Why is the determinant like that?

A simple explanation for the determinant formula starting from the concept of area.
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Timestamps
00:00 Introduction
00:17 Act I: Flatland
06:36 Act II: Going to the 3D
11:06 Act III: Can we get much higher?
17:15 Cofactor expansions from the Leibniz formula

Note on technical details
My intention was to convey the visual intuitions surrounding the determinant rather than provide rigorous proofs, of which many can be found from many textbooks. As such, I left out more technical aspects of the mathematics involved. In the video, we showed that if there exists a function satisfying the five rules, it must have the form given by the Leibniz formula (uniqueness), but we did not show that the Leibniz formula actually satisfied those five rules (existence). Similarly, I chose to avoid any discussion of defining area and volume via the Lebesgue measure and proving that the determinant does indeed measure volume in this sense — these formalisms detract from the intuitions I am trying to convey. For similar reasons I avoided mentioning the exterior algebra and geometric algebra — every abstraction comes with a pedagogical cost.

Also, (-1)^sign is often taken to be the definition of the sign of a permutation, rather than just the sign function I introduced.

References
My treatment of permutations was adapted from https://math.ou.edu/~nbrady/teaching/.... The perspective on determinants presented here is standard in mathematics, though it is often only taught to students in pure mathematics, perhaps owing to its abstraction. One possible reference is Chapter 3 of Sergei Treil's book, amusingly titled Linear Algebra Done Wrong (available from https://www.math.brown.edu/streil/pap....

Further reading
John Hannah, A geometric approach to determinants, American Mathematical Monthly 103 (1996), 401–409. [A modern exposition that is similar to my presentation.]
Karl Weierstrass, Zur Determinantentheorie (notes prepared during the winter semester of 1886–87), published posthumously in Mathematische Werke von Karl Weierstrass 3 (Mayer and Müller, Berlin 1903), 271–287, J. Knoblauch, ed.; available from https://archive.org/details/mathemati.... [The characterization of the determinant as the unique function from R^{n^2} to R satisfying the standard multilinear axioms presented in the video goes back to Weierstrass, who formulated these axioms and proved existence and uniqueness sometime before 1886 during one of the mathematics seminars at Friedrich-Wilhelms-Universität Berlin.]

Q: How did you animate this video?
A: I used Manim Community (https://www.manim.community), which is a Python library for creating mathematical animations, created by Grant Sanderson of 3Blue1Brown.

Q: Were you really rejected from art school?
A: For each time I applied to art school, I was not successful.

Music by Vincent Rubinetti
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