Why is the determinant like that?

so why is the inverse matrix like that?

Thank you for this video 🙏🏻❤️

Amazing visuals

Nice trick

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! 😊

😅😅😅😅well information good show you 😅😅

Thanks!

Is there any geometric explanation (in terms of vectors) why determinant is unchanged under transpose?

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.

Too fast.

The missing part: why do you assume that determinant is a volume in space of some shape?

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

geometric algebra?

Nice

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!!

I've been trapped in this for years. The moment of understanding make you my father.

Hey bro ,you just stole the concept of vector spaces and linear algebra but it's kind of nice the way you presented...........

Det.= delta...! Area vector perpendicular to surface!!

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

Like that what ?? 🫠

This right and left handedness of the det. For me it's a demonstration that higher dimensions are implicit. They're facing the other way in a plane in 3d. Likewise a 3d object can face two ways in 4d and on we go.

In fact, the convention of dimensionality is wonky. If a 2x2 matrix is linearly dependent then we say it's 2d (with a rank of 1 admitedly) but the cofactor will never project into the 2nd dimension unless you take the bizarre view that the line is 2d because it's not flat. Anyway, that's only really a debate on terminology, it doesn't change anything fundamental.

That is the clearest explanation of determinants I have seen!

Arrays can be used for cutting-edge encryption where two points are constantly exchanging information to continue a connection. Second, they can make cryptocurrencies non-mineable and ultra-light since the only type of information that matters in this system is not the history of positions.But only the current positions in the matrix, millions of operations can be done per second. If X number of errors occur, the operation is terminated. Cryptography as a living body in motion. And beyond Furthermore, it can be used in biometric signatures in some way, I also foresee.

Not gonna lie - I got lost along the way, but still watched it and I really appreciate all the visuals and work that was put into this video. I missed though explanation what is actual determinant. From video it looks like it's the area, but is it just an area, or something else? I would watch a video explaining what we use it for. For example how it relates to systems of linear equations? Or how it relates to Jacobians? Matrices is some wild shit for me.

I don't remember much from matrices, but I remember determinants refered to some matrix properties. For example, if 2 rows are the same, then the determinant is 0. Is it because they basically "overlap" so that the area is 0 as you said? Or also, when there's column or row of zeros, then the determinant was zero. Is it also because the area basically becomes zero?

I remember some time ago I teached one guy how to calculate 2x2, 3x3 and 4x4 determinants and some other matrix operations. I just knew how to do it from the college and he just started and needed it to pass some test. But when he asked what it's used for, I just said I have no idea. All I could tell him that matrices can be used in 2D and 3D graphics and that there are things like rotation matrices. So I knew they are used there. But I had no idea what to tell him about purpose of determinants. And I felt bad, cause I always like to explain something intuivitely, and I lack this intuition for matrices.

mannn that's deep

such a well animated video! i never thought of the determinant like this

Great video

Can you give Source Code ??

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.

To really define determinants the right way you do need to see tensors and multilinear forms first. When you do that you learn that for an n-dimensional vector space V, the space of alternating multilinear forms V(alt) is 1 dimensional. So given any endomorphism f from V, you get an endomorfism f' from V(alt), and you can write is as f' = c I, where I is the identity and c is a unique constant (that is because a basis is precisely the identity). That constant is the determinant of an endomorphism, and that approach provides a basis-free definition that simplifies a whole lot every proof involving determinants. The harder the definitions, the cleaner the proofs will be. When you mess with that horrible formula everything becomes obscure and cumbersome.

this was really enlightening, thank you.

Very interesting video of the topic, will come back to watch it again.

Nice

very intuitive

THE ONE PIECE
THE ONE PIECE IS REAL

Idont think that A(v+u,w)=A(v,w)+A(u,w) in that example

Think of the 3x3 matrix as a cylinder. Then its determinant is the sum of its down-and-right diagonals minus the sum of its down and left diagonals.

uhm i think u should prove why determinant is the span area or volume of basis vector

what a video this is

Nice... Which software?

Excellent treatment!! I will share this everywhere. Also, I almost fell out of my chair with your closing statement. So funny.

Can somebody please tell me why A(v+u, w) = A(v, w+u) in Rule 4. I've been trying to graph them like in the video, but the areas don't seem the same to me. can't get my head around it.