本书在总结多年实践教学经验的基础上编写而成，主要内容包括行列式、矩阵、线性方程组、相似矩阵。全书内容简洁、实用，同时注重数学知识与具体专业的结合，精选专业案例，将专业案例转化为数学问题并提供解答思路。本书每章后都附带复习题，本书的最后附有参考答案。在本书附录中，设置数学实验指导、行列式数学实验常用指令、简单的线性规划问题等拓展内容，另外还配套部分知识点的数字化学习资源，供学习者选学或参考。在实际教学和学习中，教师和学生可根据具体的课时安排，对书中的内容进行一定程度的增补或删减。

胡煜，中山大学应用数学专业研究生毕业，现任职于广东工贸职业技术学院，副教授，长期从事数学课程教学与研究工作。

第 1 章 行列式·······························································································1
1.1 行列式的概念····················································································1
1.1.1 二阶行列式 ·············································································1
1.1.2 三阶行列式 ·············································································3
1.1.3 n 阶行列式 ··············································································3
习题 1.1···························································································5
1.2 行列式的性质····················································································6
习题 1.2···························································································8
1.3 克莱姆法则及应用··············································································9
1.3.1 克莱姆法则 ·············································································9
1.3.2 运用克莱姆法则讨论齐次线性方程组的解 ···································.10
习题 1.3························································································.11
复习题 1 ······························································································.12
本章知识精要 ·······················································································.12
第 2 章 矩阵·······························································································.14
2.1 矩阵的概念····················································································.14
习题 2.1························································································.16
2.2 矩阵的性质及运算···········································································.17
2.2.1 矩阵相等 ·············································································.17
2.2.2 矩阵的运算 ··········································································.18
2.2.3 矩阵的转置 ··········································································.22
2.2.4 方阵的行列式 ·······································································.24
习题 2.2························································································.24
2.3 矩阵的初等变换与矩阵的秩·······························································.25
线性代数
VI
2.3.1 矩阵的初等变换 ····································································.25
2.3.2 矩阵的秩 ·············································································.27
习题 2.3························································································.29
2.4 逆矩阵··························································································.30
2.4.1 逆矩阵的概念 ·······································································.30
2.4.2 逆矩阵的性质 ·······································································.30
2.4.3 逆矩阵的求法 ·······································································.31
2.4.4 用逆矩阵解线性方程组 ···························································.34
习题 2.4························································································.34
2.5 矩阵的应用····················································································.35
2.5.1 编制通信密码 ·······································································.35
2.5.2 投入产出分析 ·······································································.37
复习题 2 ······························································································.40
本章知识精要 ·······················································································.42
第 3 章 线性方程组······················································································.44
3.1 向量组的线性相关性········································································.44
3.1.1 n 维向量空间········································································.44
3.1.2 线性相关性概念 ····································································.45
3.1.3 线性相关性的判定 ·································································.45
习题 3.1························································································.47
3.2 向量组的秩与矩阵···········································································.48
3.2.1 极大线性无关向量组 ······························································.48
3.2.2 向量组的秩 ··········································································.48
3.2.3 矩阵与向量组秩的关系 ···························································.49
习题 3.2························································································.49
3.3 线性方程组的解··············································································.50
3.3.1 消元法解线性方程组 ······························································.50
3.3.2 线性方程组解的结构 ······························································.53
习题 3.3························································································.54
3.4 应用·····························································································.55
复习题 3 ······························································································.60
本章知识精要 ·······················································································.61
第 4 章 相似矩阵·························································································.64
4.1 向量组的正交规范化········································································.64
目 录
VII
4.1.1 向量内积及其性质 ·································································.64
4.1.2 正交向量组及其性质 ······························································.65
4.1.3 规范正交基及其求法 ······························································.66
4.1.4 正交矩阵与正交变换 ······························································.67
习题 4.1························································································.69
4.2 方阵的特征值与特征向量··································································.69
4.2.1 特征值与特征向量的概念 ························································.69
4.2.2 特征值与特征向量的基本性质 ··················································.72
习题 4.2························································································.72
4.3 相似矩阵的概念、性质及应用····························································.72
4.3.1 相似矩阵的概念 ····································································.73
4.3.2 相似矩阵的性质 ····································································.73
4.3.3 矩阵与对角矩阵相似的条件 ·····················································.75
习题 4.3························································································.77
4.4 实对称矩阵的性质与对角化·······························································.77
4.4.1 实对称矩阵的性质 ·································································.77
4.4.2 实对称矩阵的对角化 ······························································.78
习题 4.4························································································.82
复习题 4 ······························································································.82
本章知识精要 ·······················································································.83
附录 A 数学实验指导···················································································.84
附录 B 行列式数学实验常用指令····································································.99
附录 C 简单的线性规划问题··········································································101
附录 D 习题参考答案···················································································106
参考文献·····································································································114
线性代数