Math for Economists - Lecture 4

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Math for Economists - Lecture 4
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OK or war on Section 9 . 1 now Chapter 9 in general is that it's a big chapter there's a lot of stuff here the 2 topics that we needed
Solidified in Chapter 9 are determinants and hinders their related but in order to get to this we have the study this and that and then it's a little bit differ with 3 by 3 reason is with Cuba accused We spend a little time onto by choosing to both of those things and we go back and study determinants by themselves and then we go reimburses again so we can bounce back and forth between these 2 ideas So Section 9 . 1 wouldn't talk about a 2 dimensional her 2 by 2 matrices Aviva wanted you don't wanna go back to high school level Maybe even before I score Pamela as system equation better we've been setters The studying head if considered to exit polls now when you're younger that's just an equation by itself a new solid you find out that Exs equaled 3 hats but now it clear in this new context and we've been steady we could say well That's a system of equations anchored Eckert extract off the coefficient would I get I get a 1 by 1 matrix is not very interesting but there There's a culinary right that's as I could think of it as a 1 by 1 matrix times X which is also a little 1 by 1 matrix and that equals I 1 by 1 nature books It doesn't help us to look at it that way except that were leading up to looking at more complicated situations were building an analogy OK so we solve this will not solve multiplied by a half and when I multiplied by have had to do on both sides that's not mysterious owner right down the next at the next steps you want while they choose a half the reason I chose a half to isolate the X rays and and what would I mean by isolate multiplying by something so that when I get this product to the vote 1 that's the whole point so I can write down sex Pier 1 equals 1 x equals 3 hats and then go and most of that nothing really interesting er Nuri saying there were no 1 thing I wanna do as I rewrite this 3 hats there's another notation for 1 hack you you guys know about it you you've experienced it but it doesn't help us to ready we always rather Red Hat but in this case I wouldn't go out of my waiter rated inking immediately to demonstrate where we're headed so that this step step here are the unit this step I could think of this as Chiu There's 3 is the same as 1 habits to the negative ones you could say that is to intercede you don't you don't say that usually you'll say to the negative wonder one-half but I want Collett to endorse any idea to Ingres is Hernandez that's why I use the word hinders may have to Then I multiplied by 2 embarrassed and the result is 1 basically kills So in general I can write down 5 2nd rate down a N all linear equations of that in the universe there And if I wanna solve this Exs equal a university For all cases except when can't Avenue Enders When a 0 rate could just reciprocal so they would 0 I can't do the reciprocal I can have 1 over 0 so I have to add here a little Carter that a man was This is all systems of equations with 1 variable 1 equation is not like we did know how Do ready but now we're gonna do You see a connection between that and what we've been doing this So now consider this a step on the to X plus Y equals 3 and X plus 2 y Equals the 30 similar if I get real and backed up in that situation to x equals
3 so high wage increase the complexity that had a variable and then I also add another equation Way have 2 variables into equations and now what I wanna do is right this turn matrices owner gather all the important information and stick them in matrices that's where we put information get on 1 hand I get that Coefficients of I extract all the coefficients I've got a two-year a 1 . 1 and 2
And what should I do with the excimer wanna put those in a matrix in the same way that I had the exit matrix there
Now I've got to variable so I should store them together in a single matrix and then what should I do about the solutions will over here ahead a 3 and I put that in the in the matrix containing all the
The right hand side information so this 2 things their equivalent the same thing I could write this directorate said the same thing but we can see that by multiplying it out if you like filets Mobley cannot attitude matrix multiplication so you can see if I take this this vectors a column vector I turned on its side and then do the multiplication AX times to 2 X plus Y times 1 and then I get X times 1 is Exon than tends to to Y and then that's equal to 3 3 but you see that This is the reality what it means for matrices to be equal of this majors as you will at nature means this thing here is equal that of that equation and this is equal to that of the 2nd quarter so they really are equal and now what I wanted to Ozona make the connection between this and not by riding out I think this since Terms of symbolism so this was 1 example 9 generalized this represents just an example of a X equals be this 1 here Represents an example of capital time some variable vector you think of these either vector or variables that that's a list of all the variables and then that call some other sectors which has a list of numbers So if I wanna so now that he is playing the role of the expert that he is a list of variables So where my do V I wanna be able to find In So that I can basically divided by the matrix just like they did here I divided by the 2 0 0 1 divided by the able we don't know what that means that that's the issue if I have to buy 2 matrix What is it mean
The divide that matrix and then what's what happened when a divided here when a debate here I got 1 so what's the 1 in the matrix world that's the next question what what represents the 1 here in the matrix for cancer is the identity matrix Phillips
Just look at that 1st at the identity matrix plays role of Wan as Iike to what what is this feature of 1 of their The thing about 1 is if by multiply won by any number I get that number back so if this if this has the same features the that's what I mean by plays the role of what was just look at that if I take any matrix Dean And then multiplied By the identity matrix with see what we get back on minute take this column turned on its side Hagar 1 0 here I match it up I get 1 times a day Plus 0 that set and then a one-time C plus 0 let's seat and then I switch Move over this column 0 1 4 0 8 plus D and then 0 C plus so that's what I mean by plays the role of the identity and I go through but if you multiplied on the left side as well that's also equal to check yourself so that's what I mean by this language Europe plays the role of 1 in identity India to buy 2 matrix World so what's gonna happen at this stage is when I divide by aII the result is the the identity will see that now without trying to tell you had Do this redesign is going to give you an inverse Russia's play around with that 1st set foot in duty the inverse of that nature So it's a is equal to 2 1 1 2 now we don't have signed this yet Teller make a play around the little there then a endorses 1 over 3 times but to negative coordinated 1 Now let's see what happens when multiply these 2 A times a and that's equal to 2 1 1 2 that's multiply the left but he hinders But instead
That way at the constant up from rarity A 2 1 1 2 legal practice with our multiplication here OK so take the to 1 Put side that is me for minus 1 its 3 there the the upper left-hand corner than I have 2 times negative 1 negative 2 plus 2 0 that's looking good and then then take the 1 to hear have 1 times 2 is too Minus 2 0 and have negative 1 plus 4 history and then remember we had Scalar multiplication last time A matrix times a number on the outside that just means a multiply everything I see inside by that one-third so that's equal to 1 0 0 1 which created it so I haven't
Produced this so-called Anders Anatoly I get it will have to work on that release that's our operates its just like dividing by the nature you see we talk about dividing in this case because comfortable with that but really what I'm doing is I'm multiplying living inverse division is is a former multiplication multiplying by this hinders up there I get 1 here I get the identity matrix Chris
So now I have a VAT
The situation over here
How could I sulfur X and Y would suffer if they want to see where sulfur X here So I'm gonna multiplier both sides by
1 3rd to negative 1 to that's the 1st time you want to know why and here we have to pay attention have to actually multiply on the same side remember in nature in the majors WORLD AT doesn't equal BAA so multiply on left here at the mop of well over here on here and I happen to write
That way we could have written a half on that side nobody would've been upset about it used to that here we have to keep track of everything so I need a happy 1 3rd to make it 1 2 and then times this guy here 3 3 going I don't need to do this multiplication because I know this is easily the identity of all right at the edge matrix and then to the multiplication 1 3rd time
When I do the 3 the 3 three-year I get 6 minus 3 3 that we can tell what size matrix should be because this is 8 2 by 2 4 and this is to buy 1 This will be a to buy 1 as a result OK and the the last 1 is negative 3 plus 6 years 3 of them radicals want 1 of my final result we see this if I were a multiply that addresses X Y This is a very major behaves like 1 so Multiplied It just you see it doesn't change anything So that that implies that X Y equals 1 1 and you can go back and see that's the solution here by plug-in x equals 1 and wife equals 1 and 385 billion x equals 1 Wykle's Omega 3 solution is Mexico's want why equals 1 and I got it in a new way my new way was to pretend like the Matrix is just a regular number
Can I essentially Divided by Divided both sides by than I was However was able to solve multiple variables not just that situation So that our new trick we need to learn look put put some steps in here so we start off with a view equals and than this that hero was multiplied side by a interests
And then there's canceled just like that 2 industry the 3 made too The 2 women and that the one-half made 1 now I to calls will be removed cause a university and and finally the Vectra's interested and is equal to a D and you see this notation here
It is reminiscent Of this notation is really the same thing as want communicate that to you that
We're doing the same thing it's just It's a big boy man rather than to a child infant yes All edges up well OK So But the variables that I line up in the 1st 1st column here that's going to be the 1st variable and their tenant in order so I X then why have X and Y if I were to put 1 over here than I would put like this would ever order they come in here I put them in vertical I can't It's only once But you mean this particular case any identity matrix is a square matrix votes to buy 2 or 3 by 3 and then it's always gotta have the 1 was down the main drag it's a diagonal matrix has all ones so that it it's just an extension here's the 1 by 1 and there's a 2 by 2 5 3 by 3 members of the crew a
So what we wanted do now is we wanna get a former Olympic ill-defined in more told you what it was in that example only saw that it work now the goal is to
Figure out how you get that hinders matrix from a given nature and a 1 . 2 That I guess they just to restate that In a X equals example Then we had have that they not equal to 0 so It's not always the case today inverse exist even in this case so Maryland matrix situation is going Even more complicated even more situations where that hinders doesn't exist a lot about that too much Come to this thing called the determinant start off with it saying it is And by and matrix and a nurse exists then here's the situation with the the university I have made in Mercer a black people to their identity and but positions and that and similarly by multiplied together side that also gives us the again So what we were doing was 2 by 2 case and then we got I said to that's what we The bus wreck let's write down every 2 by 2 matrix in the universe than they are specified what ABC in DR you can feel free to put in whatever you want but it represents all to by 2 matrices in universe a B and C a B C and D are a real numbers if you like they are real numbers he alright so there is a time I wanna findings made in what I know in advance that he hits It's also a became it's another matrix I have to get the identity out again but I have no idea what's in here However what I do know is if I multiply these 2 I get the identity of the give me a little equation to work with a naked reversed that most the 2 by 2 identity so I can write that down that tells me that a B C D Is equal to times X W Y and Z I know what that nasty is outlook There were those variables at least I can say that that's equal to 1 0 0 1 And that is the opportunity to find out what I want is I give you this 6 numbers is given nature just represents 1 in Munich but then here what I wanna get is X W Y In terms of that amount formula For what the That's my goal
Take a little bit Oracle have to to concentrate pretty hard here to stick with this time pocket so let's get started let's do the multiplication what I wanted is I just wanna focus on aren't really on just 1 part here on a multiplier
Those severely multiplied X Y and will I get here thought when I do that I get a X plus BY that's the 1st OK the 1st at her left hand corner 1st slot here and an array out that way because each 1 here is a little bit more complicated than just a number because that'll letters floating around the next 1 is in a VAX time c Bell b c x and then plus D Y C X that's that 1st Call multiplied through the use me this year and then on the right side I still have 1 0 0 1 so this stuff he has Equal this member quality of major cities if 2 2 major Caesar equal and every location is equal so that means that this guy is equal 1 and this is equal to 0 I can finish this off by it Had taken the W Z multiplying it through this write it down just for practice and that's a W plus Z and then I get CW last season but the nice thing here is that all the X is a wiser over here and W disease over there So I can considered 2 systems of 2 variables talk at so User saying a 2nd ago that this war is going to focus on the Orion do the whole thing it's a lot women do this party redefined X and Y and I'll tell you what W Z This upper left-hand corner must equal 1 so I can write down this equation plus BY liquid 1 and this lower left-hand corner must equal that 0 there select right down CTX Plus DYG 0 For such masters now at this row Echelon form that weekend but his rented out what what would that look like and augmented nature Said at a as Michael efficient As The 2nd coefficient and then 1 here but I don't see an Indian 0 there that's the augmented sure my variables owners assault while they remind you that severe summoning variable is around her summary letters solve for X and Y at what's that this produces here A B C D 1 0 ahead of what should we do with this Augmented major What's the thing that we do we chatted makes zeros right hand and in Minneapolis reduced row Echelon form I want this upper left-hand corner 1 so start their car but I replaced Role 1 was 1 over a rope just making a 1 and I have wanted to be over overhead and then dividing by a here so it want over a C D Rhodes to doesn't change that making progress may be go I get over 1 get worse it's true it is a little worse OK but what we do next we use this 1 has appeared To get rid of the other numbers and its called who move would really like to have a 0 here if I get a 0 here at essentially eliminated very born Arkansas for the other variables so that's my next move is I'm gonna replace Roach who or what should I replace it would have about minus seed rolled 1 would that do the job would make this a 0 I don't care what else happened there's 1 0 OK the top row doesn't change the bottom row I get 0 there that Goodnight at to Bill careful multiplying this by minus see here And then they Adding it so that would I get is D minus V over If you believe that And multiplying row 1 by negative cities of negative see here That's this term And I'm adding it to do so I get That Ugliness The isn't so bad after multiplied by negative see here is negative see over NIZ said 0 so I get paid the started it pretty ugly let's take a moment here Then and combine these 2 fracture barely get us get a common denominator so common nominator means that that this means that I am about to get a common denominator Were working really hard here the reason for working so hard is because at the end ended it would a formula that says Here's a matrix here's and Soto were set to work through this Get a formula OK my common on areas A on the bottom of 0 And then there's gotta be a D over A D minus Z may have 1 over a over a noble either now or home free 50 little careful about how we saw you this enables me the sulfur Why did you see that that's if this was a 5 in this was a 10 I'd say 5 White Eagles that was not 5 has not said All that stuff but we can still write it down the says a D minus BCU over a times I was might over it The Nikon consult knows is canceled So now I get the credit over here that implies that equals 2 suitcase canceled and you get minus Z over A day minors B. C Matsui always do it we get it down with knew 1 variable solve that got it them back substitute defined with the other variables So there's Y And then we can use the original equation the easiest original equation use years this 1 has 0 and will choose this CD X equals 0 That implies that Z X plus D OK 12 foot NY wires this whole thing trying seize over a D Please see that as equals 0 So I had to see taxes will too Now when I put the D CD in the area to the other side it's times over Beijing minus And now I get X canceled Eckersley We go over a D minus that's a lotta work but we we've made some serious progress here member what we were trying to do is I wanna know what this matrix says That causes this to become members of because become the identity award in Vershire so we just found this value in this value in terms of those that were given the pretty powerful we could do the same thing for W I would just say this sequel 0 and now 1 equal on go through the same process save you torture there so here's what we What we determine here as the solution is X equals to X is equal to D over a duty miners See you got that then why is equal to a minus seat Older A series not tell you what other values WTO is equal to minus beating over a D & Z is equal to a over a mind at this point you should say what the heck is 80 PC quiet
Thank goodness rightly We have edema is easy in every 1 of those that nite but what is that will eventually get What that is right down the inverters we have a formula that would give him a penny Dubai to matrix commuters a universe is equal to well let it all out that X was deed over 80 miners BC and this was minus V over a land and minor sealed a seat and had a boat or a finest PC that's had a rate 18 miners BC every single time I don't like that were elected to since every single thing is scaled by that number a mother was bring it out front 1 over a D minus times Deve China's minus a analogous act because not all exposed
Us A D minus PC can confuse things and bit confused the minor knee look at it he did you not able to see what's going on what happened here for what looks like a D N A D yet switched the the d Aaron opposite order of these 2 guys switch and that what happened in the B in the sea they stay the same you you make a minus Looking I have committed a terrible mathematical error here but give a give 1 per cent bonus at the end of the course annually until they were married But you have a 89 per cent of the castle 90 plus Just tell me right now where my errors It's not in this this is the right for me but I'm missing something here Yes No no was a problem here I need a state something extra Right The sale of the Bonds UFT e-mail me tell me that you're the winner of this guy isn't is kind of important fact that plays the role member back we're doing just regular numbers and I wanted the inverse Ikey Gillian Anderson anything except when it was 0 and now I can get in inverse trainee matrix except when this is equal to 0 So now we have to study with that thing versatile example here so But know we just got here with this is pretty powerful tell a couple examples Go back to their original 1 that I told you with the inverse was let's check that 1 see if I wasn't that I was lying we got the the formula so I can just say immediately that gave the 1st people to want over OK what's the ADD minus PC let's get that straight when you do in a minus PC just do multiplication diagonally this way minus multiplication diagonally the other way In this case 1 over its 1 or 2 times to minus 1 times 1 That's the 1 over 80 miners BCE then it says switch today in there and said D that this case you notice It is staying here but I did switch on and then the The make those 2 negative that's what it said if I had B and here I have minus
They're in their urine burst upon their time to negative 1 . 1 2 would if you like you don't put the 130 and Whitaker's is direct to 3 on the bottom every single time that just keep
Hinders that's 1 we you didn't get into those Avenue 1 at 4 3 2 1
Too well lately Israeli again immediately this is like learning died by matrices 2nd right down the numerous immediately it's wandered over 8 0 4 times 1 that AT and then minus 2 times 3 times OK what I do here for these 2 guys a switch and social 1 4 and these 2 as lever Lorelei Makeham negative that Native one-half times want negative 3 4 The muzzle check it a day in time is a that's about half time 1 2 3 Included 2 4 4 3 2 1 lead that out there For the 1st just do the month matrix multiplication then put the happened afterward OK so when I do this idea for minus 6 is native to the idea negative 8 plus it is 0 3 minus 3 0 negatives exposed negative too This is a negative at half out front had been cut so I want to see again OK so we have in fact learned defined embarrass you don't ever have to do this process again memorize the formula If someone says here's a 2 by 2 major find the inverse than you do Go in and do it the formula that the advantage of putting all our work into getting a former now we've got it this Smith if now we need to investigate this
What is that day Have all But And That should be the question What is this magical a D minus BC it's important because we divide everything by it
1st of all we knew had a name for it with the use of vocabulary this is called Determinants of Inc And the notation for that like absolute value So Both have a little example the sea with this really represents There's a matrix we could find the inverse who opalesce all registered red now or look for this BC was also pay attention to language isn't it true that
All I have to do to find out whether there is an inverse are not is just check this because this is Eagle 0 no inverse but if it's not equal to 0 that I haven't so it's the thing that determines for me whether there's a new version of its I check Please give me a matrix I don't in a solution if it turns a certain color and I can identify whether or not it has an inverse determined is also a reason why we use this absolute value notation and will see that right now so what I wanna do here Where we're witness right now what's going on inside the Matrix I alluded to back to this fact last time I said said matrix Ursem geometry right now just a block numbers Phenomena show you the picture that goes with this major News a few pictures is not just 1 that will focus in on 1 thing so what I wanna do is think I use columns yet Exxon focus in on these columns 2 0 3 5 and demographic McGrath How to 0 If I graf 2 0 I can put it right here is 2 0 Rangers a truck There's 2 0 Rate that that's the Parliament 2 0 Just thinking is the 1st foreigners X in the 2nd component is and then due 3 5 0 here's 3 and 1 2 3 4 5 years 3 5 Jail them to that but little arrows on there because they're they're called back tears we haven't gotten into that yet but they are vectors member if I had just a single column this call the column vector and vector several Arizona now women do is I'm Pembina create a parallel Using the list 2 vectors and this area which shoot it equals and determinants of a witch We calculate what's the determinant of a it's 2 times 5 is 10 minus 3 times 0 So determinant analysis of the area Here you get theory of a parallelogram its base time high figures of the base the height is thought to transfer which is so that's the hidden information instead of a matrix if if I trained myself by practice hard enough I can look at a matrix and see the picture in my mind is not always necessary that the exact picture but it would be helpful when you see a major cities have some parallel pop India's that's what it represents represents parallel the area of apparel Now it's even more remarkable that I do the same thing with just the Rose It's the same area By changing the the roses being 2 3 in the 0 5 also get the same of knowledge think about Patients were worried that reimbursed Hinders doesn't exist when this thing is equal to 0 . 0 area so it's 0 area that means that Is that the Rose must be the same there must be on the same line as soon as off on the same line you're giving it an area When they're together you don't get any area You will see a little bit more of that go but will now woman that this determinant by itself because the goal is to get the inverse of larger matrices but as you see in this example here the inverse of 2 by 2 required us to know that can do this What pop that so now will I wanna do is to study sending it would do this for 2 by 2 3 by 3 4 by foreign even But whatever and can and then the ultimate goal is defined the inverse of those larger matrices but we have to stop along the way and a steady determinants but as a subject that were in Dunellen learned how the Duke determinants of larger matrices and toward that and we have to go through a list of theorems in the book in order to find more complicated matrices you wanna learn how manipulate matrices and so we go through these and see what this determinant of them
We can do more complicated wouldn't do it these there and it is well worth a look at the statement is the fear that they the U.S. terms this picture that conceded with your head and don't have to prove algebraically you can just see what's happening is the first one is the determinant of data transfer
So if I were to take the trance poses to get the same determinant but no you wanna think of others I'm saying timely say were determined I'm saying a word area so the area this matrix his area the dot-matrix represents What I'll do is I'll keep using that same matrix will have the same picture will just keep seeing how it doesn't Age What if So if a person is 2 3 0 5 of them A transposed appear what's transpose gives us a chance to remember what Trans poses I take the Rose and tournament of the columns called adjournment the Rose wary light so that road needs 1st column that needs to be 2nd the transposed the picture for this guy is the same as that 1 Quick Air One 2 3 4 5 3 That same picture that has an area of 10 and that if I do this guy here Please The Now I guess this is my 1st call on 2 3 I don't want to I want to 3 And then I have 0 5 that's 0 1 2 3 4 5 areas also base time that The basis 5 and the is too That carries 10 so that the Tonight because that's what transpose does it it changes Picture that Secord transformation of the picture if you really good you can see these things in your head that that but on your own want that
Hey Luckily go through every 1 of but I want you to see What you'll have to know how to use these
A little bit of 9 . 2 where are manager now them when a star with an original matrix and Eminem modify it Using these role operations member the role operations and we learned we could switch Rosa we could add a row to another road or we multiply row by constant all of those role operations only through them say what happens is determined and when do operation Keep track of it what that can enable us to do is take a matrix and changes into its reduced form make it a lot simpler 1st before we do turn that that's point of all OK how doyou write those of mathematical it's a little bit challenging to do that so if beady Saying on getting a new hearings from an old matrix and what might attack pyramid attacked the interchange it to rose as 1 of the role operations so I'm going to obtain B from a buy a changing True grows in you could think of your colleagues We've only really learned role operations of largest won't stick to that focus If that then what happened to the determinant of 12 inches the area of a B minus determined So this was an interesting Willis checkered I'll stick their same example a 2 3 0 5 and let's get picture for a again I know it's that we keep doing it that practice so I have to 0 and then 3 5 We know that areas can now would do his own interchange 2 rose so that just means which these 2 rows on the goat draw 1 change to a wrote to see the double means an interchange in 2 rows of B is 0 5 2 3 and to the picture for that Now it's 0 2 That goes a yellow here That's 0 2 for the 1st and then 5 3 1 2 3 4 5 3 that's there but you see that's the same It's got a base of 2 and a height of 5 with the same area oversights negative Hundreds of that should be weird preyed on what the heck is negative from what's going on So that's a question that comes up
It is I might take the determine negative numbers and I'm telling you that it's area waters negative area will voters with that can you think of a have a situation our you might negative area in the real world
For you might think of something is negative area think about that 1st It's a little hard up with flu leading a little example OK if I take this this is an area right now if I have a pool of water a here what would you say the area of its reflection This is 10 what's the area of the reflection you might call at negative tenor is not the same credits its doesn't even exist a material World it's justice reflection that's 1 way to think of it the other way think of it is we've got to think of this as a little but that transparency Got the front side and at the back so this is the area of the backside That's the way it that's More of a correct way of thinking and all of this comes from our oil you have the experience to talk about this but this all comes from the right hand rule so when I do the right hand rule here and I do this lecture tonight and I close Towards that vector points out this direction that gives me a positive but if I take 5 years my 2 vectors here if I start with respect to the 1st sector And I moved forwards that vector than the bomb points in the red and will be taken physics something you may have seen if you haven't been got to just let them go not really part of the class service so that it comes up regular confusion at that question what the heck is this
Negative Area well there go get it to give it is the backside of the rectangle If forever part of this classes to just try to develop a little intuition for these things that we reach I'd do that these little someone out intuition work with that He is a theory 9 . 3 0 off air for a has 2 identical
Then the area 0 where the determinant 2 0 of course it is if that if the 2 Roser exactly the same than than these 2 columns again exactly the same Solana drop the vectors Amiga the same vector bullets 7 example of that if I had a vehicles 2 3 2 3 that's what it says here there's 2 identical rose sexually or columns to rumored tried shelter from too much information That's common back And we do it rose true truth columns to come to terms the IRA Hundreds Show with free graft that got 1 2 2 2 2 2 There and then 3 3 asserted their yellow The earth To see no area and if they exactly the same thing and there's no area so get the X terminal and if you calculate the determined
In do algebraically chewed don't think you have to look at the areas factors get a habit of taking This says it's 2 times 3 That's these 2 plus or minus 3 attempts to his way back This is a report here there 9 . 5 visit morgani used to do more complicated nature see a larger majors that gang leader is obtained Dominic new matrix from a
By a row over operation of the form so what's what's our typical role operation that would be down in general might take a role I can replace it With something times some other OJ plus they had time relax that's what a typical role operation looks like When his Rome all down all in the universe for the good news is really good news because if I do a role operation of this type which is the type of thing they use to reduce matrices Right This is what we keep using over over again we do a row reduction then the good news is that determinant of equal to the terminal of this leaves the determinant unchanged right that This type of role operation it leaves determinants unchanged I do those until I'm blue in the face area cheeses The day the idea that is that I have a really complicated nature so all these numbers they're not convenient not easy to compute I could reduce it down to a bunch of ones and zeros and things I can and I'll still have the same determine so I could reduce down to a simple complicate computation will see this in action I wanna get all these years How will see it in action so by a is my typical matrix editing using every single time Thursday and then we're might German a dual role operation of that so let's of replace wrote 1 with negative wrote to 1 around the something that's the type by describing up there if I do that The new Duma replaced role once wrote doesn't change on multiply wrote to buy a negative ad There to multiply this by native is native 5 add that a negative to lose my new matrix policy with the area this week
The standard picture here this drive quickly that's the that's the picture we keep adding for that 1 and then what's the picture over here OK well it's still 2 0 and now it's negative to 5
What's that My group it's still base times I think 2 times 5 said that left it unchanged
But remarkable that I can do this kind of thing with seems to jumble up the Rubik's Cube Ranieri messes things that but every time you do that you preserve this area he gives you different parallelogram but the area stays the same the 1 that's why stayed within these matrices uses geometry which states fixed even when you do these But it sometimes gets modified but even this case it states 6 The next arose It's so that you can use the determinant in a figurehead basically it's a shortcut for special type of nature's call triangular major You come up by diagonal matrices are part of a triangular matrices as examples of this learned example in this case so And here's their 1st chance Oregon do matrices that are larger than to buy 2 even tho we haven't really looked at it Amin introduced here just as it's appropriate case there's there's my 1st major someone I always use head that happens to be 1 of these triangular matrices and the ruling means that the determinant of this is just the product
Daniels that 0 down there it dozen contributed thinkers notice I go to times 5 minus 310 0 so it is that parts not not included this is difficult to find a buyer to step outside and more Exchanges like this now now this is and it's it's now that 0 0 but you see what's called triangular is triangle here and try The trying every outside the triangle 0 determinant of being here It is also just 2 types virus was the product of the main egg rolls with 4 brokers a lodger matrices
This is an upper training a triangular matrix is called upper train because the triangle of version that's just for your purposes uses Favors triangular the determinant of this would just be the product of a magnum German and is equal to his face negative 1 tribes time 6 negative 24
OK here's another triangular matrix this 1 is full of hope of a copy both on the same time on the since a of triangular area another triangles on the bottom and what you need to know is you don't have to go through the normal process some eventually give you the formula for how you do these determinants but these ones if you know the short shortcut you don't have to do all the cup competitions here the determinant of negative 1 transport and 6 months out of 4 4 by 4 just so you start to see were added here our goal is to build some large matrices backhand so there's there's a the triangular matrix there's the triangle and determinant this Case is just the product of a meeting I am source Each
Look at it a couple more reason will be done with that quite bit
The 1 rule operation we have uncovered is multiplying a row by constant member if I had a have been there and multiply the rope by to steer the half find out what happens determinant That is so good Now He is being obtained from ahead by multiplying may grow by a constant them what happened determinant of both their constant landed here this is lam that the Greek letter L that's the lowercase versions so that the determinant of be busy books Alam time German users Bigger implications and so on it A is that example and keep using Now what happens if I multiply that's described this in a row operational and replace wrote to wear 3 2 times too When a replaced rose 2 with 2 times that That is to be which is too 3 0 10 legislative in terms of Picture where it I just do I just multiply that side of the peril of and so what I did was I multiplied I had this little side here
5 That's to them and now the height 10 instead of 5 so I'm to German and is equal to 20 10 2 years which is illegal although it handy here so that reflexes so multiplying a row causes the area Which makes sense that I just multiplied 1 of the sides of high by tennis and her bag to
Here's a little trick that goes with this year's here will give it as an example given by that determinant of a is equal to
I thought I was telling us I don't tell you the matrix But I tell you what the determinant of a matrix I can tell you that a visit to buy matrix So what it is As a determinant of 3 years The list this past her 2nd by multiply A matrix by 3 that means News can you believe me if I said that means you multiply every row by 3 remote like everything in thereby 3 great everything in the matrix by 3 that's 0 3 years but Over here all I did was multiply 1 then it caused the area to change by that amount so now in this case I multiplying both were so was back in the car Each time a multiplier role it turns the determinant In 2 3 times it so this is going to be equal to 19 times the determined however but 3 square so that you know The 9 3 squared times they would mind 5 45 and the Since I multiplying both rose by 3 each 1 of those needs to get 3 I've only multiply 1 road B 3 times it But if I'm old but both rose its 3 times 3 that's that's how that theory gets used as we can ask you can say Well here's the determine the matrix and manipulated a certain way what the new determined without ever giving you the matrix That's an example that will some others as well as this was also this is kind of a combination of everything 9 . 8 says that determined that ADB is equal to the determined that can't prove it turned to the think of that that in terms of area so if I take 2 major cities this guy has an area this gazetteer 6 2 separate pictures area area
And if multiply too The area of the new matrix is the saying that somebody asked last time or when you multiply it this new meters what the new numbers mean what's happening is these areas are preserved and that's why we multiply and that funny way multiply in a funny way so that we preserve these various that's what's really going on preserving the geometry in major cities and that's why we multiplied funny way OK so let's see this road quick here 2 3 0 5 that's our standard 1 of he negative 1 negative 2 1 and the determinants made here that's equal to 10 in an overnight Oregon being the determiner this guy
It is for minus 2 4 times 1 my negative 1 tend to tag this subtract back so that for minus 2 which is to the ballots OK The product at least you guessed that product basically induced quickly it's 4 times to his each minus 6 2 and 4 times 0 minus 10 net negative 2 What's It is 1 of men have 0 times native 1 plus 5 And the determinant here The of a beating That's equal to 10 spread out its 2 times By minors 1 times negative 10 20 And then That's equal to a German so when I went and multiply 2 matrices what it does is accused you knew matrix that has the same area as the product of 2 original products of various that's really what's going on with the multiplication as were preserving the area that's in advance will say that is that you could say the multiplication of matrices is the same as or the area of the 2 multiply is the same as the product of the period
If was taught to sit there and go through all those years everything now wait to get back to
The nuts and bolts of actually doing these things I If you want you you could draw the pictures out there and see that the areas I didn't do that for you but it's available now a do we have Did We had to do larger matrices that's the inquiry showed no introduce determinants of and that and major Where you use 3 by 3 as an example of an attack because it's the that's the first one that's more complicated and the 4 by 4 of the 5 Basil follow along with is too complicated thing about 4 by 4 or you can buy and when your 1st doing it a focus in on a 3 by 3 6 who will learn attitude thing with that and will expand that to 4 by force maybe so here we go if I write down every 3 5 3 Matrix in the universe how can I do that will I write down a little indices so that I can keep track of everything that a 1 1 1 2 3 and then I had a 2 1 8 2 2 8 2 3 3 1 3 2 A 3 3 1 introduced a new notation if I wanna talk about the determinant of saying I can't take this matrix and I could just put straight brackets around it and that means take the determinant there's what highlight that that this is the notation hiking brackets which is means the matrix but if I put vertical lines in the same way that I have there that state determined loyalty is to wreck the things over and over again but this is this is a notation
Along Would Ovalle where called might
Years That's miners with an old like the miners in the mine shaft the miners cover matrix excellent thing about what miners psych major and minor and so on minors Something miners something smaller and smaller or small we must start with the big matrix and the relief create these miners Witcher that's what we know today with an M & M I J had correct this out in words in urine and dissect all words in red down in him and that each day so this is and believe it determined so it's gonna be a determinant the women some things 1st called the Matrix obtained from a aid being the 1 we're deal back there's generic matrix who were Alleghany due to obtain this is I'm gonna removed What should I remove well they did the notation indicates something right that 1st number always means the road for the BVI row and the J. always mean something that's gonna do they did call for the miners in India Obtained by removing the And J. call That's why we use that indicate that notation This to thinks they were in a star with a runner remove the Roma column and radiative but some leftover numbers and then take determined Right down to it a year's enough you wanna write again next time that thinking of those days right there do in these woods calculate M 1 2 and 1 to fear so that means I take his only the determinant visitors a 1st for determine it's not just a major determined some news the straight brackets and then on a play in their little tedious better get used to to these notations the indices if you make these out of time so there is my original matrix and then what it do it says to remove well on this case it's the 1st row will remove the 1st row and the 2nd column And so it says it's the determinant of the matrix obtained from 8 by doing that so once remove those I now have a smaller matrix it's now a 2 1 8 2 3 8 3 1 8 3 3 5 reduced that's scholar minor and then the straight brackets means take the determined now determined headed into the determinant its you multiply those that also that a a 3 3 miners A 2 3 2 1 so that is the determinant of the minor obtained from a book by removing the 1st row in the 2nd column We can do enough 1 year Do another 1 real quick and to 3 3 days that's gonna be Nana remove under scanner rented out and in terms of the lines were crossing at summit take the 2nd In the 3rd column And so we end up with here is that took up the 2nd row in the 3rd column under a 1 1 they want to Left it at these 2 guys left And then I have those 2 has bought a 3 1 3 2 That determinants is A 1 1 8 3 Chiu minus a mortgage to a 3 like it's all real quick you probably pretty sick that do some real numbers of actual matrix with some numbers and slow Grave There's a matrix Let's go calculate those guys there so and want to and 1 2 It the determinants of the Matrix leftover if I remove the 1st row in the 2nd column the 1st row 2nd column for 6 9 and that's equal to 36 . 2 negatives So it not too bad when you calculated when I'm writing about general looks pretty nasty if I had em to 3 that means I remove this 2nd row in the 3rd column so I just have the 1 2 left over in the said That's equal to 8 minus 14 candidates as well as just coincidence that those 2 numbers are the same They are always turn out even for that largest you chew that's enough let's do 1 more just to prove it is adamant notes and so am 1 1 day and the 1st row in the 1st column 5 6 8 9 That gives me 45 that this 1 minus 14 8 negative broker as the big step the 1st big step in finding the determinants through most ready did say whether determinant is
His next concert which is building on the scarcity of the minor still take a step back further we have a matrix that we can go calculate all these miners and the now organ a calculate this thing called
Kohl factor which comes from these miners anyway intact it's almost exactly the same as minor has a little extra things so close that factors a there there labeled the label is city IGA The abuse saying they at the role here in the column here as defined as IGA is equal to negative 1 high-cost times Focus enough I just get rid of this part of its critical factor is the same as a minor It really get a calculator miners physical factory is the minors Victims get as little fudge factor This guy this guy looks like But like it's a lot more intimidating it really Holidays theater plus or minus 1 because of this Power years I've of minus 1 of this power years even in only a positive 1 so the only thing this could ever be Is plus or minus 1 and enters attaching that plus minus 1 to these miners were in a computer that
Local factories so you know it's like this is a bigger icon on your desk by double click on that idea that all definition over there and then if I double-click on the negative wonder the IPOs to take care that is so low that look at em
1 1 Figure out what I want the IPOs Jay's RC ignore uses it would do them in your head real quick so C 1 1 means I go negative 1 of the 1 that trading even-numbered Aaron so that's a Sky B 1 man and 1 1 but OK so that's positive ones who doesn't show up at the M 1 1 we have over there that's 5 6 8 9 If I use that example so that's going into negative What's make A year products Phillips to see which 1 north of here see to what they see 1 means I've got to hear a 1 there which means a go to their own one-man to close 1 of them again And 1 day but in this case negative 1 to the 3rd power that is a negative 1 so this is equal to Negative and have got the determinant of what I take out a 2nd row in the 1st column right hand to won his 2nd row 1st columns of 2 3 8 9 left over so that's equal to the negative on outside 18 minus 24 18 minus 24 a kind of myself and a quarter of you're the top if not this 1 is equal to negative he seemed minus 24 which is negative BAGHDAD 6 positives Let's see 2 1 cofactor
And Are now ready to ride out with the determined Indeterminate today this is for and my and that's out in English as is
Enemy hard read personal go back and make sure we see what it's going to be determined Defying to beat this process I'm going to multiply figures with me so far multiplication and this is keep that in your head multiply don't take not by the elements of this is key here for big box here any for any of this will be your choice any role work column SO so far are you with me multiply the elements of any row column where my fuckin about so far picked the top row 1 2 and 3 those are the elements of the Top the 3rd column 3 6 9 the elements of the 3rd column on a multiply those by something not by themselves All the here so the language makes sense when a minor multiplied by its cofactor Kalisz together factors like multiplication summer multiplied by Dies the corresponding cofactor Corresponding cofactor something the elements of those and multiplying by the corresponding cofactor so as an example imitate 1 up there in the upper left-hand corner its corresponding Co factories C 1 1 the upper left-hand corner cofactor by the corresponding Co factors and then hung going to add results Products great example You know the words in Matadi Chowdhury that by you know like you're reading a novel just goes right Unless you stop and analyze every single word along way by just a runaway under multiply the elements of any column by the corresponding factors that the results he is with me still that That's a little lesson on how he learned that he shouldn't expect to build read that and have the whole idea pop into your head you have to go word by word nature Understand every OK but I can help you do that so that we can do is get to the result here so Show But to write it down a generalist do all the 3 by 3 in the universe I've got a 1 1 but just right down After rated them twice for the determinant of a OK that's a you straight bracket notation that 8 1 1 8 1 2 8 1 3 2 1 2 2 2 3 3 1 3 2 3 3 0 there it is that the determinant and what does it say was this process said it says cake Any row or column just picked the 1st row To be get The if I do the 1st row when as is say they do It says is Take elements of any road OK I pick the 1st row and that multiply them by the corresponding called back so that means in this case ended in a 1 1 its corresponding cofactor C 1 then I take a 1 to and multiplied by its corresponding cofactor 1 3 guests will multiplied by its corresponding cofactor she really do you have to know what the little indices are treated source We get around they want to know what to do if not I'll show you to write that And that's eagled German now fight if I do lead a 3rd call on a different looks like the enemy 2 completely different things weary of the same number and all demonstrate that the of I take the 3rd column what's activity that's going to be a 1 3 C 1 3 plus A 2 3 2 3 plus A 3 3 See through 3 both of these are equal to the terminal They look different giving it the same number of nature sold at all out if prices are at war finally in a position to do some determinants of larger reaches takes a lot of work to get all that out but now agrees to some concrete examples of excess about you the simplest fine thank wanted by examining it's easy copy of the and let's just do it in their let's do the 1st year but said there's 1 more thing I wanna do I wanna I just wanna do the negative won for the add J. list to see what all that
What all these locations gave you in terms of this case so when you have a 1 1 that's going be a positive there and then when I got a 1st row sacking column a wanted to use the Zenaida negative and that it is I if I just go the next on understanding want Although for muddied shrimp even God Lakers Ramadi to So a new computing the cofactor helped to just know the positive and negative location so as I'm doing the 1st row here I'm also considering the locations of those spots this is a positive start his negative but that's a positive Solicitude The determinant of a broker with the stadium says take this gagged and multiply it by its cofactor so that'll be 1 1 1 2 plus 2 0 but is not plus it's might as well let us just put his Class C 1 2 plus 3 times Now go compute these thinks So a one-time C 1 1 that's the matrix is a determinant of the matrix when I crossed off the the 1st row in the 1st column 5 6 8 9 and then this is a B minus 2 times the determinants and
Visits minuses because this location near the 1 that's where that negative comes from way switches here because when I got a computer's cofactor It's negative 1 to the 2 1 plus 2 Oh surrogates negative 1 Q which is Aggie OK And then when across off 1st row and 2nd column I get 4 6 7 time plus 3 9 take off the 1st Role in the 3rd column for 5 7 8 and then we compete them This is 45 minus 48 minus 2 times 36 Linus 42 plus 3 times 30 to minus Negative 3 miners 2 Times A negative 6 plus 3 times negative 3 and that's equal to negative 3 plus 12 minus 9 0 That was using the 1st row not 3rd call just like we get over there because I wanna show you that it doesn't matter who is part of the definition is that you can take any row or common to this process you always get the same numbers so let's see that at least in 2 cases There's still a year illustrated can finale 3rd column and also over here in the 3rd column is plus minus You can keep track of the pluses and minuses that way The determined that there is no B 3 sees this is C 1 3 plus 6 times C seat 2 3 plus nine-time sees 3 3 when you get good at it you have to write that you just go straight to what you doing so if you didn't write this Sourwood you do you reduce 3 and just right down the determinant of just take that 3 of you cross The row and column in so that You see what you have to write these down you can't just say I'm going across off The role column threes and the remaining steps It is what I put here Many have to subtract off the 6 because this is a negative location to pay attention to that And then Take the sex and distrust its role its column that you want to 780 had 9 3 3 row and column I get the 1 2 4 5 left over Now that's 3 times 32 minus 35 minus 6 times 8 minus 14 Plus 9 times by minus that's equal to 3 times negative 3 is negative signed negative 6 times negative 6 is plus 36 and then nine-time native 3 native 27 0 OK now think about this if we in the Chubais 2 cases we got 0 0 would that mean in terms of that area you have 0 area and so what does it mean rumor in 3 dimensions of the last thing I'll tell you about so winner in 3 dimensions what's the three-dimensional equivalent of the area was a one-dimensional could another area So this is hired you get It is 1 dimension is as like the new look one-dimensional like becomes area and one-dimensional becomes while you cute so if I had a I don't know enough roads here columns I in comments last saw stock of have those columns and 3 dimensions wary habits Is this lips throw them and I have a cube and are that my yellow Chuck here those factors three-dimensional vectors from these roads And then Determinant of a this volume that's acquittal of the three-dimensional version the 1 we had a determinant of 0 if our right all those vectors out would have 0 0 volume which essentially you have a flat shape or even while we can really recognize that you can't look at that girl yet but low-volume 0 s is their brains are sophisticated enough so we need to figure it out using algebra and certainly baby of four-dimensional 1 you can even see it
It doesn't exist and a three-dimensional world still that's really what you do and when you calculating volume and 1 will go back to the 1 by 1 case because it's true here too if I have a sequel to saying 3 Then that's a distance of 3 the determine if I had a 1 by 1 matrix 3 by had a negative 3
That's a negative dish 10 pages it's the distance from 0 Valner negatives says the reflection of the water if you visit all consistent adhered blank then to to Terry to to its volume for my florets four-dimensional OK that's enough for today certainly had covered a lot