det: An introduction to determinants

# 行列式入门 (An introduction to determinants)

## 介绍 (About what is the book?)

这是一本关于行列式的书. 行列式是一个初等的, 有用的工具. 行列式在现在不如在过去那么重要了, 但学习行列式不是不好的. 行列式在代数, 微积分, 几何是有用的. 于是, 我写了本书.  
This is a book about determinants. Determinants are an elementary but useful device. Determinants are not as important now as they were in the past, but it is not bad to learn determinants. Determinants are useful in algebra, calculus and geometry. That is the reason for which I penned this book.

我假定您有中学代数基础 (包括但不限于有理数, 实数, 代数式, 2 元 1 次方程组, 整式的加减乘, 因式分解, 分式, 1 元 1 次不等式, 常量与变量, 函数, 2 次根式, 1 元 2 次方程; 我列不完). 我希望本书不会是难的, 但是, 因为我几乎无法作实验 (用我的书教初学者行列式), 故我只能希望如此.  
I assume that the reader is familiar with elementary algebra (whose topics include but are not limited to rational numbers, real numbers, algebraic expressions, systems of linear equations with 2 unknowns, basic operations of polynomials, factorisation, rational expressions, linear inequalities with 1 unknown, constants and variables, functions, square roots, quadratic equations with 1 unknown; I am sorry that I cannot name all). I hope that the book is not too hard, but I can only hope so, because I can hardly verify my guess (= I do not have the opportunity to teach beginners determinants with this book).

本书有一些创新, 但不多, 也不重要.  
There are novel ideas in this book, but there are just a few, and they do not seem really to matter.

若您发现本书的错误, 请使我知道它. 谢谢.  
If there is any mistake in this book, please do not hesitate to tell me about it. Thanks.

这是封面; 我从[这儿](https://github.com/oetiker/lshort)抄来的. 副标题 "或 376 分了解行列式" 的数字没有什么特别的意思; "版本 2π, 2024" 也没有什么特别的意思.
这些是孩子不懂事儿瞎写的.  
Here is the cover of the book, which I copied from [this project](https://github.com/oetiker/lshort). The subtitle "or determinants in 376 minutes" does not signify anything special, and neither does "Version 2π, 2024". I wrote them on the cover... "mischievously".

<img src="bildoj%20en%20Markdown/kovrilo.png" width="62.5%" height="62.5%" alt="kovrilo" />

**我承诺我未使用由人工智能作的内容写此书.**  
**Mi promesas, ke mi ne uzis enhavojn generitajn de artefarita intelekto por verki ĉi tiun libron.**  
**I promise that I did not use contents generated by artificial intelligence to write this book.**

## 想看书? (How to read the book)

- [https://gitee.com/septsea/det/raw/cxefa/libro.pdf](https://gitee.com/septsea/det/raw/cxefa/libro.pdf)
- [https://gitcode.net/septsea/det/raw/cxefa/libro.pdf](https://gitcode.net/septsea/det/raw/cxefa/libro.pdf)
- [https://www.123pan.com/s/QvKUVv-K4WHA](https://www.123pan.com/s/QvKUVv-K4WHA)
- [https://www.bananaspace.org/wiki/讲义:行列式入门](https://www.bananaspace.org/wiki/讲义:行列式入门)
(在线阅读; 不必下载文件. Read it online, and you do not have to download the book explicitly.)
- [https://github.com/septsea/det/raw/cxefa/libro.pdf](https://github.com/septsea/det/raw/cxefa/libro.pdf)
(但因为 "网络安全", 您不一定能从这儿下. You probably cannot download it from here because of "cybersecurity", though.)
- [https://codeberg.org/septsea/det/raw/cxefa/libro.pdf](https://codeberg.org/septsea/det/raw/cxefa/libro.pdf)
(但因为 "网络安全", 您不一定能从这儿下. You probably cannot download it from here because of "cybersecurity", though.)
- [https://bitbucket.org/septsea/det/raw/cxefa/libro.pdf](https://bitbucket.org/septsea/det/raw/cxefa/libro.pdf)
(但因为 "网络安全", 您不一定能从这儿下. You probably cannot download it from here because of "cybersecurity", though.)

<!-- 本书甚至还有一套[教学片](https://www.bilibili.com/video/BV13D4y1p7QW/);
不过, 我想, 您不一定会喜欢它.
毕竟, 我作教学片的目的是修改书的错误.
我不一定改正了所有的错误;
但至少比不作教学片时要好多了.
**当然, 若您能告诉我本书的数学的错误,
则这是好的.**

There even exists [a series of videos accompanying the book](https://www.bilibili.com/video/BV13D4y1p7QW/). You probably do not like it, though, because the purpose of making those videos was to proofread the book. I did not correct every error, but I did correct every error that I spotted. **It is be good that you tell me mathematical errors in the book.**

我也作了一套[习题](https://www.bilibili.com/video/BV1cG4y1u7jG/).

There is also [a set of exercises](https://www.bilibili.com/video/BV1cG4y1u7jG/). -->

<!-- <details>
<summary>点我看书的目录. Click on me to see the table of contents of the book.</summary>

这是书的目录. 第一章里, 带 `(*)` 的是选学内容. 您不学选学内容, 并不会影响您对任何必学内容 (也就是, 未声明为 "选学内容" 的节) 的理解.  
Here is the table of contents of the book. Sections with `(*)` in the first chapter are optional sections. Not learning optional sections does not affect one's understanding any compulsory sections (= sections in the first chapter that are not optional sections).

**The table of contents has not been translated into English yet, since doing that would require much time and effort.**

- 序
- Acknowledgements
- 依赖说明
- 第一章: 行列式
    - 缺项定位
    - (\*) 排列
    - -1 的整数次方
    - (\*) 逆序数与符号
    - 阵
    - 行列式
    - 按一列展开行列式
    - (\*) 按多列展开行列式
    - (\*) 完全展开行列式
    - 阵的一个新记号
    - (\*) 完全展开行列式 (续)
    - 单位阵
    - 行列式的性质
    - 用行列式的性质确定行列式
    - "行" 列式
    - (\*) "行" 列式 (续)
    - 阵的积
    - Binet–Cauchy 公式 (青春版)
    - 按一列 (行) 展开行列式的公式的变体
    - 线性方程组
    - 由 n 个 n 元 ≤1 次方程作成的方程组 (1)
    - (\*) 由 n 个 n 元 ≤1 次方程作成的方程组 (2)
    - (\*) 由 n 个 n 元 ≤1 次方程作成的方程组 (3)
    - (\*) 由 n 个 n 元 ≤1 次方程作成的方程组 (4)
    - (\*) 由 m 个 n 元 ≤1 次方程作成的方程组 (1)
    - (\*) 由 m 个 n 元 ≤1 次方程作成的方程组 (2)
    - (\*) 由 m 个 n 元 ≤1 次方程作成的方程组 (3)
    - (\*) 由 m 个 n 元 ≤1 次方程作成的方程组 (4)
    - (\*) 由 m 个 n 元 ≤1 次方程作成的方程组 (5)
    - (\*) Binet–Cauchy 公式
    - 杂例
    - (\*) La lasta leciono
- 附录 A: 2 元 2 次式
    - 准备
    - 2 元 2 次式
    - 2 元 2 次方程组
    - 1 元 4 次方程
- 附录 B: 求和号、求积号、数学归纳法
    - 求和号
    - 求积号
    - 数学归纳法
    - 结合律、交换律、分配律
- 附录 C: 后日谈
    - 我要如何定义行列式?
    - 我要讲阵吗?
    - 行列式的性质
    - 排列
    - Binet–Cauchy 公式
    - 按一行展开行列式
    - 方阵与其转置的行列式相等
    - (关于列的) 多线性
    - 按前二列展开行列式
    - (关于列的) 交错性
    - (关于列的) 反称性
    - 反称性的应用
    - 用行列式的性质确定行列式
    - 类行列式
    - 绝对值的性质 (1)
    - 绝对值的性质 (2)
    - 绝对值的性质 (3)
    - 关于实数的大小的几个事实
    - 行列式为 0 的阵的性质
    - 判断阵的行列式不是 0 的方法
    - 判断实阵的行列式大于 0 的方法
    - 消去律
    - 古伴的性质 (1)
    - 古伴的性质 (2)
    - 古伴的性质 (3)
    - 阵的积, 分块地写
    - 转置的性质
    - 辛阵
    - 反称阵
    - 奇数级反称阵的行列式为 0
    - 阵的积与倍加
    - 反称阵与倍加
    - Pfaffian
    - Pfaffian 的性质
    - 阵的积与倍加 (续)
    - Pfaffian 的性质 (续)
    - 辛阵的行列式为 1
    - 杂例
    - 总结
- 参考文献
- 记号表
- 词表
- Colophon: notes on typesetting

</details>

-->

## 关于著作权 (About the copyright)

我用 [0BSD](https://opensource.org/license/0bsd) 协议发布了本书:  
I made this book available under [0BSD](https://opensource.org/license/0bsd):

> Copyright © 2024 Determinantulo
>
> Permission to use, copy, modify, and/or distribute this software for any purpose with or without fee is hereby granted.
> 
> **The software is provided "as is" and the author disclaims all warranties with regard to this software including all implied warranties of merchantability and fitness. In no event shall the author be liable for any special, direct, indirect, or consequential damages or any damages whatsoever resulting from loss of use, data or profits, whether in an action of contract, negligence or other tortious action, arising out of or in connection with the use or performance of this software.**

我给一个汉语翻译作为参考 (我从[这儿](https://commons.wikimedia.org/wiki/Template:0BSD/i18n/zh)复制过来的):  
(Here is a Chinese translation of the licence; I do not have to retranslate the following text into English, do I?)

> 著作权 © 2024 Determinantulo
>
> 特此授予出于任何目的使用、复制、修改和/或分发本软件的许可, 无论是否收费.
> 
> **本软件按 "原样" 提供, 作者不承担与本软件有关的所有保证, 包括对适销性和适用性的所有默示保证. 在任何情况下, 作者均不对任何特殊的、直接的、间接的或后果性的损害或因失去使用、数据或利润而导致的任何损害负责, 无论是在合同诉讼、疏忽或其他诉讼中与本软件的使用或性能有关.**

(注: 虽然本项目主要是一本书, 但因本书是用由纯文本文件作成的代码文件生成的, 故本书与生成本书的代码也可被认为是 "软件".)  
(Note that this project is basically a book, but because this book was generated by source code files in plain text, this book, along with the code generating this book, can be also considered as "software".)

通俗地, 您可以自由地用、改、发我的书, 且您**不必**标出本书的来源 (您当然也**不必**保留著作权声明); 不过, 若出了问题, 则我这儿没有 "售后", 且您必须自己负责 (法律地).  
Briefly, you can use, modify and distribute my book without interference or restriction, and you **do not have to** declare from where you get this book (and you **do not have to** preserve the notice of copyright, either); however, there is no warranty, so if you cause any trouble, you will be the one who will be responsible.

**本仓库的艺术作品不一定用本书所用的协议. 您已被提醒.**  
**La arkverkoj en ĉi tiu deponejo ne nepre uzas la permesilon, kiun ĉi tiu libro uzas. Vi estis avertita.**  
**The artworks in this repository do not necessarily use the licence that this book uses. You have been warned.**

## 关于书的标识 (About the emblem of the book)

我感谢 [Petrica](https://github.com/PetricaT) 为本书作了一个 VTuber 风格的标识:  
Mi dankas [Petrica](https://github.com/PetricaT) pro fari VTuber-stilan emblemon por ĉi tiu libro:  
I thank [Petrica](https://github.com/PetricaT) for making a VTuber-fashioned emblem for this book:

https://github.com/PetricaT/ProgrammingVTuberLogos-Addon/tree/main/Determinant

---

可在 [`dokumentaro`](./dokumentaro/) 了解本书的更多情报.  
Multaj informoj estas en [`dokumentaro`](./dokumentaro/).  
More information is in [`dokumentaro`](./dokumentaro/) ("documentation").

Determinantulo  
2024-06-28

Septsea/det

简介

发行版

贡献者

近期动态

Septsea/det .gitee-modal { width: 500px !important; }

简介

发行版

开源评估指数源自 OSS-Compass 评估体系，评估体系围绕以下三个维度对项目展开评估：

贡献者

近期动态

搜索帮助

Septsea/det