C.39. Until the re-seeing

The title of this section is a literal translation of “Ĝis la revido” in Esperanto, which idiomatically means “See you” in English. Yes, I am bidding farewell—it is not the case that everything that has a beginning has an end, but it is the case that this book, An introduction to determinants, which has a beginning, has an end.

I do not really have anything interesting to say, but I still choose to say something here.

It is said that there are at least five ways of defining the determinant:

•	as a function satisfying certain axioms;
•	explicitly by the “big formula”;
•	recursively;
•	as the “oriented volume” of a parallelepiped;
•	by the exterior algebra.

I define the determinant recursively in this book. I believe that this is the easiest way for beginners. Some other ways are hinted in this book as well.

This book is mainly about determinants, and it is over four hundred pages long. I know that I have written too much in this book: the number of pages in Appendix C (the current chapter), which consists of reading materials, is greater than the total number of pages in all other chapters. However, I really wanted to share my knowledge, hoping that it might be useful and helpful.

This book is available under the Zero-Clause BSD License, which means that any person or organisation is allowed to use this book in any way, for any purpose, with or without charge. I have not yet earned a single cent from this book, but I did not, do not, and will not prevent any person or organisation from making a profit from this book. I want to propagate the values of “open knowledge”, and I want to prove its profitability, if I can ever earn money from this book.

I thank mathematicians and their work; I thank artists who made artworks for this book; I thank my friends, who were good brokers; I thank everyone who read or reads this book.

Parenthetically, the day on which I wrote this closing remark is 28 June 2025. Both six and twenty-eight are perfect numbers, in the sense that each is equal to the sum of all its positive divisors, excluding itself. It is also worth noting that $6.28$ is a decimal approximation of $2\pi$ (or $\tau$), the ratio of the circumference of a circle to its radius. Therefore I think of 28 June as a “mirinda” (meaning “worth marvelling at”) day. Moreover, $2025$ is remarkable:$$\begin{array}{rl}2025=& 1^6\cdot 3^4\cdot 5^2\cdot 7^0\\ =& 5^2+20^2+40^2\\ =& 27^2+36^2\\ =& (20+25{)}^2\\ =& (1+2+3+4+5+6+7+8+9{)}^2\\ =& 1^3+2^3+3^3+4^3+5^3+6^3+7^3+8^3+9^3\\ =& \det \left[\begin{array}{cccc}1& 5& 14& 12\\ 9& 2& 7& 6\\ 13& 8& 3& 11\\ 15& 10& 4& 16\end{array}\right]\\ =& \det \left[\begin{array}{ccccc}1& 18& 9& 6& 10\\ 5& 24& 13& 7& 12\\ 8& 22& 21& 23& 14\\ 16& 2& 15& 4& 17\\ 19& 11& 20& 3& 25\end{array}\right].\end{array}$$I have only listed a few here.

Fartu bone. Kredu, ke ni renkontiĝos iam, ie, iel, ial. (Do well. Believe that we will meet sometime, somewhere, somehow, for some reason.)