18. Properties of Determinants

[@sepehrs6098]

Gilbert Strang is gifted in two ways. Not only he possesses the knowledge and expertise necessary to be a math professor, but also he has the charisma that encourages people to both listen and enjoy what he is talking about.

[@DanielCoutoF]

Sepehr S Totally agree with you. I feel so lucky to live in an era that allows common people like me to learn directly from such amazing and taletend people like pf G. Strang. Sheers from Brazil

[@Nakameguro97]

QING XIE Agreed. Herbert Gross on Calculus and Gilbert Strang on Linear Algebra are equally amazing.

[@nofiltariq1726]

exactly

[@tolunaytemiz]

disagree, the former has to do with a lot of work not with giftedness.

[@PhucLe-qs7nx]

00:00 det(I) = 1 03:16 det(P) = 1 or -1 07:00 The determinant is linear in **each** row. 11:34 2 equal rows => determinant = 0. 14:34 det(A) = det(U). 19:00 Row of 0s, determinant = 0. 22:20 det(U) = product of pivots 28:30 det(A) = 0 A is singular 37:40 det(AB) = det(A) * det(B) 41:40 det(A^T) = det(A) 47:00 Odd / Even number of row exchanges

[@stearin1978]

DEFINITION PROPERTIES: (1) Identity matrix, det= 1: no proof. (2) Rows exchange reverse the sign: no proof. (3) Linearity on rows: no proof. COROLLARY PROPERTIES: (4) 2 equal rows, det = 0: proof based on (2). (5) Basic Gaussian elimination transfromation (i.e. row_k - l*row_i) doesn't change the determinant: proof based (3) and (4). (6) row of 0's, det = 0: proof based on (3) and row of 0's=row_k-row_k. (7) Upper triangular, det (U) = d_1*...*d_n: proof based on elimination process (5) and (1) and (3). (8) det(A) = 0 A is singular (invertible). Proof: approximate, based on Gauss elimination process (A -> U -> D). (9) Homomorphism of groups property: det(AB) = det(AB): NO PROOF!!!. So... Integer power works: det(A^(-1)) = det(A)^-1, det(A^2) = det(A)^2. Scaler pops out with power of n (Volume property): det(kA) = k^n det(A). (10) Transposition property: det(A')=det(A): proof based LU decomposition. As always honorable Gilbert Strang showed only the nice proofs.

[@Soapluvva]

Greetings from New York City! Prof. Strang was my Linear Algebra 18.700 professor during my sophomore year at MIT in Spring 1973. I loved his teaching style, did well on the tests, and received an “A” in the class.

[@thangibleword6854]

and what are you doing here? Revise

[@ChaithanyaPv1]

😂​@@thangibleword6854

[@DaWanderer]

i didn't learn anything about linear algebra at my own university. then I found these lectures from MIT. this stuff is GOLD!!

[@henrypeterson8497]

Watching 3b1b's series and then this one paints such a rich picture of the connections between the symbolic and geometric interpretations for these ideas. So fascinating to see the differences in their approach, yet so illuminating to piece together the equivalences and build a deeper conceptual understanding.

[@Mark-nm9sm]

Yes!!!! literally , also i did a bit of khan academy prior to this . Im beggining to like math and not be so immensely intimidated by it. All 3 teachings have so far been top notch

[@GavinoFelix]

I love when he's building up to property 3. You can tell how excited he's getting (adjusting his hair, taking little pauses). Very cute

This is the best introduction to determinants, that I have seen so far.

[@martintoilet5887]

Same

[@Gritpawa]

#facts

[@therealsachin]

You must watch 3blue1brown’s video on linear algebra.

[@kingplunger1]

@@therealsachin His videos are great, but I actually think that seeing this first and then the geometric interpretation is better

[@bfl9075]

Determinant is the first chapter in some textbooks, which makes students lost in linear algebra. Dr. Glibert made everything easy to understand, thank you!

[@hubomba]

I was in the Sheldon Axler camp that linear algebra should be taught abstractly with focus given to linear transformations from an almost exclusively algebraic point of view and hold off on determinants until the last possible second. Strang's lectures change all of that, these are the best lectures on linear algebra taught from the angle of matrix algebra. Both approaches are valuable but Strang really nails it. Deriving the determinant from properties like this provides excellent motivation versus just jotting the nasty formula down, it also helps with mathematical maturity since it forces understanding by the student.

[@abramcz]

I am so very grateful that MIT decided to make these courses available on the web. A very generous and civic-minded thing to do. The fact that a middle-IQ, middle-income person like me, living very far from Massachusetts, can get this level of teaching for free on my computer is almost too good to be true. Thank you, MIT!

[@karthik3685]

I learnt these properties in my 11th grade. Had no intuition for it, nor any particular interest. I knew the rules, knew how to apply them, knew how to solve 'tricky' problems. This is just sublime! So lovely! I am totally in awe.

[@technoshrink]

Another way to prove that the determinant of a matrix with a zero row is zero would be to add one of the other rows to that row. That would create a duplicate row, which he proved the determinant of was zero.

[@ozzyfromspace]

clever

[@integralboi2900]

Big brain

[@sur1kor]

I am so addicted to these lectures.. Thanks MIT and Prof Gilbert Strang.

[@kinghassy334]

I am currently a software engineer in 3rd year. I watched a couple of these lectures back in first year when I was taking linear algebra and found it extremely confusing because that was my introduction and these lectures expect you to know the basics. Now that I know the basics and am currently reviewing what I've already been introduced to, these lectures are super insightful.

[@vlogsofanundergrad2034]

40:30 The 3B1B perspective of areas and volumes for determinants.

[@kanikabagree1084]

You are the best maths professor i've come accross thankyou so much Dr Gilbert Strang you're a blessing for all the students who struggle with linear algebra

[@Viggen66]

Thanks so much Prof Gilbert Strang, I wasn't understanding Determinant usage, and now I fully understand owing to the proprieties, which makes it so simple. you have just a gifted skill to teach so well and clear, before answering the questions, putting the shoes of a student, to think of a possible solution and applying what u have just teach, and applying to problem solving instead of mechanizing meaningless math rules.

[@drgothmania]

I've never had a chance to learn determinants this way. It was always that teachers gave out a bunch of formulas and methods and made students memorize without further explanations. Thanks for walking us step by step through this wonderful concept, Prof. Strang.

[@vozzen]

One important thing determinant says about a matrix is how much the volume/area/length is changing. Everything become much clearer with determinants when you learn this fact. There exist very good videos on KZbin about it, where you can see it in action.

[@btsjiminface]

Oh, that's explains the meaning behind how we know from previous lectures that for projection matrix P, P^n = P, which means that det(P^n) = (det(P))^n is true only if det(P) = 0. The multiplying the projection matrix to b multiple times does not change the length of the projection of b.

[@thedailyepochs338]

especially 3 blue 1 brown , the image from his video just came to my head

[@freeeagle6074]

This is taught in Lecture 20.

[@hj-core]

The way Professor Gilbert teaches the determinant is just amazing!

[@sumitgupta6905]

Best way to teach determinants. I used to worry about how this is the worst part of linear algebra since it involved a big formula that was thrown to me. I loved the intuition about the property 9 about volumes of n dimensional cubes. Never thought determinants would get me this excited. Long Live Prof. Strang. Thankyou MIT.

[@aleksybalazinski]

A quick observation for 39:45. We can also prove that det(2A)=2^n det(A) by noticing that 2A=2IA=DA, where D is a diagonal matrix with twos down the main diagonal. Then, det(2A)=(by 9)=det(D)det(A)=(by 7)=2^n det(A).

[@jerryzack]

Yeah, bro, you wrote what i want to post.

[@chuntelee4371]

This is the most correct way to learn linear algebra. Establish direct sense but not be buried by thousands of definition and proof.

[@swapnils6902]

While in high school (in India), I used to hate matrices, determinants, and vectors. They taught it like they were just a bunch of mindless, random calculations. Prof. Strang gives meaning to all of them and linear algebra has suddenly become wayyyy more interesting!

[@Alex-bc3tt]

This deserves to be in the Guinness book of records as the best introduction to determinants 🙏🙏🙏🙏

[@replicaacliper]

That was actually the coolest introduction to determinants ive ever seen and will ever see probably. Hopefully he brings in the physical intuition later

[@quirkyquester]

ayeee back here again!!! This is a great lecture! amazing details about the properties and proofs. I was just wondering how the hell det A = ad-bc. can't stop thinking about it. And here we areeee. Professor Strang proved it!! Thank you Professor Strang and MIT OCW!! U are the best!

[@rinkaghosh7961]

How valuable these lectures are !!! Kinda Addicted !!!! 🙏 Thank you very much Prof. Strang and MIT 🙏

[@salehjamsaljames]

my prof. can come and learn from this prof.

[@adhoax3521]

Mine can't even learn since he's dumb . they are full of dumbness.

[@ndonyosoko5680]

Especially those professors who read math slides derived from textbooks.

[@trevandrea8909]

😭 😭 😭 ​@@ndonyosoko5680

[@mrpotatohed4]

Never learned determinants like this, always just given the formula and the applications. Very enlightening.

[@muratbayraktar4114]

one way to prove property 4 is to use property 2: ( a b; c d) = ( tc td; c d) then the determinant is t* (cd-cd) which is 0. I love the way prof. Strang teaches it's inspiring

[@thomasrad6296]

33:36 "That's what she said."

[@danieljulian4676]

~22:10 "now I have to get serious" so, what was all that other stuff?

[@shakirulislam2477]

I learned more about determinants within the first 5 minutes of this video than I did in my 3 hours of lectures on the topic so far.

[@wduan1225]

this is 100x clear than the linear algebra course i took back in college, good teach does make a difference

[@veenashalini4904]

I blessed to see the prof.Gilbert strang lecture. Very thankful to u.

[@kavanavvasishta4692]

Love you Professor. You are such an adorable person and a great great teacher!

[@cache-re8if]

21:43 "Your idea is better" - very humble!

[@shailenderkumar7315]

Sir, thank you for your inspirational lectures, your style of delivery really motivates us to appreciate the structure and beauty of mathematics developed in a step by step way

[@supersnowva6717]

Everything is so clearly explained and laid out! Thank you so much Prof. Strang!

[@tongeason1235]

I start learning ordinary differential equation and Laplace transforms and I found the method of teaching was decent and clear( better than many uni professor)

[@ouahidlakhal5934]

Great teacher. thanks you MIT for the high quality courses you share.

[@nota2938]

In the last bit, there lies the fact the alternating group A_n could be viewed as being a normal subgroup of index 2 in the corresponding permutation group S_n. Algebra is a fun topic.

[@yevonnaelandrew9553]

The best explanation on the determinant of matrix ever! Thank you.

[@bhatiaayush11]

18:23 "I'm ready for the kill", amazing!

[@DareDefyMe88]

These videos are amazing for test review. My linear algebra teacher is awesome but these videos are nice to watch since hopefully I already know everything going on.

[@julietteaqui]

GS: "the determinant of an upper triangular matrix would be just (d1) times (d2) times ...*hands rotate* (dn)." me: *writes (dnd)*

[@lovvyparhar393]

The best determinant lecture I had!

[@radicalengineer2331]

🎯 Key Takeaways for quick navigation: 02:46 🔄 *The determinant of the identity matrix is always one.* 03:31 ➖ *Exchanging two rows in a matrix reverses the sign of its determinant.* 04:55 🔀 *The determinant of a permutation matrix is either one or minus one, depending on the number of row exchanges (even or odd).* 05:51 🔄 *For a 2x2 matrix, the determinant is ad-bc.* 08:11 🔄 *If a row is multiplied by a scalar 'T,' the determinant becomes T times the original determinant.* 09:34 🔢 *The determinant behaves linearly in each row when other rows remain unchanged.* 14:38 🚫 *If two rows are equal in a matrix, the determinant is zero.* 19:11 🚫 *A complete row of zeroes in a matrix results in a determinant of zero.* 23:44 ✖️ *The determinant of an upper triangular matrix is the product of its diagonal entries.* 28:48 ⚖️ *The determinant of a matrix is zero if and only if the matrix is singular.* 30:15 🔄 *The determinant of a matrix A is non-zero if and only if A is invertible. This is established by the connection between invertibility (full set of pivots) and the determinant being the product of non-zero pivots.* 31:12 🧮 *The determinant of a 2x2 matrix is found through the elimination process, and the formula is ad-bc. This is derived by understanding the steps of elimination on a 2x2 matrix.* 35:34 🔄 *Property 9: The determinant of the product of two matrices (A and B) is the product of their determinants. This property is valuable and distinct from addition properties.* 36:31 📉 *Using Property 9, the determinant of the inverse of matrix A is 1 over the determinant of A. This is derived by considering A inverse times A equals the identity matrix.* 37:59 🔄 *Property 9 extends to diagonal matrices, providing an easy check for determinant of products involving diagonal matrices. The determinant of A-squared is the determinant of A squared.* 40:49 📏 *Property 9 relates to the volume change when doubling the sides of a box (represented by a matrix). The determinant of 2A is 2^n times the determinant of A, where n is the dimension.* 41:38 🔀 *Property 9 aligns with the concept that if the determinant of A is zero, A is singular, and the inverse doesn't exist. The property is crucial for non-singular matrices.* 42:07 🔄 *Property 10: The determinant of a transposed matrix equals the determinant of the original matrix. Transposing does not change the determinant, but it affects the sign in the context of exchanging columns.* 43:03 🔁 *Property 10 indicates that exchanging two columns reverses the sign of the determinant, similar to how exchanging two rows does. This is demonstrated by transposing and using the properties 1-9.* 47:11 🔢 *The determinant is well-defined by properties 1-3, and it maintains its sign for an even number of row exchanges. This is a key fact established through algebraic reasoning.* Made with HARPA AI

[@danishji2172]

Idk if anyone else felt this way but this man has the charisma of a fatherly figure. It is hard not to like him, and a lot.

[@SindhiScienceChannel]

Rule 6 can be proved with rule 4 and rule 5. If a complete row j is ZERO, one can subtract -1 times another row k from row j (rule 5) so that row j and row k become the same, making rule 4 applicable.

[@karimkaan8700]

Thanks to you we really start to see what s going on in Algebra

[@spacesher3135]

"Now how do I prove that?" Me: Gilbert bruh, chill not necessary. I trust you with my whole life.

[@anangelsdiaries]

For those curious about the last fact he stated, that`s proved in Group theory.

[@FabledNarrative]

The way he says "Kill" makes him sound serious @ 25:56

[@pd1769]

He's a serial killer Ok bad joke sorry

[@morganjones7428]

What an absolutely incredible lecture!!!

[@juancruzparada8690]

absoluty beautyfull, im loving those clases, congrats from Argentina, keep giving us those amazing clases, plz

[@akaci-w3k]

Thank you Professor Strang.

[@cafe-tomate]

For rule 6, take the matrix 2×2 {0,0, c, d} and write the first 0=c-c and the second zero=d-d

[@Zesty_Soul]

22:04 Actually, det D (the nomenclature I'm using for the determinant below) can't be proven to be = 0 by choosing 't' = 0. Let's say, det D = | 0 0 | | c d | This 'det D' (having an entire first row of 0s) can be proven to be = 0 via property #3a by the use of any real number 't' (other than 0), which can be '5' too (as randomly chosen by Prof. Strang). Let's see it work for t = 5. Let's multiply both sides of the above equation by 5: 5 x det D = | 5x0 5x0 | = | 0 0 | = det D | c d | | c d | => 5 x det D = det D => (5-1).det D = 0 => 4.det D = 0 => det D = 0 [works similarly for any real number 't' (other than t = 0).] (Ankur Kundan Bansal)

[@vonfred54]

the most interesting algebra ever. thank you professor!!

[@dAntony1]

He’s more effective with that one HUGE piece of chalk than my professor with her ipad and Zoom. She should just assign this as the lectures and be done with it.

[@ThePositiev3x]

At 45:30 he describes the "L" matrix as "Lower triangular matrix with 1s on the diagonal". In this case "L" is not exactly lower triangular but a special form of lower triangular matrix. Isn't it? Because I think the diagonal of lower triangular matrix doesn't have to consist of 1s ...

[@txtlu6898]

+ThePositiev3x A can be factored to LU in which L is lower matrix with 1's, check his book in page 97.

[@NuncNuncNuncNunc]

+ThePositiev3x, from your question I assume you already understand this so this clarification is really for others Here it would not matter if the diagonals were all 1s or not. L and L' have the same trace and all zeros below or above the diagonal so they have the same determinant. Making L unit triangular forces the LU factorization to be unique. The particular form of L here, Prof. Strang shows in earlier lectures, comes from combining the inverses of elimination matrices that have 1 on the diagonal. As a practical matter, using 1s in the diagonal makes showing the linear algebra clearer by eliminating a lot of arithmetic.

[@afrakilic5672]

I just enjoy to watch these videos... so much...

[@murat99977]

46:00 can anabody help, What is LU here? In which video did he teach these and triangular matrices?

[@Labroidas]

I'm really thankful to the people going through the trouble of making these subtitles, but in this video the subtitles were so full of errors of all kinds that it was really irritating :( I had to turn them off eventually because they distracted me so much.

[@Meritjamtsho]

Man! you are an inspiring mathematician. If i become one, full credit to you.....

[@rampabayir]

all the properties of determinants are like his little babies for Prof.Strang. He cants choose one over the other, they are all key properties :).

[@arkitkabir5086]

I have one doubt. If EA = U where E is elimination matrix and U is upper tringular , then det(EA) = det(E)*det(A) = det(U). However at property 7 it was discussed that we first carry out elimination , get U and then use property 7 since it is easier that way and determinant of U will be same as A. That's only possible if det(E) = 1 (by using property 9). Is that always the case?

[@APaleDot]

He already proved that every row addition leaves the determinant unchanged, meaning all row addition matrices have determinant 1. And he already proved that row permutation matrices have determinant -1. Since E is just a product of these matrices: E = (E_1)(E_2)...(E_n), where E_i is either a row addition or permutation matrix. Then det(E) = det(E_1)det(E_2)...det(E_n) and we get that det(E) = ±1 depending on how many row permutations there are. Thus, we conclude that det(U) = det(EA) = ± det(A) If you want to see more clearly why every row addition matrix has determinant = 1, look at its form. A row addition matrix is a triangular matrix with ones down the diagonal. The determinant of any triangular matrix is the product of its diagonal entries (as shown in the lecture), so its determinant is 1.

[@marverickbin]

best det class i ever had.

[@serkanvai]

@slatz20 he explains that if you multiply only one row. so he did it correct.

[@anhkhoangothe6793]

The definition is just brilliant

[@adhoax3521]

Can anyone tell why those three properties exist and why only and specifically them?

[@D3tyHuff]

Determinant has been determined. Thank you MIT!

[@rowechenzhong8950]

800th upvote. Very good video!

[@karimkaan8700]

I can see why American people are by far well educated . HAT OFF TO YOU

[@greekman234]

Dterminants is my fav part of linear Algebra xD

[@jaswanthbyjus1288]

Great guy in the planet

[@CadrinTheWerecat]

And for that matter: why are these comments repeated in every single lecture from the course, with the exact same wording and just the title of the lecture swapped?

[@suleimanshumba9182]

wonderful presentation but i wish next time you increase figurative examples

[@rodrigomaximo1034]

I love these lectures, but it's really annoying how, in a lot of them, the professor's mic stops working at some point and the subtitles also start to get wonky. Makes it really hard to understand sometimes.

[@serkanvai]

@slatz20 he explains that if you multiply only one row! so he did it correct

[@TrevorKafka]

Does anyone have a good proof for property 9?

[@coltonmelnick8301]

The determinant measures the scaling of the area between the basis vectors. So, it follows that this scaling of area can be written as det(AB) = det(A)det(B)

[@ZiJyuWang]

don't know what he meant in last minute about 'Odd / Even number of row exchanges', cause when we do seven row exchanges and then ten exchanges, that's seventeen exchanges in total, hence sign does change, right? just don't get it😢hope somebody could help me please🙏

[@georgesadler7830]

Although the lecture on determinants are great, in the beginning of math 18.06, DR. string seems disinterested in determinants. Determinants play an important part in solving linear equations.

[@LennyLeonard85]

@seisdoesmatter The chalk's really awesome. Looks thicker than ordinary chalk. It seems a bit like the chalk kids use to paint on the pavement!

[@Z-eng0]

How do you prove the single row linearity of a determinant, I mean how is in matrix A + matrix B we sum ALL rows but in det(A) + det(B) we don't?

[@ZYau-lc5ql]

I really want to go to MIT😍

[@publicanimal]

Same comment as everyone else. This was a much better explanation than my professor gave.

[@User-cv4ee]

Did we prove det(AB) = detA*detB in the lecture?

[@SilverArro]

User 10482 Not fully. It’s a pretty lengthy proof, which is probably why he doesn’t take the time to do it. He does tell you that you eventually get there using what he calls “property 3”. Namely, that the determinant is a linear operator on each row separately. If you understand what that property means, then you can reason through why det(AB) = (det A)(det B).

[@vishaljain4915]

at 27:00, how come you are just able to factor out the diagonals? If you take out d1, arent all the diagonals in the matrix from the 2nd row to the nth row, dn/d1?

[@zenobenetti9641]

I was also confused at first, but I guess that once we factor out d1 (so that the first element is 1, in virtue of property five we can apply a linear combination of the other rows on the first row so that all but the first element turn zero. We can then procede with the second row and so on. It's just a guess but I hope it's correct.

[@avishshah2186]

All the elements below d1 are 0 hence we can simply divide the row by d1 and it will not have any impact on other rows Also, the row picture won't change so we can definitely do this

[@mikecohen5887]

For property 3b should the top row be (a+a' b) --> (a b) + (a' b)

[@landsgevaer]

Property 2 wasn't needed as a fundamental property, since it follows from property 3.

[@ashutoshtiwari4398]

could you explain how?

[@bhaskarpandey8586]

........... ............. ........... ............. ...R.... => ....R.... => .(R+(S-R)). => ....S.... ....S.... ..(S-R).. ....(S-R)... ..(S-R).. Then substract row S from row S-R to obtain -R and then take the minus sign outside the matrix from row -R

[@miyamotomusashi4556]

Ooooooooooooooooh my god this is soooooo beautiful. Thank you.

[@berkeunal5773]

Can sb explain what he is talking about at the end? (It is sth about permutations but I cannot understand coz of the sound)

[@DanielSColao]

Fantastic lecture!

[@giri2344]

Thank you sir🙏🙏🙏

[@balugowthaman7619]

Isn't proof for property 10 circular?

[@bridge5189]

but then WHAT IS DETERMINANT?

[@dashnarayana]

Perhaps Gil Strang would have replaced Newton if he were his contemporary