19. Determinant Formulas and Cofactors

MIT 18.06 Linear Algebra, Spring 2005
Instructor: Gilbert Strang
View the complete course: ocw.mit.edu/18-...
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19. Determinant Formulas and Cofactors
License: Creative Commons BY-NC-SA
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[@vishalpani5044]

"Ok...we're 'executing' a determinant for formula here"...quality content!!

[@danieljulian4676]

zero determinant *is* the grim reaper...

[@theworkethic263]

What are the prerequisites in order to take this course ? I just passed out standard 10th , now i am in standard 11th. School has not started yet . Please guide. Can i take this course ? Please guide me 🙏

[@yorgunkaptaan]

@@theworkethic263, you can start from lecture one directly. kzbin.info/www/bejne/kHyWgGdmZ9yZZ8U Prof. describe the concept from first steps. Formaly, calculus1 is prerequisit, there are just 1 or 2 lessons about derivatives and they are unimportant for understanding "the whole picture"(I watched first 22 lectures btw, I dont know 23-34). Though, you should consider, why I need to learn Linear Algebra. If you want to learn university maths(proof based maths), I think Linear Algebra is not the best choice to start.

[@jas4768]

@@theworkethic263 You definitely need an introduction to group theory before you can fully grasp linear algebra. Pm me and I can give you some resources.

[@sahilnegi4326]

@@theworkethic263 i dont think u can take this course. You need basic knowledge of matrices beforehand which is taught in 12 std. Also u need to know about vectors. You can give it a try but frankly it ll be a waste of time.

[@BenRush]

Seriously, it just doesn't get much better than this. What a well-delivered series of lectures from a clearly gifted teacher and mathematician.

[@andrewshea8052]

.

[@BenRush]

I have seen the problem sets, in fact. I still believe the lectures were very well presented. To each their own, I guess.

[@anirudhagarwal950]

u have no idea how bad the teachers are everywhere else. The person who taught us didn't even tell us why we were doing what we were doing. The whole determinant thing is delivered so beautifully and also his information can be directly used in robotics or regression or coding in general because he presents it so nicely. No teacher has ever explained determinants like this ever, they simply gave the formula and that is it. The problem sets seemed theoretical but I didn't really think they were unsolvable. He cannot teach how to solve each problem in class but the essence of linear algebra is captured very well here especially if you need to use this to code. If everything is simple that wouldn't be fun now would it.

[@jeoantapo]

+Anirudh Agarwal I agree with you, I love teaching math and I always try to improve myself seeing these pricelees lecture. for not doing the same mistake that my teachers did.

[@alenjose3903]

15:13 I don't want to make that precise - Gilbert Strang - 16:33 Big determinant formula 22:34 Example 24:48 *DEATH PASSING BY* 29:26 Cofactor 3x3 36:00 Cofactor features 39:45 Cofactor formula 43:54 Tri-diagonal matrix Example

[@rithvikyagnamurthy6560]

I love the casual „Death Passing By“ lmao

[@zhuoruizhou799]

I was wondering what the heck is "death passing by" lmao

[@bikespike4150]

Raised an eyebrow. Remembered where I was in the lecture and fast forward to "DEATH PASSING BY". Go back to where I was and keep studying.

[@rubabfatima3095]

your lectures make me understand linear algebra soo well i was starting to dislike maths beacuse of this but now iam motivated to study it , thanks alot :)

[@Saganist420]

This guy is so brilliant, that it makes ME think I'm brilliant just by listening and understanding his thoughts.

[@faizanmansoori]

wow! Watching computational formula for determinant emerging out of the three basic rules of determinant is the most wonderful thing in this series.

[@adamc973]

The permutation definition of determinants gave me so much strife, but Strang makes it crystal clear. What a legend.

[@aksei5786]

Seriously, the details of little things shape the masterpiece, Prof. Strang Rocks!!!!!!

[@sergekhalimovskyy5467]

So underestimated approaches have been revield through column space matters more then rows space. So genius.

[@arkhalykherjan6642]

For 48 differet reasons, I love prof. Strang

[@sunritroykarmakar4406]

fantastic cameo from mr. death

[@sebastiandelgado6263]

Incredible award-deserving lecture

[@ubermensch5472]

Am considering dropping out of college and learning from OCW

[@thelastcipher9135]

you don't see many youtube videos with 100k plus views in 7 years with 0 dislikes

[@joewalkerr5027]

look again, your welcome

[@alihodge6200]

FUCK YOU ! WHAT HAVE YOU DONE

[@batman9937]

It's 4 now

[@bfl9075]

Dr. Gilbert made these cofactor things are so obvious!

[@serkanvai]

haha :) "as long as it is not periodic" 25:00 very mathematical comment

[@luongtainh9733]

"the apparition just wanted to be sure that we got the right answer.." MIT is super cool XD

[@subhadeepmandal8513]

Best linear algebra lectures

[@radicalengineer2331]

🎯 Key Takeaways for quick navigation: 00:00 🧠 *The lecture focuses on finding a formula for the determinant of an n by n matrix.* 01:24 🔢 *The speaker reviews three key properties of determinants and mentions the goal of deriving a formula.* 03:42 🔄 *The method involves using linearity, exchanging rows, and splitting rows to simplify the determinant calculation.* 05:59 🧮 *The speaker demonstrates the method for a 2x2 matrix and extends it to a 3x3 matrix, emphasizing the systematic approach.* 08:24 🔄 *Survivor elements in a determinant correspond to entries from each row and column, forming a permutation matrix.* 14:53 🔄 *The lecture concludes with a three by three determinant formula, laying the groundwork for a general formula for an n by n matrix.* 16:47 🧮 *The general formula for the determinant of an n by n matrix is expressed as a sum of n factorial terms, each representing a permutation with alternating plus and minus signs.* 20:31 📜 *The speaker discusses the significance of the determinant formula, including its connection to properties such as the determinant of the identity matrix being one.* 23:50 ❓ *The speaker explores a 4x4 matrix example, applying the determinant formula to calculate potential non-zero terms and discusses the possibility of a singular matrix.* 27:07 🤯 *The lecture concludes with the speaker expressing uncertainty about the outcome of the example and a humorous reference to an unexpected event.* 28:56 🔄 *Cofactors break down an n by n determinant into determinants one size smaller, revealing a pattern of plus and minus signs based on the indices' sum (i+j).* 30:15 🧩 *Cofactors for a three by three matrix involve taking the determinant of smaller matrices formed by excluding corresponding rows and columns, with the sign determined by the i+j rule.* 35:57 🔄 *The cofactor formula for any element aij in a matrix A involves multiplying aij by the determinant of the (n-1) matrix obtained by excluding the row i and column j, with the sign based on the i+j rule.* 37:46 🔄 *Cofactors are plus or minus the determinant of smaller matrices, forming a checkerboard pattern determined by the i+j rule.* 41:00 🔄 *The cofactor formula allows building up an n by n determinant from smaller determinants, simplifying the process by breaking down complex determinants into more manageable parts.* 42:52 🔄 *The cofactor formula serves as an intermediate step between the efficient pivot formula and the more complex n factorial term formula, providing insights into the structure of determinants.* 48:28 🔄 *Using cofactors, the determinant of a tri-diagonal matrix of ones (A4) is expressed in terms of the determinants of smaller matrices (A3 and A2), showcasing a systematic approach to calculating determinants of different sizes.* Made with HARPA AI

[@user-hk3ej4hk7m]

I like more the A(i,j)*-1^(i+j)*C(i,j) form

[@alberto3071]

Let Aij be the cofactor of aij in A. Then by definition detA= ai1Ai1+...+ainAin. What would happen if the Aij were taken from another row? let B be the matrix with the same rows as A except for the k-th row that is the same as the i-th row. Then, bk1Bk1+...+bknBkn=detB=0 cause B has two same rows. But bkj are aij and Bkj are Aij (cause Aij =(-1)^(i+j)Dij and Dij eliminates the column and the row so it'll be the same) Consider C= A*(adj(A))^t. The position cii will be ai1Ai1+...+ainAin= detA. cij with i!=j will be 0 because of the lemma proved above. Therefore C=detA*In, and with some algebraic changes, A^-1=( adj(A)^t)/detA

[@michaellewis7861]

det(A)=aij*det(A-columni,rowj)*(-1)^n-1

[@zack_120]

The blackboard writing is ready to be copied into the textbook!

[@nitinpatel5324]

A+ for the teacher

[@anynamecanbeuse]

sort of like solving sudoku, like it.

[@hektor6766]

Give Laplace his due.

[@danielalmeida7140]

i'm sorry but what properties make the initial split (around 4:00) possible? linearity says that det a+a' b+b' c d equals det a b c d plus det a' b' c d or yet that you can factor a scalar out of a row in a way that det ta tb c d becomes t * det a b c d and I don't see how this makes it possible to split det a b c d into det a 0 c d plus det 0 b c d am i missing something?

[@thelastcipher9135]

because det a 0 c d can be split into det det a 0 plus a 0 c 0 0 d same for det with 0 b c d. that's the reason if we follow the rules of computation but geometric intuition is almost zero if the non-pivot entries are non-zero. while determinants with only pivot entries are intuitive since it's just areas of squares and rectangles into higher dimensions.

[@jtlvhpublic]

You can think of | a b | = | a + 0 0 + b | = | a 0 | + | 0 b | | c d | | c d | | c d | | c d |

[@abhishekdugar73]

calls the first row a 'HE' the second row a 'SHE' 18:35

[@nalinikantamaharana3932]

nice!!! This way i never study determinant

[@lucilius9491]

nazgul wanted to learn determinant.

[@francoisvictor8]

who's also watching on halloween? 🎃🎃

[@jeevan288]

y not try diagonal method? its 200% more easy right?

[@shoumikghosal]

Easy stuff times horrible stuff

[@gp2111]

Is there a "Gilbert Strang Fan Club?" If not, there needs to be! I effin love this dude.

[@pubgplayer1720]

Yup, there is.

[@9888565407]

@@pubgplayer1720 kaha hai naam bta

[@Adrian24134]

yeah, he is so cute!

[@dalisabe62]

Professor Strang is an authority on this subject and inspiring to many mathematicians who teach this course or use it in special applications with his unique intuitive approach.

[@sergekhalimovskyy5467]

So true

[@SteVeTurchin22]

24:40 a Physicist walks into a math lecture

[@coding99]

determinant really matters...!!

[@tldesign1808]

lol the camera man zoomed out and followed it

[@huseyinkursatcan]

"keep walking away"

[@ozzyfromspace]

"Okay, we're executing a determinant formula here" 😂

[@mehmetaliozer2403]

maybe he was coming from stanford :d:d

[@theflaggeddragon9472]

"for like 48 different reasons, that determinant is zero. It's dead" OMG I love this guy

[@jatinthakwani6224]

Tell me those 48 please

[@mehmetaliozer2403]

gilbert's missing secret "the 48 reason why the determinant is zero"

[@AlokParikh]

I'd never thought I would binge watch anything apart from movies and seasons but this course got me wrong. All Hail Prof Strang!!!

[@RC-bm9mf]

Same here. It’s getting more interesting than a TV series.

[@_HJ_K]

24:40 was it halloween? They seemed to have linear algebra on Monday, Wednesday and Friday, and this lecture seemed to be on Monday (professor said "see you on Wednesday" in the end) and I searched halloween of 2005 was also a Monday, so I guess it was halloween then☺️

[@rosadovelascojosuedavid1894]

Lol I appreciate you did that. I was wondering the same.

[@bariqfirmansyah1702]

it's a spring semester

[@nebulamakinnexplaininstuff3527]

bro you saved us a lotta time researching that real matt pat there😄

[@Samurai_Jack__]

U should be a detective

[@lehoumusic2633]

He's leading you to think, rather than proof after theorem.

[@RC-bm9mf]

I love his pursuit for practical insight.

[@WotanFinrod]

my english is not good enough but this guy teaches better than my teacher speaking my native language

[@debanjandhar2995]

One more reason for you to learn English

[@zguan6397]

subtitles~

[@abramcz]

Thank you, MIT, for so generously making these lectures available free!

[@clottedscream]

can’t believe i’m paying thousands of dollars for a guy who’s terrible at this stuff to teach me when Gilbert Strang is out here teaching things excellently for free

[@jas4768]

This is absolutely incredible. The 'big formula' was thrown at us with absolutely no context and no mention of arithmetic with determinants; this cleared up so much for my university algebra course.

[@rishabhshirke1175]

18:49 Now he won't run out of pronouns

[@seeker6300]

So all these concepts are taught in my college course, but they don't really tell how different topics are related. Then come exams, where you are supposed to link everything together on your own. I think this is why Dr. Strang's lectures are special!

[@TheTechAdmin]

24:38 My fellow Imgurians, this is what you're looking for.

[@3rdpartyU5er]

Lol just came here

[@kierandavis8054]

Ey joining the squad.

[@MarlsbysDragons]

Wassup?

[@Mrx679]

You are a beautiful person

[@MarlsbysDragons]

Why thank you, I like to think so.

[@adjombite9735]

i believe strang is the best professor i've ever seen. you make things easy to understand adjombi

[@sahilnegi4326]

His classes get u hooked up like its some sitcom that u wanna binge watch.

[@HappehLlama]

LOL "for like, 48 different reasons, that determinant is zero. so this one is dead." Heehee. Amazing video and series - thank you!! :)

[@nitinnilesh]

52:15. "I'm gonna be stop by either the time runs out or the board runs out." Ha, Ha, Ha

[@tathagatanandi5813]

Board is used as a Resource in MIT!!! Hahahahaaa

[@benescaglayan]

50:37 Why did not he also calculate the cofactor for a23 in the smaller matrix when he was looking for the cofactor for a21? I know it is still going to be zero because of the zero column on the left but I think he should have made that clear.

[@canmumcu9804]

Ur god damn right my Turkish friend xd

[@varun5059]

actually, he does mention there that instead of taking the cofactors by row, you can also take the cofactors by column because of the transpose rule. In this case it would be convenient to take the cofactor of the smaller 3x3 matrix by column since the other two terms in that column are zero.

[@CopyFox7]

"Whether the rest of the world will realize I was in danger or not, we don't know." XD

[@mainakbiswas2584]

Who was the person that entered the room; and we always Prof. portraying his sense of humour, "Whether the rest of the world realize that I was in danger!!" Such a great mathematician with such great sense of humour!

[@studyonly2136]

Happy Halloweens guys, from 2022 with love :D still very very very useful for a struggling student like me.

[@nguyenbaodung1603]

I give up everything, watching film and sufing web just to comprehend this amazing content given by this gifted professor omg.

[@tehinfidel]

Last segment is interesting: |A_n| = |A_n-1| - |A_n-2| looked awfully fibonacci-like. It turns out that the determinants of the symmetric tridiagonal matrices with i in the non-diagonal entries exactly follows the Fibonacci sequence. Mind blown yet again.

[@MaryKateBlack]

24:40 hahahaha

[@tauceti8341]

+Mary - Kate That was hilarious haha

[@SteVeTurchin22]

+Mary - Kate hahahah priceless

[@adityanarendra5886]

The way he asks Why? Wins my heart.Long Live Strang Sensei

[@AngeloYeo]

Was it like Halloween or something? In spring? :p

[@tathagatanandi5813]

its like a witch or something

[@ahmedtamer4620]

@24:50 I was really dumb to think that my time has come and angel of death was just trying to show his presence.

[@manisharadwad3538]

You ppl gotta give some credit to camera man who decided to zoom out when the death was passing by

[@samberg5145]

Cofactors are at 28:30

[@luvon1114]

We need an identity for the dude that passed by lol

[@abdulmukit4420]

I love his lectures. I don't know if they will help me in my work but it's still worth it.

[@fateplus1]

i have to say greatest lecture of any lecture out of all the MIT opencourseware ive watched, thanks a bunch

[@hypnoticpoisons]

the professor is not only brilliant but humorous!

[@seanguo9366]

"*executing*"

[@nguyenbaodung1603]

We're "executing" a determinant of formula here, ahhh I f-ing love professor, wish that I could attend his class huhu.

[@georgesadler7830]

This is another masterpiece by the grand dragon of mathematics DR. Gilbert Strang. I did not have any professor at the University of Maryland College Park in Mathematics and Electrical Engineering in the 1990's that makes math and science easy to understand. This professor is incredible in every math lecture/video at MIT.

[@merttsenell]

we love you very much Gilbert Strang :)

[@abhimanyudubey338]

@ 24:39, that epic moment. :P

[@alimhasanagic5895]

On our faculty they didn't tell us anything about 3-diagonal. GREAT video and thx for the lecture.

[@restlessxx]

What a lovely teacher...

[@nguyenbaodung1603]

24:42 I really hope camera man zoom in professor's face. His face is always a mood XDD

[@souravanand99]

Prof. Gilbert Implies God's Gift

[@TheMickaelPT]

What an amazing teacher, thank you!

[@AsiqueIkbalAnsari]

Watching this on 16th July, 2023.

[@tzaidi2349]

Youve found the GS Fan Club my friend

[@fretzT_T]

18:48 People made more pronouns so he can continue.

[@吴瀚宇]

24:40 what kind of coser is that?

[@Brandon-yk6st]

the formula at 51:32 does it work for all Matrices, or it was just for this case..please help

[@fretzT_T]

I think it only works for that tridiagonal matrix. As most of the terms vanished because of the zeroes in the columns/rows.

[@nawidkeshtmand9925]

At 47:20, why does he need to subtract row 3 from row 2 in order to calculate the cofactor of a21 ?

[@Brien831]

i+j is odd

[@jackbradley4737]

24:42 what in the!! XD. So random for no reason. This was so unexpected literally caught me so off guard and by surprise lol. Hilarious

[@hypnoticpoisons]

he's gonna kill all of them, what an assassin:D

[@sadfjlkjsdivje]

24.41, this guy is a legend. I wonder where he is now.

[@rolandheinze7182]

He just released 18.065 matrix methods for deep learning

[@bautistabaiocchi-lora1339]

Prof strang brings a smile to my face every lecture. Without fail.

[@fishmathematician]

Holly.. I am actually shocked.

[@tathagatanandi5813]

|A(n)|=|A(n-1)|-|A(n-2)|. Does this hold for only tridiagonal matrices??

[@doge-coin]

Same question here.

[@ashutoshtiwari4398]

Yes. Only for tridiagonal Matrices.

[@gBomfim]

Muito obrigado.

[@chordsequencer001]

That formula for 3 X 3s looks very different than the one I learnt with cofactors all in row one and their corresponding "minors", the dot product that is and with the middle term negated.

[@aminullah2666]

r

[@familiagomezvillarreal610]

Factor a -1 and you are back to your old formula.

[@throwawayidiot6451]

Do we ever use cofactors for anything else besides computing the determinants?

[@phil97n]

Awesome lecture thanks. Seeing executer crossing the floor (@ 24:40), I'm guessing the lecture was on Halloween day, or just read to execute determinant formulas

[@Cadellinman]

I also wondered if the lecture was recorded around Halloween, but this was the Spring semester that year apparently.

[@afrakilic5672]

🥰🥰🥰🥰🥰🥰🥰🥰🥰

[@genius_39]

apparition😂😂

[@Learn_With_Anmol]

😮

[@reid300]

ha that is not as bad as at my school when the grads are having their iron ring ceremony they all run through the 1st year classes hammered and spraying silly string and whatever else they find yelling and causing a general ruckus.

[@michaellewis7861]

46:31 Use the Cofactor Formula from the beginning.

[@compresswealthdivideeconom3757]

Some things require only a textbook.

[@thedailyepochs338]

i know i know its juvenile but that's what she said @ 42:34

[@fabianzambrano2744]

I would like if the subtitles also put in Spanish, please