Cramer's rule, explained geometrically | Chapter 12, Essence of linear algebra 

Perhaps some of you are wondering why, 2.5 years later, I've come to insert a video into this series. Does it mean the start of an extension to the series? Er...no. Or rather, not yet. I'd been sitting on this video for a while, thinking I'd wait to put it out until there was a larger batch of new linear algebra content. But other plans have risen above that in the project list, so it seemed a bit silly to keep it unpublished for too much longer. In a few weeks, I'll start putting out some content for a miniseries on differential equations, so stay tuned for that! And after that...well, actually, I have a bad habit or breaking promises, so I'll keep the forecasting to a minimum here :) Fun little challenge puzzle: Use Cramer's rule to write down/explain the formula for the inverse of a 2x2 matrix. What about 3x3? 4x4? ---- Edit (correction): In the video, I describe matrices which preserve dot products as "orthonormal". Actually, the standard terminology is to call them "orthogonal". The word "orthonormal" typically describes a set of vectors which are all unit length and orthogonal. But, if you think about it, dot-product-preserving matrices *should* be called orthonormal, since not only do they keep orthogonal vectors orthogonal (which, confusingly, several *non*-orthogonal matrices due as well, such as simple scaling), they also mush preserve lengths. For example, how confusing is it that we can say the columns of an orthogonal matrix are orthonormal, but a matrix whose columns are orthogonal may not be orthogonal. GAH! Maybe my casual mistake here can help nudge the tides of terminology towards something more reasonable, though of course that wasn't the intent.

6:52 immediately got so excited when you went to 3 dimensions, because I knew I was going to get to hear you say "parallelepiped"

God, I love how I understand everything but after 5 minutes after watching the video I forget everything.

This channel is truly amazing- so original and so much work put into it. Keep up your amazing work!

taking linear algebra this semester with an extremely difficult professor. your whole series has helped me in ways you will never know. thank u so much.

Cramer's rule (written as the product of A and its adjusted equalling the determinant of A times the identity matrix) is not just important for the reasons given in the beginning of the video but also for other reasons. For instance, if your matrix is made of integers and the determinant is +/-1, then you know that its inverse is also made of integers. This is useful when dealing with matrices whose entries belong to a general ring.

This channel is a continual reminder for why I love math.

In Poland we used to have spoken math exams when we needed to explain everything that we're doing and why. And the method from this video is called "metoda wyznacznikowa" (determinant method). When one student was asked why it's called that, he answered "it's because Viznachnikov invented it".

i love how 12min can become 5 hours

You're getting so close to Geometric Algebra! (Oriented volumes: just get the wedge product involved and you're basically there.) Take it all the way! We're ready! We need it! :)

I studied Cramer's rule since my high school days including determinants and matrices but never took it seriously thinking that its just a fancy way of writing numbers and performing operations and now I realize how important it is to the world of mathematics. love this channel.

I was struggling with this video at first. I don’t know why but I found this idea a little bit hard to grasp but after watching this video for four times I finally understood what you were trying to state. And it was utterly beautiful.

Such an intuitive explanation of what we learn only in the abstract mode in our schools. Thanks for existing 3b1b. Can you also do Hilbert Space and its application in Quantum Mechanics?

Just when I was going to sleep Sleep can wait

You just made sense of a lecture I struggled through 35 years ago. Thank you, it now makes sense.

Man I always wondered in my math class how this was possible. We never had any visual intuition, neither our teachers wanted to show us. That's how freakish bad educational system is here. Thank you man. Grant, I wanna thank you in person! 🙏

You are an artist !!! Kids in grad school everywhere will learn so much faster because of how visually you can communicate ideas.

This was just brilliant! Couldn't ever think cramer's rule could even have such a relation with geometry!

a SEINFELD REFERENCE in a 3b1b video MY LIFE IS COMPLETE

I am so dumb, I need to listen to the video several times to get the whole idea but I love it 3b1b, thank you so much for your work!

I wish I had teachers like you in school and at the university. You present everything in such a fascinating way with the visualizations. Maybe I wouldn't have lost interest in computer science program, if I knew how this all relates to geometry and space. Keep up the work man, your videos are gold!

This video is sooo good!! We just briefly rushed over Cramer’s rule in one day in my precalc class, with no actual understanding at all. This makes it so much more clear and satisfying! Keep up the amazing content :)))

Wow, I never thought I would be this early for a video. Sure it's gonna be great. The whole Linear Algebra series is fantastic!

1:46 - 1:57 - That simple visualisation alone would be a perfect way to teach kids the meaning of those equations in school. Instead, children are often told not to visualize algebra, which is a missed opportunity for many of them, if you'd ask me.

My teacher teaches me how it works and you teach me why it works. HUGE THANKS TO YOU SIR. please keep up the good work Thank you, love from India

why u upload at 12:35 now I gotta deprive myself of sleep.

Everytime I watch your videos all my sadness and depression goes away. I'm very happy and amazed at the amount of clever ness went into these concepts. I wish i learnt all this in my high school. "Ur videos make me wanna live to see this beauty".

What makes me really REALLY happy, is that other than most math channels, you have black background. Almost every other math channel has bright white background and i am just trying not to go blind. I got my phone on lowest brightness settings and i still have to flinch my eyes to be able to watch them without physical pain.

the essence of calculus and linear algebra series(serieses?) are truly amazing ...extensions of them would be so cool and appreciated ;D

Can you please, make a video on Hilbert Space and its application in Quantum Mechanics?

The ability you have to convey almost anything in a clear and intuitive way really shows how smart you are. You and Richard Feynman made me love mathematics

I was just thinking about this topic yesterday, and how it works. Thanks for the in depth guide.

I learned matrices last year and never understood Cramer's rule. thanks for the vid :)

lol I was just telling someone about a 3blue1brown video and here comes another one! And it explains Cramer's Rule!

Rainy Sunday morning, coffee, chocolate and this video. There is nothing else I could ask for! This is perfect!

This explanation is gold. Much more illuminating than the straightforward but obscure proof using the properties of determinant.

0:00 Intro 0:50 Why learn it? 1:28 The setup 2:37 Types of answers 3:14 A mistake to learn from 5:26 The take-away

I watched the whole series again because this video came out, and it just so happens I’m also concurrently taking a rigorous linear algebra course. It’s thrilling to me how in depth this series goes (and how little of that depth I picked up when I watched this 2 years ago) and seeing these topics I understand in a very different perspective. I’m very excited for the differential equation series to come, since I’m taking that in the fall!

Your channel is the only place where we can see and feel Mathematics rather just scribbling equations! Just loved it! Good job!

Oh dang dude, you were completely right! Did a few Cramer Rule exercises and the concept of the dual vector clicks! Appreciate it!

I'm not sure if you are into Mathematical Logic but I 'd really love to see a video from you on Gödel's Incompleteness theorems. Your channel is amazing, thank you and keep up the good work!

3b1b's contents have always been really articulate. The topics in the past uploads have been very complex :(

this is really good explanation I mean i was never taught why cramers rule works and its really comforting to understand these things

Could you please do a series on Abstract Algebra? (Groups, Rings, Fields etc.). Thanks!

Hi Sir!! could you please make video on convolution and correlation?

Brilliant. I wondered why Kramer's rule worked since highschool and I finally got (and understood) the answer

Your videos of explaning these concepts in the simplest and an intuitive manner will have such huge positive ripple effect in this world.... Thank you for your selfless service to the humanity. 👏🏼👏🏼👍

Please do a video on Laplace Transform!

You can generalize Gaussian elimination from solving systems of equations to finding inverse matrices, just by doing it on multiple columns at once. Applying the same with Cramer's rule gives A^(-1)=adj(A)/det(A). Neat!

sweet. Just started Lin Algebra1 at uni and your videos are a big help

When You figure it all out, it feels like suddenly someone just taught you the magic of nature, Thank you Sir, You are doing a wonderful Job.

2nd best moment of the day: "Ah, a new 3b1b video!" Finger on the mouse button goes _click_ Best moment of the day: About halfway through the video, "Ah, I see where you're going with this!" Brain goes _click_

Enlightening. Just purely enlightening! I think the key to understanding here, as pointed out in the video, is that under linear transformation all areas (or volumes in 3d case) change in the same way, so that the RATIO of change is the same. Cramer's rule is really all about this change. Rearranging the equations to reflect this ratio of change really helped me digest this one. I've never taken any linear algebra class before, but this brilliant series makes me really want to learn much more about the subject. To enlight, not to daunt, students, is the only golden standard of teaching. Can't imagine how much happier and more satisfactory students could have been if they were taught this way in school. Oh man, this even makes me want to become a teacher like him. Keep up the enlightning process, please!

You truly are a blessing to mankind

I have probably covered a month of coursework with this channel within a day (counting exercises). I have donated to a couple of videos but honestly, I cannot pay you enough for your service

ax+by=c where a and b are vectors wedge each side with b(find the area of the parallelogram formed between the vector and b) (a^b)x+(b^b)y=c^b a vector wedged with itself is 0 x=c^b/a^b repeat with wedging a on the left to get y=a^c/a^b

Was taught this in high school out of context, felt pretty detached from reality, glad to see the sense behind it

For 3 x 3, we have z=det (i, j, mystery) y=det (i, mystery, k) x=det (mystery, j, k) Then after the transformation, we have that x det A = (output, j, k) ydetA=(i, output, k) zdetA=(i, j, output) And the rules follow for x, y, z

Ok - this merits a second viewing when it's not bedtime, AND where I have time to do the 3d exercise at the end! Thanks so much for making this Grant, it's awesome.

Laplace transforms and ODE series!!! (I loved the Fourier transform too) please...

considering i got this right, it is amazing that cramers rule also works in 1D, where it comes down to just a linear equation (a*x=b), where x=b/a. x=det(b)/det(a)=b/a

As lovely as always :) Could you perhaps cover tensors in the future?

We study all these things in highschool but we're never told about their use in this field, for this reason I find these videos mindblowing.

I will be delighted if you start a serie about differential geometry and curvature

More linear algebra for the win! Thanks for the awesome video Grant! You should do a video on differential forms and the generalized Stokes' theorem. That would just be fantastic! I just finished reading Vector Calculus by Hubbard and Hubbard and would love to see some geometric intuition into the crowning theorem of that text. Thanks again, you're the best!

Just when I was about to say Cramer's rule was impossible to understand geometrically, 3b1b has come in to save the day!

3b1b video! About Cramer's rule! Explained geometrically! On my birthday! 🎉

I always questioned myself about why doing that proccess i can correctly calculate the variables values, It is not intuitive, its so beautiful to finally understand It, i just feel like some kind of gift has been given yo me, thank you!!!!!!!

The best explanation for it! I have never seen such kind of explanation but the old "a matrix is a function of a determinant"...

A 3b1b and a Mathologer video on the same day! What have I done to deserve this?

8:39 the determinant before transformation is 1. for unit vectors so the change of signed area is y*determinant of transformation 1 goes to 1*determinant value y goes to y*determinant value

This argument works directly in exterior algebra (or its generalization, geometric algebra). Ask your professor about exterior algebra today!

This channel truly is a bless. I remember I watch this series when they were posted, just before entering engineering school, and it really gave me interest in math, and in particular the intuition you give is great. Thank you

I love this series so far... It's clear, interesting and encouraging! Sometimes I even pause the video and try to figure out by myself beforehand, which I never do during class. All thanks to the enlightenment of this video. You really make me change my way of thinking maths. Frankly, this is the first time ever in my life I think maths is actually interesting. Thank you.

Just saw this new video when I was reviewing for my linear algebra final!!

Amazing video! So the key idea is that the determinant of any matrix M basically represents the area of the shape whose edges are the column vectors of M. And we learned from previous lesson that det(A) is the area scaling factor of any shape in the original vector space. Combining these two principles we have det(T(i), T(v)) = det(A) * det(i, v), where det(T(i), T(v)) represents the area of parallelogram whose defining edges are T(i) and T(v). and since det(i, v) = 1*y - 0*x = y, we get det(T(i), T(v)) = det(A) * y, and consequently y = det(T(i), T(v))/ det(A). Quite amazing how the formulation of this rule is so easily understood under visual interpretation. Keep the videos coming please!

I love your channel, it certainly makes me enjoy learning and visualize everything. An small quotation. Crammer's Rule is actually awesome when you dont have a numerical matrix but one that uses variables, such as the ones we use to define regressions

this is channel is the best thing happened to maths

Usually you can think of simultaneous equations as 2 lines and finding the point of intersection, but you can also think of it as 2 points and finding the line that connects them. I did some calculations for this a while ago and ended up with a determinant on the denominator and it’s nice to see why that happens

Usually when I'm watching lectures on RU-vid I would turn on 1.5x and watch as fast as possible. However when watching 3b1b's video, I never skip a single second.

One of my life wishes is to have one video like this explaining laplace's theorem on determinants

It amazes me how humans can think of these sort of nuances and actually discover something while doing it

Guess what? I probably will never forget Cramers Rule again. Thanks a lot. Amazing lectures.

Brilliant exposé as usual. I struggled around 9:27 with the reasoning leading to the numerator Area to be understood as a newly constructed determinant. It took me too long to grok that any parallelogram shaped area corresponds to a stretching of the i- plus j- hat square by an amount defined by the determinant of a square matrix whose column vectors define the parallelogram . So just as y is unknown so also is Area unknown. But y is equal to Area/det A. Area is the determinant of a new matrix constructed as the known transformed i-hat column vector (first column of A) with the known transformed {x,y} which is the RHS of equation ie. the known coordinates of where unknown {x,y} ends up. Very obvious: after my struggles. These videos are priceless because they offer beauty also and even to those with my very modest math skills.

after meditating over this for a while with pen and paper, straightforward and really cool explanation

the linear algebra is so beautfiul! thank you for showing us that!!

5:18: that is because in an orthoNORMAL coordinate transformation, no streach occurred. So the projection of yellow vector on vector i (base 1) is the old x, you can rotate the entire picture clockwise back and see. This episode tells us that: On the old coordinates, every base (i, j, k...) is size 1. Old volume of parallelepiped (form by X0, j0, k0) is x0. New volume of parallelepiped (form by X, j, k) can be easily calculated using the given numbers. New volume is stretched by det. So: x0 =oldV = newV / det

I'm very much looking forward to an "Essence of Differential Equations" series of videos if that's what you're planning on.

11:53 never knew it took this many people for one episode. No wonder the top-notch quality!

I watched your playlist before the beginning of my Linear Algebra course this semester and it gave me a great geometric intuition, but at this point in the course the intuition has started to get swallowed by all the weird math stuff. Excited to watch (when I get the chance) this video to hopefully reignite that wonderful intuition!

9:35 the idea is to create a specific area( two variables: one known and one unknown) and of course we need to assign it to the vector that we looking for so we make (unit length)*(one coordinate of vector) why is unit length (i)? because it will be transformed and we know by how much ( the rate transformed green vector) why do we create area? because we can use rate of change of the area ( the determinant) and the transformed area( known transformed green and red vector ) to know assign area( with one uknown 1*y)

I quickly forgot about Cramer's rule when I was taught it, but now I'll never forget it. Thanks 3b1b!

to digest 12 mins content i need 12 hrs of rigorous stduy . This is god level explanation

This channel is such a gem

It should be a crime for a series this thorough and amazing to leave out SVD

this is the first time i have understood why cramer's rule works. i have looked for explanations for ages and nothing got through. thank you so much for this. this is so freaking clever

Your channel & content represents an essential milestone in the evolution of maths education

A very nice geometric understanding of Cramer's rule, that I didn't see at all until now. It was just algebra for me. Thanks. 1:34 But Gaussian elimination is also pretty geometrically! You change the basis of the target space to the standard basis so that finding the solution is easy, but at the same time since you're doing row operations you don't change the row and null spaces so you're left with the same solution to the re-posed problem. I think that's rather neat.

Hi @3Blue1Brown, I took some time to add Spanish subtitles to the video. I found that the English subtitles are misplaced too, and some are missing, if you want to take a look at them. Cheers!

More linear algebra videos? Christmas came early !