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。

前 言 本书依据高等职业教育人才培养目标并在总结多年实践教学经验的基础上编写而成. 在编写过程中，编者团队立足高职特色，以应用为目的，本着“联系实际、淡化概念、加强应用”的教学原则，强调数学概念与实际问题的联系. 本书不强调灌输数学逻辑的严密性、思维的严谨性，不追求复杂的计算和变换，而重视数学应用意识，以培养学生灵活运用知识和解决问题、分析问题的能力. 编写内容力求简洁易懂、突出实用性，教师在教学中可根据专业和学时区别在内容上有所取舍. 本书充分考虑高职、高专学生的数学基础，在处理复杂的高等数学计算问题时，简要介绍了Mathematica数学软件的使用方法，以帮助学生更好地理解相关概念和理论. 本书的最后附有参考答案，可供学习者参考. 同时，部分习题配有视频解答，学习者可扫描二维码后观看. 对于本书中标有*的内容，可依据实际情况灵活掌握，另行安排学习. 本书由广东工贸职业技术学院胡煜担任主编，吴立炎、李惠珠、傅秀莲、刘芳担任副主编，编者均来自教学第一线，并有多年的教学经验. 限于编者水平有限，书中不妥之处在所难免，敬请广大读者批评指正. 编 者 2020.9

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