一道高等代数竞赛考研题的一题多解研究
A Study on Multiple Solutions to a Competition and Postgraduate Entrance Examination Question from Higher Algebra

矩阵理论是高等代数课程教学中重要的基础内容，而矩阵秩相关问题更是高等代数竞赛考研中一类常见问题。针对一道竞赛考研试题，我们给出了6种求解方法：利用矩阵的分块初等变换、利用多项式互素与矩阵的分块初等变换、利用矩阵秩的不等式、利用矩阵的特征值与特征向量、利用线性变换的值域和利用线性变换的核，这些方法对于理解高等代数矩阵问题的代数证法与几何证法，提高逻辑思维能力大有裨益。
The matrix theory is an important basic content in the teaching of advanced algebra courses, and the issue of matrix rank is a common problem in the competition and the postgraduate entrance examination. Six methods for solving a competition question or a postgraduate entrance exam question were provided, including using block elementary transformation of matrices, polynomial coprime and block elementary transformation of matrices, inequality of matrix rank, eigenvalues and eigenvectors of matrices, range of linear transformations, and kernel of linear transformations. All the methods are of great benefit for understanding the algebraic and geometric proofs of advanced algebraic matrix problems, and improving logical thinking abilities.