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

This rule seems random to many students, but it has a beautiful reason for being true.
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Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld

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If you want to contribute translated subtitles or to help review those that have already been made by others and need approval, you can click the gear icon in the video and go to subtitles/cc, then "add subtitles/cc". I really appreciate those who do this, as it helps make the lessons accessible to more people.

Music by Vincent Rubinetti.
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Various social media stuffs:

[bluebrown]

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 But other plans have risen above that in the project list, so it seemed a bit silly to keep it unpublished for too much longer.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?


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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.

[dcs_]

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"

[IulianAxiomAVI]

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

[MrJoshie_]

This channel is truly amazing- so original and so much work put into it.

Keep up your amazing work!

[crosserr]

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".

[meaninglessjunk]

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.

[johnchessant]

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

[OnTheThirdDay]

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.

[zubaidakarimjuthy]

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.

[apoorvmishra]

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.

[MattWoelk]

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! :)

[BigJohn]

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

[quahntasy]

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?

[guiselic]

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

[NavjotSingh-dyiu]

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

[MrDaanjanssen]

Just when I was going to sleep

Sleep can wait

[hemanthkotagiri]

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! 🙏

[esnaw]

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!

[snowy]

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!

[halyon]

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