Evaluating the Determinant of a Matrix

When it comes to matrices, beyond addition, subtraction, and multiplication, we have to learn how to evaluate something called a determinant. This is a new concept that only applies to square matrices, so let's learn what it means and how to do it!
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[@katiepickrel]

if you all are are looking for a quicker way, you can put your matrix in upper triangular form and take the product of the diagonal of your square matrix. (e.g. make sure all your entries under the diagonal are zero and then multiply all your diagonal entries=det of matrix.) Its way easier for larger matrices.

[@ashflint23]

Ty so much!

[@eldersprig]

What's a permanent? Actually, *what* is a determinant? yeah, I know its *area*, but I don't know what it really means or what you do with it.

[@jds189]

kzbin.info/www/bejne/f6GWiWyChM1lms0

[@nebulousdraconic4642]

@@jds189 Incredibly helpful series, thank you

[@sivakumar-oh2if]

Same question here why we need determinate

[@dominus_ignaviae]

would highly recommend 3blue1brown's linear algebra series! give it a try!

[@emlmm88]

Oh man, I'd love full courses in algebra and linear algebra from you. I have a feeling that your exposition of Galois groups and multilinear forms would be legendary. Great work, as usual.

[@debilista]

MY PROFESSORS SUCK SO BAD they do not know this method it appears I learned most maths in english because in any other language the resources are too scarce or too low quality for effective learning, not even my professors it seems try to find any other methods. Nobody here speaks english? Its first world after all, for real? While others suffer finding and learning from local trash resources wasting half of their lives i can just turn the thing on and pass the semester that easy. 5 minutes of this channel explained more than 2 lectures of local fat professors. The easiest way of finding determinants, unlike professors that encounter unnecessarily huge strings of numbers at 5x5 matrices. So much free time for games, youtube and cola remains at my disposal. Unbelivable.

[@madypheus]

Same here lmao

[@hansisbrucker813]

I really miss the why in this picture 😳

[@MarleneWalker-su8ku]

For a system of two equations it's easy to see wha the determinant of X and y are but when I use the same method on three equations I get a determinant which is composed of four other determinants. Kind of like determinants within a bigger determinant, not yet found the answer why it works the way it is shown for a three by three matrix and not seen it covered in any book yet or on u tube and it's an obvious question one would ask , not spent much time trying to figure it because I have been doing some other maths . Think I'm missing something really simple in my algebra as I isolate x y and z in a system of three equations or it needs to be looked at from another perspective that I am not aware . Gonna give it some more time soon so that if I die I may rest in peace lol😊. Sorry I couldn't help any more

[@MarleneWalker-su8ku]

For a system of two equations it's easy to see wha the determinant of X and y are but when I use the same method on three equations I get a determinant which is composed of four other determinants. Kind of like determinants within a bigger determinant, not yet found the answer why it works the way it is shown for a three by three matrix and not seen it covered in any book yet or on u tube and it's an obvious question one would ask , not spent much time trying to figure it because I have been doing some other maths . Think I'm missing something really simple in my algebra as I isolate x y and z in a system of three equations or it needs to be looked at from another perspective that I am not aware . Gonna give it some more time soon so that if I die I may rest in peace lol😊. Sorry I couldn't help any more

[@MarleneWalker-su8ku]

For a system of two equations it's easy to see wha the determinant of X and y are but when I use the same method on three equations I get a determinant which is composed of four other determinants. Kind of like determinants within a bigger determinant, not yet found the answer why it works the way it is shown for a three by three matrix and not seen it covered in any book yet or on u tube and it's an obvious question one would ask , not spent much time trying to figure it because I have been doing some other maths . Think I'm missing something really simple in my algebra as I isolate x y and z in a system of three equations or it needs to be looked at from another perspective that I am not aware . Gonna give it some more time soon so that if I die I may rest in peace lol😊. Sorry I couldn't help any more

[@MarleneWalker-su8ku]

For a system of two equations it's easy to see wha the determinant of X and y are but when I use the same method on three equations I get a determinant which is composed of four other determinants. Kind of like determinants within a bigger determinant, not yet found the answer why it works the way it is shown for a three by three matrix and not seen it covered in any book yet or on u tube and it's an obvious question one would ask , not spent much time trying to figure it because I have been doing some other maths . Think I'm missing something really simple in my algebra as I isolate x y and z in a system of three equations or it needs to be looked at from another perspective that I am not aware . Gonna give it some more time soon so that if I die I may rest in peace lol😊. Sorry I couldn't help any more

[@hansisbrucker813]

@@MarleneWalker-su8ku thank you for your answer :)

[@navinsahu2946]

I know the concepts but I cannot calculate sorry

[@thatkingdomheartsguy9615]

I love you and hate this math. This is by far the worst.

[@aurora9183]

ahhhhhhhhhhh the method is easy but the calculations are killing me! I wish i was allowed to use a calculator :/

[@omarel-nemr6506]

Ok, I do understand what the matrix is, as it is all about some equations with same variables.. but but but what is the determinant? why to treat it in that way or that method not another? what is it trying to tell us? what the hell is that even suppose to mean? i spend all my day trying to understand what it is and i didn't get any fucking clear answer, it's like a universal Conspiracy on us to not understand what it is; I hope to get a response.

[@jds189]

kzbin.info/www/bejne/f6GWiWyChM1lms0

[@jds189]

ill take your word for it that the product is -338 lolol

[@evanbrunner3589]

This makes a great recursion problem

[@shaz-z506]

Why is while we calculate the determinant, we take a subtraction first and then addition?

[@awesomecraftstudio]

Can anyone recomend good ressources on why this works?

[@123enat]

Literally saving my life!! Thank you so much, explained and shown so beautifully it's actually so hard to believe! Can you pls cover linear independence/dependence and singular matrix decomposition? Thank you!!

[@zhikai6228]

For a 3x3 matrix, "Sarrus rule" can be used to find the determinant For a 4x4 matrix, one can re-arrange the matrix and find the diagonal to get the determinant (Link: m.kzbin.info/www/bejne/nmecmKl6aquBnck&pp=ygUTNHg0IG1hdHJpeCBzaG9ydGN1dA%3D%3D) Alternatively, one can express the 4x4 matrix as "a (3x3 matrix determinant) - b(3x3 matrix determinant) + c(3x3 matrix determinant) - d(3x3 matrix determinant" using the algorithm thought in this video for finding determinant of larger matrices and use the "Sarrus rule" to find the 4 individual (3x3 matrix determinant) to get the solution with less steps I hope this helps!

[@laiaplaiap1688]

👍👍👍💛

[@omarel-nemr6506]

فهمت المحددات ؟

[@mathcritic]

I'd have to write code for anything bigger than 2x2 😀

[@codewithdevhindi9937]

programmer things

[@vaiterius]

My linear algebra gave us a 6x6 matrix to find the determinant of for homework...

[@chiemxerxobi]

Man what??(Lil Durk voice)

[@saththiyambharathiyan8175]

For 2x2 matrix i understand.... it comes from solving of 2 variables linear equations by elimination of one variable.... But what is the intuition in the calculation of determinat of 3×3 matrix .... Why it is calculated in this method...? When I try to same elimination of variable method... I didn't get the same result.... Could you explain...?

[@haydenbryant328]

Took my test on this last week dude.

[@beaugalbraith3242]

So in practical terms what does the determinant tell us? What do matrices with the same determinant value have in common?

[@Casper76]

3Blue1Brown has a great video on this. The determinant defines what size the 1x1 unit square has in the matrix's coordinate system. So, if the determinant is >1, everything's enlarged. If it's negative, the system has been flipped.

[@_WubbleU]

I very much appreciate this. Thank you!

[@QwertyUiop-hf4ez]

The answer is I got is -866. And I used a calculator to check it.

[@glowiedetector]

same

[@libelldrian173]

Nice, I got -338 first try without calculator! 🤗

[@RadoHudran]

I got -336 but it's so hard to see where I made the mistake lol

[@blue3094]

i got 684....

[@teothedoor]

Got -348

[@PriyanshuSen-l4u]

This was exhausting

[@asetrahmgaliev1149]

Using the algorithms described in this lesson I get the answer: 1*4 - (-2)*54 + 9*(-62) - (-6)*(-3) = -418, and I still don’t understand if this is a mistake?

[@jenm1]

Hey professor Dave, thank you for the videos. I keep getting -320 rather than -338 since I’m getting -3 for the efgijkmno 3x3 matrix and subsequently getting +18 rather than -18 for the d portion of the determinant of the last 4x4 matrix question. I’m not sure what I’m doing wrong

[@teothedoor]

Got -348, perhaps mistype. I figured out. I forgot to mention -1 2(0 - 1) instead of 2(0 - 0) in first and last 3x3 matrices.

[@leandrobayonito]

I try the exercises in the paper. I got -436 instead of -338. 😢

[@16psycho]

6:23 how tf did i get -53060 ?

[@Tausif-n7x]

Love you professor. You're doing a great work❤❤

[@ToanPham-wr7xe]

😮

[@mohammedvakil4611]

India

[@LisaPhilip-p5t]

Am an IT Boy but am new to this, referring to my current class am talking

[@jaclynrosenthal6939]

In what cases do we need to solve for the determinant? In other words, what are some examples of reasons as to why we do this?

[@zeinabchouman7770]

It helps with getting the inverse of a matrix

[@carultch]

One application of the determinant is the cross product, and its similar operation in vector calculus called the curl of a vector field. Another application of the determinant, is the second derivative test, in multivariable calculus. You have a stationary point where a function of multiple variables is locally flat. You want to determine if it is a local maximum, local minimum, or a saddle point. There is a second derivative test that uses the determinant of a matrix of second derivatives, that evaluates this, to determine whether you have a saddle point as opposed to a local max or min. Another application is Cramer's rule, for solving systems of linear equations. In the event you just want one of the solutions, Cramer's rule can be easier to calculate than some of the alternatives. The denominator in Cramer's rule is the coefficient matrix's determinant. The numerator swaps a column out for the matrix of constants.

[@handsumm]

thank you. its still not clicking but thank you lol

[@randymartin5500]

Thank you this is the best series on matrices. I went through 3Blue1Brown's series on linear algebra and his was a bit too much on the graphics and not enough on the computational. He even says he does not do the calculations because a computer can do that for you. The whole point is how to compute manually so you understand from the ground up.

[@thotarojoestar3045]

I've noticed the same, 3B1Bs videos tend to be a bit on the "heavier" side taking (me) quite a few rewinds to adequately grasp each one, though they're a lot more theoretical and should imo serve as a solid foundation for students to move onto the computational side of things. TL;DR: 3B1Bs videos are excellent, unless you're in a time crunch

[@MuffinsAPlenty]

@@thotarojoestar3045 Exactly! The 3Blue1Brown videos are not intended to be a full course in linear algebra. They are intended to be a theoretical/conceptual _supplement_ to a course in linear algebra. 3Blue1Brown's playlist is intended to show you why we define things the way we do and _why_ certain techniques work. That way, when faced with a new problem you have never seen before, you can use your conceptual and theoretical understanding to come up with a proper solution.

[@malikataizhbek]

I got -374

[@davidmurphy563]

Yeah, but you didn't really explain the purpose of it...

[@adhiyanthaprabhujeyashanka2091]

I think you will use it in Physics

[@davidmurphy563]

@@adhiyanthaprabhujeyashanka2091 Actually, I think I've got it. The penny dropped after I thought about it. The determinant is the scaling factor of the area. When you multiply the x with the y and vice versa you're just working out the area/volume, the same as you would on a graph. Say if you had a basis matrix with x, y and z vectors all running in the same direction. Then the determinant is zero, there's no area covered. If you multiplied a matrix by a scalar then you scale the determinant. So, it makes sense that we take the expanded determinant of two vectors with the identity vector we can output a third vector which is orthogonal to the other two: the cross product.

[@raytheboss4650]

the comprehension took me 3 hours of suffering..... but i finally did it

[@antiquarian1773]

what if the length of the row doesn't equal the length of the column ? Is finding the matrix the same way?

[@hassanahmed2275]

no, because it's no longer a square matrix

[@jenm1]

Then you can’t multiply those matrices if the row of A is not the same length as the length of the column of B. If you’re talking about a non square matrix multiplied by some other matrix, you just have to make sure the length of A’s rows are the same length as B’s columns

[@abriellh]

im not getting the right answer. im just too disorganized

[@veerrajuannamdevula9939]

Sir why don't we define an angle in terms of line or line segment why only rays?why anti clock wise measure is positive and vice versa? Is it a convention that we follow in maths?

[@ProfessorDaveExplains]

You can define angles with line segments. Just look at any shape! As for the counterclockwise thing yes that’s just an arbitrary convention.

[@veerrajuannamdevula9939]

Tnxx sir

[@ahhhhhhhhhhhh6742]

Thanks, was trying to program n by n matrix determinant calculator. ^^

[@ahanrjgaming8866]

The answer is ...334....please check...

[@emangthuso4383]

Why do you SUBTRACT the product of the second entry from the first row, which is two, and the determinant of the two by two matrix and instead ADD for the next one?

[@simplyaweys6968]

thanks

[@veilsnzills]

I got -330 😅

[@abdullahn9827]

I like how he goes straight to the point without bullshit

[@haripriyadebnath7969]

Thankyousomuch sir!

[@rpdeka]

Why do we put alternat signs plus minus in evaluating determinants

[@griggpev]

Has to do with the sign of a given permutation corresponding to swapping columns. Look into S_n and the Laplace formula for the determinant.

[@aarushking7702]

damn dude,u r awesome

[@Tory5972]

Thank you 💕

[@PaddyMcCarthy2.1]

Great, easy to understand.

[@divvy1400yam600]

I hope this gets read : as a very old fogey I occasionly return to maths . I was taught what is in this vid a long time ago but I have NO interpretation of what is obtained when the determinant is calculated. I notice a few others have asked the same question.

[@stephend.4342]

Determinants can be used for various purposes, such as determining the solutions to a system of linear equations using Cramer’s Rule.

[@griggpev]

Math PhD student here The determinant is characterized uniquely by the properties of a function which measures volume of boxes spanned by a set of vectors (what makes them unique is that the determinant of the box spanned by the standard basis vectors is normalized to 1.) The video gives algorithms for computing them, but never defines the determinant with a formula or characterizing properties. There is only one formula for the determinant, involving symmetries on a set, and these are just mnemonics.

[@pasindumadushan3048]

Thankyou

[@mikezhang2266]

Super! thanks Dave.

[@codevader4177]

Thanks❤️ sir

[@synonymous123]

When you get the determinant of a 3 by 3 matrix, why is the middle a2 in negative, whats the intuitive reason?

[@vladimircankov1492]

It's more complicated then you think and for non-mathematicians, it's best to just remember that the sign alternates. One thing that most KZbin videos don't say, for some reason, is that you can take ANY row OR column and follow the same algorithm for the determinant. You just need to be careful with the +/- signs.

[@synonymous123]

@@vladimircankov1492 where can i get more information regarding this to learn?

[@vladimircankov1492]

@@synonymous123 I have no idea. I have a set of old math books from the 70s, that my father used when he was student (he was a mechanical engineer). The explanation in there is about 10 pages long so... Btw, the sign is calculated by raising (-1) to the power of, the sum of the indices of that element. For example: The sign of the 3rd element of the 2nd row (a_23)? (-1) to the power 2+3. (-1)⁵ = -1. So this one is a negative. The sign of the 5th element on the 3rd row (a_35)? (-1) to the power 5+3. (-1)⁸ = 1. So this one positive.

[@griggpev]

@@synonymous123any rigorous text will have your answer. I recommend Jerry Shurman’s Calculus and Analysis in Euclidean Space

[@STKeTcH]

My answer is -340 !!!!!! I've spend over an hour trying again and agian, could you please show the solution to the comprehension problem?

[@johnnyrosenberg9522]

I tried once with paper and pen only and I got -340 too. I'm sure I missed something small (like some +2 somewhere in that mess). Too many opportunities to miss stuff … I guess I will try again later.

[@johnnyrosenberg9522]

Okay, second try, -314 … What the … ?? I think I'd better start over from scratch … Later.

[@STKeTcH]

@@johnnyrosenberg9522 good luck! I got a 7/10 for my exam, so I am happy!

[@johnnyrosenberg9522]

@@STKeTcH I finally got it right! I entered all the numbers in a spreadsheet (not using any calculations, just using it to have my numbers structured and easy to follow and edit), and I still got it wrong, but this time it was just my brain not working. I calculated -9·62 as -358 for some reason (or maybe I did a typo), so the result ended up at -138. It's -558, of course (9·(60+2)=540+18=558), and the end result is -338, just as the video clearly states. The results of the 3·3 determinants are: 14, 94, -62 and -3. Multiply each one with 1, -2, 9 and 6 respectively and add and subtract using those rules, and we get: 1·14-(-2·94)+(-9·62)-(-6·3) = 14+188-558+18=-338. Yay! 😁 It's actually easy. The tough part is to not mess up the numbers, since there are so many of them. I remember studying this in the late 1980's, but I never used them since and I remember that I wasn't too interested in this particular field of the world of maths.

[@STKeTcH]

@@johnnyrosenberg9522 If you don't mind answering, what made you look into this subject again after so many years?

[@Boxing_Gamer]

Whats the purpose of it?

[@carultch]

One application is Cramer's rule, for solving systems of linear equations. The determinant of the matrix will appear in the denominator of each solution you do by Cramer's rule. The numerator of each solution will be the determinant of a modified version of the coefficient matrix, where you swap a column for the column matrix of constants.

[@小有賴]

I am frustrated after evaluating that 4X4 matrix ..

[@HDitzzDH]

Same, spent 20 minutes evaluating it just to find out it was wrong, got -264...

[@ajtroyer7306]

There's a much simpler way of doing it.

[@maiqiding2995]

you can replicate this process in STATA using mat det(A)

[@ValorZeroAdvent]

Could you use another row to evaluate the determinant for any square matrix with size 3x3 or larger? Like say, instead of first row, I use second row or third row (basically any row which I could compute the determinant for), because I see zeroes which is easy to evaluate? And if it is the case, will I have to follow the checkerboard matrix sign convention for the determinant to be calculated correctly?

[@ardraanil9536]

You can take any row or column (horizontal or vertical), but the 'j' th term (i j k) always have a negative sign in front.

[@leonardsetiawan9603]

i think the answer is -368

[@leonardsetiawan9603]

No it's my fault, it was -338

[@angajasonr.2422]

Same bruh