Why is the determinant like that?

A simple explanation for the determinant formula starting from the concept of area.

*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*

*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.]

Q: How did you animate this video?

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
Download the music on Bandcamp:
Stream the music on Spotify:

[alifarhat]

Very good presentation. My one nitpick is at 4:58, where you should probably specify that we could pick either orientation of vectors to have positive area: we simply have to pick SOME orientation to be positive, and pick counterclockwise by convention.

[NoBobPro]

So humble of you to shoutout the small mathematicians like Leibniz.

[purplycake]

5 days ago I typed into the YouTube search bar " Why are determinants like that?" but I couldn't find an intuitive enough explanation -- you read my mind and I'm excited to watch this video!

[Utesfan]

I have a PhD in mathematics (granted not algebra) and the 10th minute (plus some wine) caused the increasingly rare epiphany as to the n! terms in the determinate formula.

Thank you, sir.

[TheArtOfBeingANerd]

11:16 "we dont have enough colors" yes thus appears the biggest problem for going into higher dimensions

[mohamedimranechehabi]

Leibniz really needed that shoutout, it's always good to see big creators help smaller ones. (great video)

[mathematicalmachinery]

This is the first time I’ve seen a decent visualization of where determinants come from without using the words “geometric algebra” talking about how it’s so much better than vector algebra! Also, I had no idea that the whole “crossing” thing was part of the standard! I thought that was just a weird tangent for talking about multivector rearrangement, I hadn’t figured out how to relate that back to vector algebra. Genius explanation!

[notexactlysiev]

I had seen the permutation formula for the determinant years ago and had always wondered what on earth it had to do with an area. This was a great explanation!

[dysxleia]

Okay this channel is VERY good. I appreciate math videos more when they help me understand how to discover my formulas.

Also, you do a good job pausing in your speech to give me time to process what I saw. This is something I hope to see more mathtubers do well, but you've killed it

[siarya_math]

I finally understood what the determinant is. Thank you for making this video

[saketsreevallabhrambhogaraju]

I am facinated by the fact that you call unit vectors i-hat, j-hat and k-hat, while I am being taught about them as i-cap, j-cap and k-cap. There can be languages inside languages sometimes.

[Khalidonian]

This is THE BEST explanation of determinants on the internet by far, and I am saying this after days of searching. Thank you so much for this beautiful video. I also started a YouTube channel explaining math stuff and I actually thought about making one about determinants, but I am sure it would not have been as good as yours.

I am so glad that I live in an era where I get to see such beautiful visualizations !!!

[AZALI]

amazing video !!

i loved your coverage of the topic, and your way of breaking apart and explaining the subject was very well done and easy to follow !!

i cant wait to see more stuff from you :)

[abramcz]

Outstanding! Nowhere could I find an explanation of why the sign on a volume in space related to the permutation ordering of matrix columns. This is the only web resource I have found that explains it. Thank you.

[akbaer]

5:54 This part made me realize the connection between the cross product and the determinant

[chemalagos]

6 years using matrix and vectors, and now I have came up why we do multiply that parameter by this and so on. Cool video!!

[r.menezes]

Most intuitive presentation of the determinant i have seen!

[yash]

12:07 > _"how many swaps required? use braid diagrams"_

niice

[eitanethan]

What a great presentation vectors are such a rich area

[nickfaire]

In Geometry I I learnt the definition of the determinant with the permutations, and it was really odd, as it just poped up, without further explanation. This video's topic should have been that class XD