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 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 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 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 (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
Download the music on Bandcamp:
vincerubinetti.bandcamp.com/a...
Stream the music on Spotify:
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[@alifarhat667]

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.

[@brokemathstudent]

Yep, it should be pointed out that it’s precisely a consequence of Rule 1 that certain orientations are considered positive and others are negative - namely, if we decided that A(j, i) = 1 in Rule 1 instead, then A(i, j) would be -1 and all the areas will be negative of what they normally are.

[@mathematicalmachinery7934]

lol it took me a few minutes paused to figure out where the “orientation” came from before I saw rule 1

[@mellowInventor]

​@@brokemathstudentnegative is just the opposite of whatever the reference orientation is. It gets more difficult to calculate at A^N dimensions!

[@pauldirc..]

What is defination of area and why it is a into b in case of rectangle

[@adityamrai3892]

wonderful presentation! How do you make these animations? Is this Manim?

[@NoBobPro]

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

[@sunnythebridger7529]

😢Yep, leibniz is so underrated.

[@jimpim6454]

small? are you joking?

[@natan6218]

​@@jimpim6454 yes, they are

[@antonioplaza8557]

"small" ?

[@0MVR_0]

@@jimpim6454 intended as a joke as shoutouts are given to high profiles of lesser prestigue as a free hand up

[@purplycake515]

5 days ago I typed into the KZbin 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!

[@Mr0rris0]

You are now a determinator 😎

[@macchiato_1881]

​@@Mr0rris0im convinced most mathematicians have a horrible sense of humor because of this comment.

[@Mr0rris0]

@@macchiato_1881 well I ain no mathitician if that helps

[@osamaqidwai7800]

You didnt find 3b1b's video on this topic??

[@TheArtOfBeingANerd]

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

[@mitchratka3661]

Bro the irony of you representing the problem of a lack of representations in another form is so funny lmao

[@TheArtOfBeingANerd]

Imagining a world where we only have two colors and hence can't imagine the three dimensions we live in

[@Utesfan100]

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.

[@fazemega1222]

I have a PhD as well, no I have let me check ... yes umm of course my kidergarten diploma here (totally the same as a PhD ;) right?) of course it is about the study of "Super Proportional Vector Fields with a distinct continuus Z Functions ". It's the real deal guys, real deal ...you know that pesky... riemman million dollar problem, the answer is right there. I found out that it stops at 10^363262363 + 2353534532^2. Totally did not invent my field of study and the solution right now. Of course not. Why should I do that. In all seriousness, congrats on having a PhD the only thing I have to brag is that I like math???. I know it is really out of place in a youtube channel called "broke math student" but still.

[@mohamedimranechehabi5735]

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

[@PinkeySuavo]

I always wonder how lives of these people would look like nowadays. Would be they become some big mathematican youtubers? Would they silently work on some research? Or would they get sucked into games and tiktok and wouldn't learn at all?

[@mathematicalmachinery7934]

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!

[@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.

[@guntera3845]

I‘ve only ever heard the i-hat version. Good to know the i-cap version but I think it is more exotic.

[@whannabi]

People saying "cap" are the cool kids.

[@mgsquared5204]

Are you from the US? I’ve never heard i-cap in my life. Cap feels like such a British word. 😂

[@guidomista8448]

That's how I was taught too. Î ĵ ƙ are all hats.

[@saketsreevallabhrambhogaraju]

@@mgsquared5204 Naa, i'm not from US, I'm from India.

[@siarya_math]

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

[@akbaer60]

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

[@PinkeySuavo]

I thought of cross product too, but I cannot see the connection. I mean it looks like the determinant of 2 vectors IS the cross product of these vectors? Or?

[@iandall9570]

That is the clearest explanation of determinants I have seen!

[@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

[@dgaul123]

Yeah he had perfect pace, not too fast and not too slow.

[@yash1152]

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

[@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!

[@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

[@AZALI00013]

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 :)

[@persik7gd]

AZALI!? Love your music!

[@ARBB1]

On the braid crossings, in exterior/geometric algebra one can represent the volume elements by means of the wedge product, and then the crossing numbers can come from NOR operations applied in switching said wedges.

[@TheDayglowcamo]

Hands down best video on determinant. Further I have yet to see this so clearly laid out in any text book.

[@jivitasagdeo6913]

I can already see your channel crossing 1M if this level of content is maintained. Cheers mate. 👏👏

[@chakinether]

If only KZbin had an option to like the video's description. Your references saved me. Also, I love your logical and understandable approaches to show the properties of determinants. Thank you!

[@Sachmun]

Brilliant video, it's the first time I deeply understand why the determinant is a volume and also a really elegant derivation of Leibniz formula !

[@DaniMadridDaniMineCraft]

Thank you a lot for this video, I have been asking that same question for the last 5 years and yet could not find anything but the simple geometric demostration or the derivation for the simple 2d determinant. Most of people stop there, but for me it feels like proof by induction, it works but you dont learn anything. After watching the video I feel like the awnser was always there! and that I was overcomplicating things, as always the genius of linear algebra is in its simplicity, you made it pretty simple every step of the way!

[@r.menezes]

Most intuitive presentation of the determinant i have seen!

[@pratyushgora]

this is so cool, u know what are the general questions in the mind of a math student, thanks a lot for making this video

[@azzibreaker]

im taking linear algebra right now, this video explained in the terms and notation the professor used!! I LOVED THIS TYYSM

[@borissimovic441]

This video is a real treasure, it is one of the best and most important videos/lessons regarding the topic of linear algebra! Thank you so much! 😊

[@bandaragunarathna4340]

Excellent Video! I really like proving all the complex applications, by first starting with simple rules. I think, this is the best way to teach mathematics in schools too. Please try to do more video like this whenever you can, as it will be a big service for humanity.

[@kaidenschmidt157]

Oh my god, I loved this. Illustrating the geometric object associated to each algebraic step during dimensions two and three was fantastic, and then the inclusion of braid groups!

[@JunioriDjazz]

you're amazing dude, never seen math that way, you blew my mind, keep going with your videos, cheers from Brazil

[@yplayergames7934]

This video catch my attention so hard that I actually stopped studying just to watch this beauty explanation and visual representation of the Determinant of the Matrix

[@timmaths]

Incredible! Even though this channel is still small, it's amazing!

[@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!!

[@divy1211]

this is one of the most awesome videos on determinants

[@SapereAude625]

Such a good video, helped a lot with understanding/recalling what i've studied a year ago :D I was particularly confused with why determinant is defined by permutaions, thank you for your effort!

[@butterbits372]

This is a really excellent explanation for one of the more arcane formulas from my linear algebra course.

[@pedroth3]

Simply amazing, always wanted to see/do this! Thanks

[@lovishnahar1807]

hloo sir ,love from india , u explained the topic which even 3b1b failed me to, this shows ur understanding of concept. furthermore i want u to keep making videos on this linear algebra topic its just seems to out of the box to understand this

[@punditgi]

Sheer brilliance! All math videos should be this good. 🎉😊

[@bart2019]

Fantastic explanation. One thing that you didn't address but I personally always found important, is that in 2 dimensions if you swap the coordinates and flip the sign of one, you get a normal vector: same size but orthogonal to the original. The determinant is the dot product of this one orthogonal vector, and the other original one. Carthesian dot products are independent of the choice of coordinate systems, and thus, are a physical value. That value is the (signed) area.

[@moonwatcher2001]

Congratulations! This a fascinating, amene, as well as relaxing, approach to a usually non well explained algebraic geometry. I love this kind of videos. Thanks for the post. +1 subscriber❤

[@mustafizurrahman5699]

More than a genius. I had never been taught in this way. WOW physical interpretation is awesome

[@thenoblegnuwildebeest3625]

I've been looking for a video on this topic for months, thanks

[@pyton1d752]

please keep making educational math content, this is magical

[@alijoueizadeh2896]

Thank you for your precious time.

[@gametimewitharyan6665]

A comment for the algoirthm. I have no words on how great of a video this is, hope more are on the way!

[@infinite_flesh]

I deeply enjoyed watching this brief explanation

[@linuxp00]

Also, it's useful to note that a circular permutation of an even sequence (ab, abcd, ....) has alternating sign, given by mod 2 formula, while an odd sequence (a, abc, abcde, ...) has unchanging sign. That's rather useful to rearrange subsequences of symbols in an order more fitting for partitions sorting, that also doesn't affect the total ordering.

[@SaidVSMath]

This was excellent, the determinant always annoyed me but this is so illustrative and makes me want to learn about symmetric groups more. Thank you so much for such a good video!!

[@KaiLu-wz7vj]

That is an very interesting visualization of determinant . It has benefited me a lot

[@artemisSystem]

Oh this is amazing. I'm taking a group theory course, without having taken linear algebra, and i get somewhat lost when they talk about matrix groups. Having recently learned about alternating groups (groups of even permutations), it was really cool to see them crop up in determinants

[@the_eternal_paradox]

bro this is EXACTLY what I'm trying so hard to read and understand in my textbook rigght now... your explanation is perfectly clear and blowing my mind, kind of :0 thanks for this although I will saay, my book is also throwing differentials at us so to me it still feels just more complex and yet also beautiful :D

[@Furlic-LastBraincell]

I've been looking for video like this for a loooooooong time...

[@connorkearley7789]

thank you for this great explanation

[@painlesskun3959]

I have done my highschool (cal.maths or add maths) and I never knew determinants can be visualized this way and that it was this simple!!

[@josephlorizzo8997]

thank you so much, this Is such a great video, i understood more the determinant thanks to you!

[@wirelessboogie]

Great video, thank you! The best explanation of where matrices come from that I've ever seen! The only request is to make the time when the final exercises are shown on the screen to be longer than one second and move them to the center. When I stop the video at the last second, I can't read the bottom lines because they're covered with KZbin video buttons.

[@ilde4]

Beautiful presentation.

[@muhamadsyakir7731]

masterpiece explanation video.

[@antonioplaza8557]

absolutely amazing video

[@latarte3931]

Great work ❤! You're killing it 💪

[@jamesking2439]

This is a fantastic explanation!

[@borissimovic441]

Thank you for this video 🙏🏻❤️

[@jakobr_]

Best outro in a math video ever.

[@prashantsharma-mc6hh]

Amazing animation with great explanations. Thanks! Btw, did you use Manim for animations?

[@pelegsap]

This is one of the best (if not THE best) video I've seen on the topic! Also, regarding 8:55 - I really hope you're not an Austrian planning on moving to Germany... 😛

[@bluemmy1886]

As someone going into their first year of a mathematics degree THANK YOU SO MUCH!!!!!!!

[@vidkids1963]

Not sure if this was mentioned, but orientation of spaces is intrinsic to geometric algebra. Love this demonstration! Thank you.

[@gazmendmavraj2994]

Best explaination!

[@leonelnicolaslizondo9406]

this is the best video of mathematics taht i saw

[@brianambelis5869]

Wonderful explanation!! Thank you so much! :-)

[@2dark4noir]

Simply amazing

[@underfilho]

I'm right now just studying differential forms and multilinear functions, I guess this video will be really good to it

[@mrshodz]

Great video. Keep it up!

[@Barba_007]

Beautiful bro, keep up with good work 💪💪

[@LambdaFeet]

Amazing visuals

[@kryptoid2568]

Best outro for a math video

[@3koningen]

fantastic video.

[@avyakthaachar2.718]

Amazing video ❤

[@alessandrodilorenzo2363]

I love the content, how did u do this amazing video,what video-rappresentation tools did u use?

[@frankconnor1529]

Nice job!

[@martinjimenez821]

In a few hours i have an Algebra exam, and this video was quite helpful, if one exercise is determinant based, i will update :)

[@martinjimenez821]

I passed 😃. Sadly, there was no determinant based exercise, great video tho

[@korigamik]

Man, this is soo cool seeing your videos. Can you share the source code for the scenes?

[@richardhu5508]

beautiful!!!

[@klevisimeri607]

Very nice!

[@thelyghter7927]

The First Time Someone Explained To me The DETERMINANT :-o

[@pauldirc..]

Seeing your videos you seems like a very interesting and smart person , I saw your last video on doing maths alone and when in the last part the quote by Robert pyar from Zen in motorcycle on pursuing highest knowledge highly resonated with me can you give me more recommendation on such books and quotes

[@red_rassmueller1716]

The ending XD Love it!!!!!!!!!

[@chinmoypal3397]

Fantastic ❤

[@blinded6502]

It's also interesting to point out, that determinant in 2d is a bivector, in 3d it's a trivector and etc. In other words it's a pseudoscalar.

[@BLVGamingY]

YES YES YES, you showed the method of calculation too, oh god I feel like the determinant has been a little neglected as a concept to be explained. If you're searching for such concepts, how about the hessian matrix and its determinant. After all it's pretty important for multivar calculus

[@mildtoadstool]

That was just continuing to apply definitions... thx.

[@rajkumararulambalam9829]

Thanks!

[@GoodMORNINGings]

You rock Thanks for this elimination

[@mr.nicolas4367]

amazing video

[@monishrules6580]

I had to slow the video down to see that last frame. Task 4 requires writing a textbook which is uhhh. Anyway amazing video, i loved your manim animations ❤

[@therealist9052]

"Some of the most interesting mathematics happens when we see where 'nonsense' leads us" . Dang what a bar. Definitely stealing this one.🤣

[@tej240]

oh how i wish these videos existed on youtube 10 years ago

[@weirdfrog1196]

Stand proud. You can cook.

[@Noname-67]

I have been thinking about the determinant lately. Using the most natural interpretation of the determinant, through exterior product, I almost grasped the determinant inside out, except a single detail. I have been especially troubled by the notion of volume, because the area we usually talk about only makes sense in the Euclidean space, where as the volume in the sense of determinant is much more general, it works for all vector spaces over any fields (and even free modules over commutative rings). Furthermore, the usual sense of the volume is extremely complicated (it requires some sort of integral calculus or measure to be defined), using it to explain something as fundamental as determinant feels very wrong to me. This video filled the final piece of the puzzle that I am missing. In mathematics, instead of describing a concept concretely, it might be useful to specify properties that we want the concept to have. So I'll just make up a definition. A (not "the") volume operator f on a n-dimensional vector space V is simply an alternating multilinear map (a function that satisfies rule 2,3 and 4 in the video) that take a list of n vectors from V to another vector space W. As the video demonstrated, if V=ℝ² and W=ℝ, then the ordinary signed area operator is a special case, the justification only relies on simple cut and paste rather than complicated integration. So here's the thing: the determinant is defined on linear maps V→V, not lists of vectors. Although the linear map can be written as matrix, doing so requires an artificial choice of basis, and also obfuscate important insight. When we think of the determinant, we think of the scaling effect of the linear map on volume, not the volume of the linear map. Scaling volume can be formalized as for any volume operator f: (det T)⋅f(v₁,…,vₙ) = f(Tv₁,…,Tvₙ) There is exactly one function det that satisfies this property, and its the determinant. (The proof for this requires the knowledge of the exterior product.) I just wrote a whole essay that probably only 5 people will ever read.