给出矩阵的求逆的方法无非三种，即：定 义法，初等变换法，伴随矩阵法。其中初等变换包括初等行变换和初等列变 换。这三种方法虽然在大多情况下都能很好解决问题，但有时候使用这些方 法就会显得很繁琐。比如，对于阶数大于 4 的矩阵我们用初等变换和伴随矩 阵就会显得很麻烦，而且容易出错。本文在这里详细讨论了 6 种逆矩阵的求 解方法，首先介绍了常用的那三种矩阵求逆方法，而且对于初等变换法，本 文做了进一步的探讨，给出了同时初等行
2、变换与列变换法。然后又介绍了分 块矩阵法、分解矩阵法、Hamilton-Caylay定理法等方法，其中分块矩阵法中 又包括三角矩阵的分块求逆法和非三角矩阵的分块求逆法。本文对于每一种 方法不仅给出了这些方法的理论依据并给出了具体应用，有的还给出了具体 方法步骤，就是为了使读者明白各种方法的特点，在使用的时候能够选择合 适的方法进行快速解题。 关键字 逆矩阵；初等变换；伴随矩阵；分块矩阵；Hamilton-Caylay 定 理 2 Six methods to find inverse matrix Abstract In this paper, on the basis of referenc
3、e, some relevant content of the inverse matrix in the course of higher algebra is analyzed, researched and summarized briefly. There are only three methods of inverse matrix in most different teaching materials referred. The methods are definition method, adjoint matrix method and elementary transfo
4、rmation method. The elementary transformation method Includes elementary row transformation and elementary column transformation. Though the three methods can well solve problem in most cases, sometimes these methods will appear very complicated. As for the matrix whose rank is more than four, if we
5、 use adjoint matrices or elementary transformation, it will be very troublesome, and error-prone. Six kinds of inverse matrix solution was discussed in this paper in detail. Firstly we introduces the three frequently-used methods, and also makes a further discussion for elementary transformation met
6、hod, giving elementary row transform and column transform method. Then this paper introduces the partitioned matrix method, the decomposition of matrix method, Hamilton - Caylay theorem method. The partitioned matrix method includes the partitioned matrix method of triangle matrix and the partitioned matrix method of common matrix. In this paper every method not only includes the theoretical basis and the specific application, but also includes the concrete steps, the purpose is to make the re
