What is a determinant?


How do we interpret the determinant intuitively? Well, here is one way!

This video was requested by Thecalculatorman on reddit!

A few quick notes:

* There are limitations to this way of thinking about the determinant, but for the most part it's solid for 3 and 2D objects.

* Finding the area of the transformed unit cube is the same as finding the area of the parallelpiped, just a little easier to explain. In hindsight, I should have added this definition too.

* There is a lot I skipped over, like how to perform the determinant. That wasn't the point of this video. I wanted to give people an intuitive feel for what the determinant was doing underneath.

As always, the simulations were done live on:



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And the music is from Josh Woodward (sped up 1.5 times):


Thanks for watching!

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Channel:  LeiosOS
Topics:  Knowledge
Giorgos Sartzetakis 


Lbertarian Armed Fight 

my friend im in serach of this Q -> Is time the determinant of all events in the enviroment?

Lee Yuan Zhi 

3 blue 1 brown does a great job as well, highly recommend watching his videos on determinants


A lot of people don’t understand Mathematics because of lack of explanation like this!

1.21 Jiggawatts 

I was trapped in the matrix once but used a determinant to get out. True story.

sanchith jain 

I love ur videos

G Sho 

Truly a genius !

Tharindu Kavinda 

very helpful video for matrix

Adwait Shinde 

I love Mathemagics !!

Narender Goud 

After so many years, i finally understand. Thank you very much

Anastasiia Havriushenko 

Thank you very much for your detailed explanation and the channel in general!

jhonatan riascos 


azad khan 

I'm not understand anything


A perfect toturial, a terrible background music. Instructer, a good lecture does not need music, because mathematics itself is a beauty.

José Tobias 




Bolor Erdene Bayarmagnai 

I wish I had this video back in school days


They never taught this in class.

Ike Exeter 

Where’s Neo in this.0

Paras Srivastav 

Well if you would simply put it one sentence instead of making a video -> *It's the volume enclosed inside a parallelepiped whose 3 sides are 3 vectors formed by 3 rows of the matrix.* Now the real question is why determinant is always associated with a square matrix?

Pushkar Mahale 

Here we are told to mug up that product of eigen values is the determinant of a square matrix. Thanks for telling why as well.


What 4 years of engineering couldn't teach ... you did it in 2.51 minutes ❤

Mr V 

How about losing the annoying background music...?

alex Lo 

I wonder when Gauss had work in matrix. Did he have this geometric description in mind?

Sa B 

This video is really great! Thank you :D

Matan Kribus 

yea but what is that thing you did in the start of the video? im leanring linear algebra

ice breaker 



What can a determinant of a 4x4 dimension represent?

Jorge Mercent 

What happens when you apply a Matrix Transformation whose det=0 to a unit cube? What will be the resultant cube look like? Is it that there will be infinite possible resultant cubes with infinite shapes?

Suyash Sharma 

The first question that pops in mind is - Aligning the unit cube along the eigenvectors..... wait...what? How do we even know that the eigenvectors are all perpendicular to each other??? Doesn't it completely depend on the physical transformation being applied as to what 3 vectors will turn out to be eigenvectors???? Like stretching a plastic cube that transforms to a new shape... To be able to apply this type of restricted transformation, you should explcitly mention that - we are applying a restriction on the transformation now to match the volume of a regular transformation (with rotation involved) on the same cube.

Hope you understand my point. Bill Smyth has already clarified it the the matrix is symmetric "so that the eigenvectors are all already orthogonal" but if i'm asked to stretch a Cube A and then take a Cube B and transform it, strictly following the orthogonality, such that the end volume is same as the Stretched Cube A's volume, ofcourse the product of eigenvalues will

Armando Duette 

Al menos haberlo subtitulado al español si el título está en español ☹️

Praapthi Praapthi 

why our maths teachers does'nt know this.


Can't get it , what are you guys doing in the beginning , are you are multiplying matrix with cube?

Sourav Mukherjee 

I learnt much more in these three minutes than the entire semester class of linear algebra. It was really awesome and it gave me the feeling that I can see things instead of just solving mechanically

Jimi Khan 


Ruolin Jia 

The person who found determinant must be a ridiculously intelligent guy.

feridun abi 

but why

vijendhar bhupathi 

How transformation is made

Lv 69 Mafia Boss [MAXM] 

Wow good video. Watching this to get the same love I have for Calculus into Linear Algebra.

Reinaldo Maciel 

Perfect explanation. Incredibly that several teachers takes more than one class to explain it (or try to) .

Patrick HotelEchoRomeo 

Didn't know before that Eigenvector and Eigenvalue have their names from the German language. We call them Eigenvektor and Eigenwert. "Eigen" means something like "its own", "Vektor" means vector and "Wert" means value.


its very fun and easy to prove this with a 2x2 matrix and two vectors u and v that will undergo a transformation. Just calculate absolute value of det(u,v) to find the old area, then calculate the new area: absolute value of det(T(u),T(v)). Then you will easily see after some algebra steps that this new area is equal to absolute value of det(A)*old area



Time for you to know the Matrix your living in,


Anonymous Official

The Crow House


To raise your Consciousness,


Mind and Magick

Shama K 

Amazing video!! I never ever imagined determinants and eigenvectors this way... Thank you so much 👌👌

avinash chowdary 

why do we want to transform a cube?

Barzhikev Il 

This is so beautiful I wanna cry

matsu sumo 

Is it useful only if it is Affine transformation??

alex Marshall 

who was Eigen..he/she pops up regular..ie Eigentones

Srikanth kola 

How to form cube with matrix

Not understood

Pladimir Vutin 

I was never taught this when we learned about determinants. We were only taught how to find one, not what it actually was.

Xav Nimportnawak 

Thank you for that video.

I've red in a book that first, the determinant was found in the pattern of solution for equations systems. You shift the equation system with a matrix*(x,y,z) vector, apply the solutions pattern, and you have a determinant... I don't remember well... would make a video about that?