It helps that the matrix is symmetric so that the eigenvalues are real and the eigenvectors are orthogonal. Not to knock, though. This is a beautiful demonstration. Generations of teachers have taught the determinant like it's just an arbitrary combination of numbers that somebody pulled out of thin air (to put it politely). The interpretation as a volume expansion is intuitive, and it also explains all those other interesting properties that the determinant has. For example, the det(A*B)=det(A)*det(B) - of course!. How about inverses? The inverse just gives you back the original unit cube, so det(inv(A))=1/det(A). And if A is singular? det(A)=0, so the cube gets squashed flat. So of course the singular matrix has no inverse, meaning that the squashed cube can't be reconstructed. Very cool :)

I always used to try to understand what I was doing during calculating the determinant in the class. Now I could understand what I was calculating. Thank you so much! I wish may I had the teacher like you who could make me feel these concepts in bones.

Your 3 minutes video just changed how I view matrix.

Hey guys, this video is meant to give an intuitive definition of the determinant. There are oodles of way to calculate it and I kinda assume that people watching this video have done a determinant calculation before. There are a few notes in the description, but I needed this video for certain videos in the future, so it was definitely worth doing. How did you guys feel about the "More info" tags that popped up? Were they too much? I think it's a good way to cite previous videos, but if you guys have a better way to do it, let me know! Thanks for watching!

This is Eureka moment. Determinant, Eigenvector, and Eigenvalue. It's like after enjoying years of ham, bacon, and pork chops without knowing their relationship, one suddenly realizes they are all from parts of same animal. And this animal could give love and joy to the human as pet, and even a new life as heart valve. Great inspiration. Thanks.

The point is just that you are taking a linear transformation of rank n, from a vector space of size n to itself, such that all the eigenvalues are real (and all eigenvectors have period 1) which means that the matrix representing the endomorphism is diagonalizable over R. Then the important property is that the determinant is an invariant and so it's the same considering the matrix of the endomorphism expressed with respect to the canonical base and with respect to one of the bases which "diagonalizes the matrix". Then you can finish knowing that the determinant of a diagonal matrix is the product of the elements on the diagonal (aka the eigenvalues). Just wanted to give an explanation on why it works, the video was great

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

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

please get rid of the background music

After learning and using determinants, eigen values and eigen vectors for 5 years, finally understood what they mean!. this was some kind of enlightening moment for me, feels like now i have seen everything and know everything that i need to know lol. Thank you!!!

This is genuinely mind-blowing. I never truly understood what a determinant actually IS, I just took for granted that it somehow exists. Eye-Opening video. Thank you!

A note that might add to this video: aligning a volume V cube along eigenvectors, you get a scaled cuboid of volume V*det(M). Do arbitrary weird objects also scale in volume as det(M)? Yes: divide your object of volume V' into a large number of little cubes oriented along eigenvectors. After you perform your transformation, the little cubes will still be non-overlapping (assuming our matrix is full rank, that is, one-to-one), so you can just add their values to approximate the volume of the weird shape. As we increase the number of cubes, the volume before transformation goes to V', and after V'*det(M), just as expected.

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

this is beautiful! I've taken linear algebra courses in college but there's so much meaning and intuition behind it that I've yet to discover!

I memorized the property that determinant is product of eigenvalues without knowing why, and this really explains it, Thank you!

This one video was enough for me to subscribe (after glancing at the other videos you have). Thanks a bunch!

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

This is good that you give a clear concept with a reasonable reality based example... I really enjoying you:)

I just started leaning vectors and this makes me wanna scream but it does help me to understand in a way so thanks.

came here to understand determinants, now I also understand eigenvectors and values even more. Wow thanks

Finally I'm able to understand way more on what I'm working with on my linear algebra class. Thank you!

This is the first time for me to be able to clearly and visually understand the relationship between determinants and eigenvalues

THANK YOU SO MUCH LEIOS , IT MADE MAKING REVISION OF MATRICIES AND EIGENVALUES MUCH MORE INTUATIVE AND ENJOYABLE !!! :) :) :)

Very well explained, and kudos for the visualization of the concept!

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

since -3 is isolated in its own row and column for the determinant you could have just taken the determinant of the matrix at the top left times its cofactor at the bottom right (-3), giving you -3(1*1-(2*2)) = 9 Originally I though you would use the cofactor trick since the matrix was so nicely set up for it but since you didn't I thought I'd mention it

seriously such a beautiful video with good description

we learned linear algebra for 1 semester and now i finally know what all of these things mean in 3 min.

Simply exceptional! This is the video i wanted to see!!

Few words , much more understanding . just amazing !

Your voice is similar to the welchlab tutor , he is absolutely amazing , Especially the way he opened the complex number world in my eye.

I wish I had this video back in school days

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

Well, done. Swift, concise, yet clear. *thumbs up

Thanks, made me link the determinant from the eigenvalue matrix with the determinant of the matrix !!

I find this algorithm for the computation very intuitive. Ty

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

This video is really great! Thank you :D

Thank you so much for that I was strugling with it for a long time. Will you please make a video on Physical or Geomatrical meaning of trace of Matrix...

I'm not understanding anything.

You helped me in getting sense of my high school matrix!

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.

This has answered SO many questions, thank you!

So in other words the determinant of a Matrix is the volume of the transformed unit cube in that matrix space 👍 After many years I finally get it )) And now I get how it's useful, like normalizating a matrix vectors by dividing the elements by determinant? Like we do with vectors x,y,z/length

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

really a great video, just changed the point of viewing matrix.

does it mean that the determinant of a matrix (in dimension 3x3) tell us how much can we magnify another matrix (also 3x3 representing a cube) if we multiply the first one by the second??? If this is it, it´s astounding awesome!!!

incredible, I was wondering for so long what was the meaning of a det. Ty

Iam not understanding it completely..but made me to realize there is much more to learn in linear algebraa...thank u very much sir.

Sweet and simple

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 be the volume. This video explains a geometric interpretation, but lives on an assumption and a restriction to get something which then becomes only obvious.

tutorial was very helpful, thank you :)

How would you interpret negative determinants then? In this particular example.

Thank you so much.your explanations are so beautiful.

Ur a very unique teacher

Well that was startlingly easy; why did nobody explain it that way in school? I'd have "got" matrices a lot quicker that way!

This is so beautiful I wanna cry

What a POV changing video!!!"❤

The last determinant where he got a 9 right? It was all inside one matrix so what was the original dimension and what are the new dimensions of the cube?

but why do we compute the determinant of 3x3 matrix like that? is there any reason of hiding rows and columns and alternative + and -?

very cool stuff, thanks for sharing!

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

Thanks 😊😘

As someone who has suffered from matices for years,and I mean yeeeeeeeears,,Thank you

I don't understand what the initial matrix acts on? on a cubic equation? how is the equation of a cube expressed with a matrix?

An amazing video. Thank you.

I searched for "What is determinant of a matrix". Now i am left with more questions.

Thanks for the great explanation. 👍

What I don't get ....whats the difference between a norm ..and an eigenvalue ...if they both scale and stretch

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.

Love these short videos! Subscribed, what sort of videos do you have coming up? I'd love something with regards to Principal Component Analysis?

determinant show what statistically like mean values ,deviation ,standard deviation, correlation coefficient etc

Wasn't what I was looking for but mind blown anyways

Ur vdeo speaks volume 😄 thanks alot

so nice and elegant!

Now I have a intuitive sense of determent only 'cause of you thank you and God bless you

Excellent Video!

even the number of elements are 9 in that matrix xd...awesome vid i was curious about the reason behind determinants and what they are used for...this video made it so much easier to understand cuz my teacher just plays around with properties what i lacked is the reason to use them...but i am still curious those elements in the matrix what do they represent in terms of the cube?

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?

Can anyone tell me. What is the acutal meaning of determinent?

Amazingg!!

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

Wow thank you so much

does this generalize? is the determinant of a 2x2 increase in area and whatever it's called for 4 dimensional objects. What if the object you are transforming is not a cube? but some other arbitrary shape. Does it still work?

Hii but how to multiply a cube by matrix

So you're saying that determinant has a connection with eigenvalue and eigenvector? I might as well learn those 😃

i just knew how to find a determinant but never has someone told me why we find it..

I love ur videos

very good video!

brilliant sir

Truly a genius !

How to form cube with matrix Not understood

how could u align a cube in the direction of eigen vectors ? Are eigen vectors of any matrix are mutually orthogonal to each other ?

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

Good job, thanx bro.

Great explanation! While I get the idea the determinate is the factor we scale the original one, but I'm still wondering how can a square matrix and its transpose have the same determinant intuitively? I can check the formula det(A) = det(A^T) by induction for a square matrix A, but how to understand the intuition behind it? Thank you!

what is the transform that you have applied to get the new volume?

You make it so obvious😭. We just studied that how to learn

Marvellous

Excellent!