n x n determinant | Matrix transformations | Linear Algebra | Khan Academy

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Defining the determinant for nxn matrices. An example of a 4x4 determinant.
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[@PaperBagMan884]

Thanks! I wish my professors were as fluent in teaching as you are.

[@lou7573]

i wish mine too:)

[@globotron27]

great for nontraditional students who didn't do this stuff in high school. I need these things "dumbed down" (I mean that in the most respectful way). Thanks for the help!

[@cheerfultrout4381]

I noticed while I was watching that the recursive rule also applies to a 2x2 matrix. therefore I would argue that even more fundamental than the definition of the determinant for a 2x2 matrix (7:55) is the determinant for a 1x1 matrix, which is itself. (and the recursive rule can also be applied to the 2x2 matrix in this way) It's the same thing really, just thought I'd point it out for people like myself, for whom looking at it like this makes it easier to understand/remember.

[@aliz9481]

👍

[@victorserras]

It's not that the recursive definition "applies" to a 2x2 matrix, it's that the 2x2 fits into the general definition. Think about the *ad-bc* for the 2x2 determinant as just a practical device so you don't have to use the definition. Kind of like a "cheat" formula.

[@jacobsilcoff]

This video was really helpful for my comp sci class. Thanks so much!

[@GodofMind3]

have an exam tommorow .. Dosent matter you there ! my savior thank you !

[@oxymon100]

Hoooly shit, you just gifted me with the ability of enjoying the beauty of math! Thanks dude!

[@moghimiiman]

final exam in the next 3 hours, thanks :-D

[@frankeinstein719]

did you pass?

[@AliShahidVlogs]

Mine in next 4 hours , and iam scared as hell :D

[@zenchiassassin283]

What year in bachelor? Maths or CS or ?

[@celeryystick]

@@zenchiassassin283 you'll never know

[@zenchiassassin283]

@@celeryystick lol

[@yeetelite9766]

I found [+|-] at the bottom of the Wikipedia page "List of mathematical symbols", it's called situational plus or minus in case you want to be really specific about the sign depending on the size of the matrix.

[@tinguspingus6729]

you are an angel from heaven

[@rengstrom]

Wow, I actually get it! First math teacher who actually taught me anything! :P

[@Kaish3k]

Cool, I was making a Class Library in C# for matrices and needed a method for determining the determinant. It was much easier than I originally thought it would be to implement (because of recursion). Sal, you should do some Computer Science videos. I'm tired of everyone asking me to tutor them.

[@mensaswede4028]

Actually the 2x2 case can use the recursive definition also. If you just define the determinant of a 1x1 matrix as the value of the single entry, then the 2x2 matrix reduces using the recursive definition to the sum and difference of the top row entries plus/minus the determinant of the 1x1 sub-matrix.

[@dilly9986]

thank you so much, this is brilliant, it made such an impact!

[@rodrigomatos2465]

Thank you so much!

[@epgmw]

Thanks a lot, very helpfull indeed. Keep up the good work

[@Kevin-jm1go]

This video is almost 100 years old, but it's still gold

[@mensaswede4028]

Looking at the comments, it seems the only way some people would understand is if you did motion animations showing the sub-matrices appearing from the original matrix. I bet the light bulb would go on. Because it actually is pretty simple, but hard to show by hand drawing.

[@kylahvatum4100]

In the second submatrix you multiplied -6 by two twice. (0x0)-(2x3) = -6 which is correctly stayed but you multiplied it by two and put it into the formula to multiply it by a 2 again. I hope that makes sense… unless I am misunderstanding the procedure. This can be noticed at the time 16:11 and on.

[@dbmasters]

Multiplying by 0 is the shiznit. Sal, you rock as usual.

[@kirammanenjoyer]

THANK YOU SO MUCH!!!!!

[@alfredenglund]

amazing! thank you!

[@georgethemagiclamb]

Thank you! Very helpful and straightforward.

[@ChristopherHunterSmith]

Quiz tomorrow, thanks for your help!

[@Itzak15]

O'boy, I can't wait to do these computations!

[@Counteris16]

Okay... The introduction to matrices and multiplation is enough for me for today...

[@iblueren]

still informative. thankyou

[@pomegranate8593]

I was about ready to drop out but now it all makes sense

[@melikaarshadi2318]

Great video! Thank you. I'm just wondering why you switch sides with minus, so you suddenly do an addition?

[@ІванПилипко-у9с]

Very useful

[@fetish1989]

sal why dont you teach me the math behind electromagnetics. could sure use your help there haha

[@milanvagner9957]

Great video but now i have the task to count the determinant of an nxn matrix. We dont know how many columns and rows are in the matrix but still i'd be expected to give a strict answer (number). Im sure some tricks can be figured out in these kinds of excercises but so far i couldnt do them. Please let me know if you know about any tutorials of similar excercises on youtube.

[@turjo7934]

This is really AWESOME !!!💋

[@Mikh5]

u rock!

[@Aghoriii]

thank you Sal you are amazing

[@TheBullyNgitPangit]

sir.Khan at the yellow color 0-9 why u didn't multiply (-2 | -9)

[@njabulomahlalela2912]

that was awesome

[@DeadKatBounce]

frickin awesome dude

[@VelAmi19]

God bless you dude I really mean it ! =)

[@cnradam]

thank you so much

[@zenchiassassin283]

Thanks!

[@shobhitkhajuria7464]

can we write the general expression of determinant in a very useful form?

[@ChrisGkay08]

Great, how would you code this on matlab using recursion.

[@MrCalhoun556]

Lol I tried solving this via transforming it into an triangular matrix and then just taking the determinant by calculating the product of the primary diagonal elements. Turned out that I copied a number wrong and I kept wondering if I have done anything wrong since my determinant always became 0. :D I also once used Laplace to calculate the determinants in some homework. Turned out I've forgotten about basic determinant rules and so I calculated the easiest stuff with the most difficult way.

[@atpham4366]

you save my life

[@williamreyes3801]

Finally!!! I swear my linear algebra teacher sucks. 20 min on KZbin is better than an 1hr of lecture.

[@dannyboy12357]

i appreciate your vids : )

[@M0vir]

I see a previous exam is computing a 6x6 without a calculator, is it really that bad? Seems really time consuming, even if you understand it.

[@TheTahlia1988]

I love you man!!!!!!

[@アナキンスカイオ一カ]

I wish I could see some demonstration or deduction of the formula.

[@legacies9041]

You can derive it yourself from the definition of A inverse

[@smitashripad9757]

But why are we changing signs???)

[@legacies9041]

Because of dimensionality. The dimensions are like you hands, your left hand is the inside out of your right hand. Think of a glove! Hence the negative

[@johnfei434]

Looking at these comments before me made me wonder "hm.. maybe my profs were in the same spot as I am now 10 years ago and looked up this video as well"

[@spyhunter0066]

What you say around 0.42 second about changing the sign of a_12 may not be correct! My book says differently. Could you please check this part! For instance for 3x3 matrix, D(A)=a_11A_11+a_12A_12+a_13A_13. Capital As are the cofactors.

[@mahiaslam6738]

Great,

[@narical]

sal u should post videos for computer science major !!!!!!!!

[@wilsonproperty981]

damn my brain wanna explode!!

[@7orqu3]

the cramer method is fastest way of getting a 3x3

[@fariskhan6027]

How we solve matrix more than order of 4 , 5,6,7 etc

[@rzrx1337]

would this work the same if there were negative coefficients?

[@mismagiuz]

RazorX53 Yes, you just need to be careful about the signs.

[@parthmistry8628]

which software do u use in this lesson and other lessons? +Khan Academy

[@Jonasmelonas]

Why is the sound gone both for videos I watch of Khan and PatrickJMT? All other youtube videos have functional sound! Why must it target the two math gurus when I am in the most of need?? :(

[@IdoZemach]

But why does this work? I mean, why does when the det=0 when calculated this way makes the matrix not invertable?

[@mattae2037]

4 years ago, but I'll reply incase it can help others. You take 1 / the determinant and then times it by the augmented matrix. If the determinant is 0 that means 1 / 0 = 0 and if you time the augmented matrix by 0 then it also becomes 0. So in a way you could say that if the determinant is 0, the inverse of the matrix is the empty matrix, but that sounds kind of weird :P

[@shravandk423]

but why is it that the determinant results in the same value regardless of the row or column from which it is expanded

[@mikevermeer1639]

nice

[@jakec904]

what about that formula?

[@APh_]

yea, QQ some more like you know his algebra teacher

[@vincentv8991]

This is Cofactor expansion?

[@sadhlife]

No.

[@killjoysplace]

16:07 how is 2x3 = -6 ? it's supposed to be 6. Then -2x6 = -12 and -2x(-12) = 24 is the final result of that part.

[@killjoysplace]

+MrAlpha545 Ow, ok I get it. Thx

[@tomasbeltran04050]

so it never simplifies?

[@yeetelite9766]

┌   ┐  │ aₙ ₙ aₙ ₙ₋₁      ⋯ aₙ ₁   │  │ aₙ₋₁ ₙ aₙ₋₁ ₙ₋₁   ⋯ aₙ₋₁ ₁   │  │ ⋮ ⋮ ⋱ ⋮    │  │ a₁ ₙ a₁ ₙ₋₁     ⋯   a₁ ₁ │ └      ┘ So I had an idea for using less wording, but now it's now turned into a question. When defining a reverse n x n matrix, would the first element still be a₁ ₁ or aₙ ₙ? Does the order flip along with the matrix indexes?

[@SergioSovi]

from Hyperlink?

[@dipankarbasak07]

This is 4x4 det. Where is the nxn det.?

[@kanck7909]

How about the determinant of a 2 by n matrix which won't have a basic (ad-bc) when you ignore the first row.Then, how do you do it?

[@ikbeneenpop1]

u can only invert a square matrix, determining the determinant of a 2xn matrix isn't defined. U can only have determinants for 1x1, 2x2, 3x3 4x4... etc matrices

[@kanck7909]

Pekky thank you but that is the deterninator of a 1x1 matrix?

[@ikbeneenpop1]

what do you mean? do you mean what the determinant of a 1x1 matrix is?

[@kanck7909]

Pekky ya because you can't do the ad-bc so you can't find the inverse

[@ikbeneenpop1]

the determinant is of a 1x1 matrix is just the value what's inside the 1x1 matrix, that's just defined. for example: the determinant of [12] is just 12, so the inverse of A=[12] is just A^-1=[1/12]

[@Syeal7]

"We are afraid of what we don't understand." ... Well I guess you dont get this then ;P Haha naah just kidding :))) Cheeers maan!

[@AK-gt8zy]

gauss-jordan elimination determinant anyone?

[@lordmcswain1436]

Yes, but only because 1/0 is undefined.

[@汤庭瑍]

u r not going to done 5x5 and 6x6 matrix question in 20 mins!

[@almaanides9127]

i keep getting -7 and cant find my error, anyone get the same thing?

[@671GrainsOfOwn]

you get, you get…..YOU GET sorry i had to make a snarky comment, thanks for the help!

[@MobiusCoin]

I know this is just the natural function of numbers and has no sentient quality to it whatsoever but somehow this feels evil...

[@ninjaturtle205]

[@lordmcswain1436]

How dare you insult your linear algebra teacher!

[@asshumanoid8438]

thank you so much, finally I understood matrices

[@eddyseid5837]

*Matrix

[@derick3253]

Eddy Seid Matrices* lol are you even in linear algebra?

[@eddyseid5837]

Derick Pinkerton​ Lol u little turd, r u even capable to understand a joke? And ur answer is yes, i m 20, i m in mathematical analysis, this stuff is for 17 yo people

[@derick3253]

Eddy Seid 20 years old and doesn't know how to speak English? Khan has videos on English grammar too I suggest you watch a few

[@ZachONeal]

Thank you so much.