What is a determinant? 

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 :)

Your 3 minutes video just changed how I view matrix.

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.

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

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

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.

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

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

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

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

This has answered SO many questions, thank you!

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

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

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

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

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

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

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!

This video is really great! Thank you :D

seriously such a beautiful video with good description

Thank you so much.your explanations are so beautiful.

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

Few words , much more understanding . just amazing !

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.

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!

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

so nice and elegant!

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

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

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

very cool stuff, thanks for sharing!

Excellent Video!

tutorial was very helpful, thank you :)

Thanks for the great explanation. 👍

An amazing video. Thank you.

Truly a genius !

I find this algorithm for the computation very intuitive. Ty

very good video!

I love ur videos

brilliant sir

Ur a very unique teacher

Excellent!

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

Sweet and simple

Ur vdeo speaks volume 😄 thanks alot

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

This is so beautiful I wanna cry

Wow thank you so much

I wish I had this video back in school days

What a POV changing video!!!"❤

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

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

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

Amazingg!!

Good job, thanx bro.

brilliant

Thanks 😊😘

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

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...

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

very nice

Marvellous

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

hey thanks for this

I'm not understanding anything.

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

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

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

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.

Lovely....thanks

Thanks for the video. Is it so that when the determinant is negative, the volume of the object always reduces no matter what? Also, if the determinant is nine , does that mean for any new volume to the transformation, the ratio of new volume to the old volume will be nine?

WOW!

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!

Amazing...

thanks a lot for info

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

very helpful video for matrix

Nice

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?

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

wow!!

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

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

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?

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

Dude cool

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?

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!!!

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.

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

Hi! Could you tell me how are you importing LaTex to a video editor?

Good

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?

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

So, if any vector is having some x,y,z eigenvalues then the determinant of that matrix will be xyz Am I correct

Thank you..

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?

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