Birne Binegar: started so first of all an announcement.
Birne Binegar: Hope assigned problems that he is now been posted there fairly easy problems to deal with calculating determinants and then two applications of the comments that will discuss today.
Birne Binegar: And then.
Birne Binegar: Friday i'll begin to talk about eigenvectors values problem but.
Birne Binegar: Normally I would give you a week to a homework assignment but done.
Birne Binegar: The calculations that are in homework set eight are sort of relevant for the rest of the Semester, so I want to make sure you can calculate some determinants readily.
Birne Binegar: See by Monday and then um.
Birne Binegar: yeah so that's that.
Birne Binegar: So what i'm gonna do today i'm gonna we talked about.
Birne Binegar: determinants for several lectures now.
Birne Binegar: One of the thing I want to do today, the theme of today's lecture is it, we can read place.
Birne Binegar: row operations as a means of solving problems if we know determinants now, of course, when I showed you in the last couple lectures that oftentimes we use real reduction to calculate determinants but, in fact, if i'm.
Birne Binegar: that's that's because that's probably the easiest way to calculate determinants playing hand but after you leave this course and.
Birne Binegar: Your instructors or your boss allows you to use anything like a mathematical software program or.
Birne Binegar: Computer otherwise to calculate determinant well, you can do that and, in fact, and so, in that case you completely avoid the reduction process, so the things I talked about today are.
Birne Binegar: You can I want you to think of determinants be completely independent of reduction process and, in fact, giving you a formal replacement for that.
Birne Binegar: The redemption process is an algorithm from what you cannot see anything about the answer until you actually work your way through each step.
Birne Binegar: Step you know, requires you to make another decision which pivot to remove which which road to to move around you don't really see the answer until you get to the very end.
Birne Binegar: But when you express things in terms of the determinant you can detect things formal things about the answer, even before you proceed with the calculation so as I go through the examples today i'll try to point out some of those instances.
Birne Binegar: Okay, so.
Birne Binegar: pause here second here.
Birne Binegar: Okay, so the first rule i'm going to give you it's a it's a formula that I won't prove but it states very simply how the sold a square linear system, so long as the determinant of a coefficient matrix is not is not equal to zero.
Birne Binegar: Okay, so here's the formula, and I see me I can display it a little better.
Birne Binegar: So.
Birne Binegar: If you have a general linear system, well, it has to be a square it's a number of equations, as you have variables and.
Birne Binegar: In addition, what i'm going to require for the serum that makes sense if the determinant of the coefficient matrix is not equal to zero.
Birne Binegar: Because it appears in the formula in the denominator, so if this were equal to zero this formula would not make sense, but given a square linear system and where the coefficient matrix does have an inverse and this formula makes perfect sense and what it tells you is that.
Birne Binegar: The solution to such a system can be calculated just using determines.
Birne Binegar: The only mystery here's what is this matrix bi well the way we get the matrix di think of it as an auxiliary matrix we get from a using the input data that this.
Birne Binegar: vector on the right hand side of the linear system and all you do to create this matrix is you, replace the ice column of a.
Birne Binegar: With the vector that appears on the right hand side of the linear system, so what i've gotten you here is the ice column of this, but everything else in this matrix bi is exactly the same as a.
Birne Binegar: So let me give you an example of how this is applied.
Birne Binegar: here's a two by two linear system.
Birne Binegar: And so what we'll need to apply kramer's rule.
Birne Binegar: which I better keep up on the blackboard so i'm sure to it.
Birne Binegar: So creamers rule.
Birne Binegar: Is that components of a solution.
Birne Binegar: The solution.
Birne Binegar: X equals B.
Birne Binegar: Has component.
Birne Binegar: equals.
Birne Binegar: Becoming bi divided by determining a.
Birne Binegar: Alright, so let's let's employee that rule for this example, the first one, we need to do is look at the linear system and identify the coefficient matrix things that are multiplying the variables and the.
Birne Binegar: Numerical vector on the right hand side be let's go ahead and calculate determinative a so that would be two times minus, this is a two by two matrix.
Birne Binegar: So it's determinants use a calculator simply by taking a difference of the diet normal products two times minus one minus one and that ends up being negative three.
Birne Binegar: Now to apply this rule, I can now start getting actually calculate the solution.
Birne Binegar: When I need to do.
Birne Binegar: Is calculate.
Birne Binegar: Great those auxiliary matrices and calculate their determines so for this one.
Birne Binegar: All I did was take this matrix a and use this as its first column, the second column stayed the same, we calculate it determinant we get negative three.
Birne Binegar: But the second one i'm getting this index here refers to which column i'm going to replace so here I replaced the second column with the vector be.
Birne Binegar: calculate this determinant this works out minus six and so now that's all we had to do there's no row reduction at all just calculating three determinants and we're able to get the solution here's the first variable.
Birne Binegar: first component works out to be one second component workout doom and so.
Birne Binegar: Our answer.
Birne Binegar: The values of X one X two that satisfied the original linear system.
Birne Binegar: or just X one equals one X two is equal to two and indeed.
Birne Binegar: here's the original matrix a here's our solution vector it's multiply arm coefficient matrix times, the solution director it works out to be.
Birne Binegar: Do that matrix multiplication it's four minus one which is exactly the original vector that was on the right hand side.
Birne Binegar: Okay, so so this method, this method really works, and in fact it not only works for two by two matrix ease but it works for three by three matrix feasible arbitrary size, you want to think you have to worry about is that this only works.
Birne Binegar: For square matrices so n by n matrices.
Birne Binegar: With.
Birne Binegar: A the minus first existing.
Birne Binegar: The way the coefficient matrix is actually inverted but as long as those conditions whole.
Birne Binegar: Then.
Birne Binegar: The determinant technique or kramer's rule is.
Birne Binegar: Really, easy to apply, and in fact it's very fast way to solve it, particularly if you're using your computer to calculate These determinants then it's it's really trivial.
Birne Binegar: Alright, so.
Birne Binegar: The next thing I want to talk about the other utility of row reduction was it allowed us to calculate the inverse of the matrix but that calculation again involves extensive role reaction, the way we.
Birne Binegar: calculate.
Birne Binegar: minus first our procedure using reduction.
Birne Binegar: wants to form the matrix a.
Birne Binegar: I joined with the identity matrix real reduce to reduce row Echelon form.
Birne Binegar: And so suppose that reduce relational on form as another augmented matrix where I have.
Birne Binegar: A prime.
Birne Binegar: A join with iPod so a prime it refers to whatever happened to the left hand side after we reduce this matrix all the way to reduce show Echelon form.
Birne Binegar: And the way the algorithm work this a prime is identical to the identity matrix then we knew the icon would be exactly the inverse of a and if a prime at not reduced to the identity matrix then we know, there was no inverse but, as we saw off, I guess, even already on the first exam.
Birne Binegar: It often requires extensive row reduction to do this calculation and row reaction, while simple enough to implement is also pretty inner prone.
Birne Binegar: If you walked by the demonstrations on the blackboard it's really easy to make errors, but what i'm going to present to you now is an alternative way of calculating the inverse of the matrix using only determinants.
Birne Binegar: The way this goes is i'm going to create.
Birne Binegar: can be given matrix a i'm going to attach.
Birne Binegar: A matrix that's formed from these.
Birne Binegar: What we call co factors this these co factors appear in already in the formula for the determinant in terms of those co factor expansions so recall.
Birne Binegar: That.
Birne Binegar: That the determinant.
Birne Binegar: of a lot of n by n matrix can be expressed as a some say from jd equals one and of.
Birne Binegar: What is it I J entry times assign factor I plus J and then we have the determine our jewelry matrix and I J where this matrix.
Birne Binegar: Is the.
Birne Binegar: I J.
Birne Binegar: Minor.
Birne Binegar: up a this matrix here, which is the matrix you obtain from a by removing it's aight calm and J.
Birne Binegar: And jake type I throw in jail column these numbers minus one to the I plus J times the determinant of these corresponding minor.
Birne Binegar: And IDA This is called the alternative like the I jake co-factor.
Birne Binegar: All right now there's another formula for the determinant.
Birne Binegar: I mindful review real quickly.
Birne Binegar: And this initial formula what I did is I fixed the row index I, and then I some overall the column index, but as we've been doing the same thing as.
Birne Binegar: Get of fixing a row index we fixed a column index and we some over the road indexes so this same thing is a J minus one to the I plus J time.
Birne Binegar: That he terminal.
Birne Binegar: Am I J.
Birne Binegar: But notice in these two equivalent formulas, the only difference between them is in one we fix the row index and some over the columns This is called an.
Birne Binegar: Co factor expansion along the ICT row, and then the other formula we fixed the column the column index and we sum over the row indexes, and so this would be the Co factor expansion along the.
Birne Binegar: Along the jake car, but what appears in both formulas in identical in an identical fashion, are these factors minus one to some of the rolling column indexes times the determinant of the corresponding miners.
Birne Binegar: The factors here are called co factors and what you get.
Birne Binegar: From this formula is a number that you can attach to any entry up a matrix a you're basically determined by.
Birne Binegar: The position of an entry inside the matrix so you choose a row index that column index you get a corresponding number so for each entry have a we have a particular, we have a particular number.
Birne Binegar: And all we do to perform the co-factor matrix is we take these numbers computed in this fashion and we reinterpret these indices as well indices of the matrix and call that matrix see so.
Birne Binegar: The matrix a.
Birne Binegar: From that matrix a you can compute a total of n squared different numbers you'll get one.
Birne Binegar: One of these numbers for each choice of I and J and those.
Birne Binegar: are running between one and end so you have a lot of numbers to compute, but if you have a fast way of computing determinants these numbers will be pretty simple to compute.
Birne Binegar: You can then user associated row and column index to arrange these guys in a matrix and then the statement I want is that I can get the.
Birne Binegar: The entries of the inverse of the matrix a simply by first of all computing to the terminal a and then using these numbers but here's The important thing to note, this is not a typo.
Birne Binegar: The I the entry in the I thrown jade column at the minus first is a scale or factor, they determine is just basically a number that here the row and column indexes are reversed now because of this.
Birne Binegar: The textbook and web assign states this theorem a little bit differently here i've sort of expressing everything in terms of these computer numbers, but the textbook takes them a slower approach.
Birne Binegar: We still you still form the same matrix see using those co factors, where the entries of the co packer matrix CIA is here, we get right here, see subscript iga is exactly.
Birne Binegar: This you can still make a matrix out of it, but what i'll do is i'll take the transpose of the matrix see that I create from that that means converting the Rose into the call.
Birne Binegar: i'll call the transpose of the matrix I agree out of these numbers i'll call that the junk matrix educate today.
Birne Binegar: And now we have.
Birne Binegar: I don't have to write things in terms of components, the inverse of a is a scale or factor one over the current a times exactly the transpose that this matrix see or the so called aggregate of a.
Birne Binegar: All right now, let me use this to show you a real Nice formula that's.
Birne Binegar: Actually, you can use quite often suppose I have a.
Birne Binegar: General.
Birne Binegar: Two by two matrix so a B, C D, you can think of these as arbitrary numbers i'm gonna.
Birne Binegar: use.
Birne Binegar: This co-factor.
Birne Binegar: This co-factor matrix to figure out what a the minus first is let's get that formula up on the blackboard again, so we can refer to it.
Birne Binegar: i'm gonna write it this way, a to the minus first is equal one over every terminal a time.
Birne Binegar: This co-factor matrix.
transpose.
Birne Binegar: Where see I J.
Birne Binegar: This matrix here before we take the term transpose is just minus one to the I play J times the determinant of the matrix I get from a bye week moving it's aight rowan call right.
Birne Binegar: Now let's apply.
Birne Binegar: This formula to this two by two matrix.
Birne Binegar: and
Birne Binegar: Okay.
Birne Binegar: So here's a matrix a B, C D.
Birne Binegar: let's start computing these co factors Okay, so the first co factor in the first slot and first row first column i'm just replacing the jays with the numbers one and then.
Birne Binegar: That minor you get that by here's your entry you delete the row in column that contains it since we started with the two by two matrix we have just a single entry D in that in that minor.
Birne Binegar: And so there's that they're not applying this formula here's our first element of the co factor matrix second element.
Birne Binegar: See one two elements refers to this one to slot of a.
Birne Binegar: The single element see is the minor the one two minor of bay and their first time factor similar ones are computed like so alright so.
Birne Binegar: After i've computed all these numbers, I can arrange them in a matrix the one one slot I had be in the one to slot I have my MC etc.
Birne Binegar: Now this matrix it's not quite the one we want what we really need is the transpose of this matrix so what we'll do is we'll take that first row and will write it as the first call take the second row right it as a second column that gives us the social matrix adequate to a.
Birne Binegar: And now we have the ingredients of the formula.
Birne Binegar: Except we haven't computed the.
Birne Binegar: The determinant of a yet.
Birne Binegar: But that's pretty simple determinant of a it's just gonna be a D minus DC so.
Birne Binegar: What you get is.
Birne Binegar: A nice clothes formula for the inverse of a two by two matrix.
Birne Binegar: Now what you can do instead of doing row reduction to calculate the determinant of a two by two matrix you can instead plug the entries into this formula and their bike calculate determine.
Birne Binegar: So let me show you how quickly that goes.
Birne Binegar: suppose I had a equal one to.
Birne Binegar: One one and done.
Birne Binegar: The terminus this.
Birne Binegar: hey it's going to equal not.
Birne Binegar: Well let's.
Birne Binegar: let's just say this means a is equal to one B is equal to to see is equal to one is equal to one and let's just apply this formula directly.
Birne Binegar: At the minus first it's going to equal.
Birne Binegar: One over the determinant a one over a D minus BC so that would be one over one times one.
Birne Binegar: minus B times C is to show that's going to be a factor of negative one.
Birne Binegar: And then.
Birne Binegar: That.
Birne Binegar: As you get of a dot matrix turns out to be.
Birne Binegar: minus one.
Birne Binegar: Sorry D is one minus one minus two minus one one.
Birne Binegar: or simply.
Birne Binegar: let's see this will be an overall factor minus one so B minus one to one minus one.
Birne Binegar: So instead of using rubber reduction to calculate the inverse of this matrix you get a nice clothes formula we're all you have to do is know the values of the entry, you can plug in this formula and directly you get the inverse of the matrix.
Birne Binegar: Alright, so that works.
Birne Binegar: Really quickly and.
Birne Binegar: As long as you don't have to use row reduction to calculate determinants, then you can completely avoid production, the soul both how to solve linear systems and how to fall a fine inverse of the matrix ease, but in addition.
Birne Binegar: It tells you something about the structure of bringing in this case the structure of the inverse of the matrix.
Birne Binegar: Now what you see is.
Birne Binegar: Basically what's going to happen now let's move back to the formula here and the original formula.
Birne Binegar: Now let's hear it is this is the one I like to use because they see more clearly what the what the structure of the inverse of the matrix is so here i'm just listing the.
Birne Binegar: Typical entry if jake entry in this matrix don't How does that depend on the coefficients or the entries at the original matrix well.
Birne Binegar: These guys are just essentially determines the returns of minors, so these guys are going to be polynomials in the entries of a because this is just.
Birne Binegar: This matrix am is created from a using the same entries of a we just removed in a row in a call, but this is just going to be this is going to come.
Birne Binegar: it's just entirely of a subset of the entries of a and when I take a determinant i'm just going to get a polynomial and the entries of this matrix which are.
Birne Binegar: Also entries of a so this factor here, it may be complicated but it's still a polynomial function, which is a lot better than, say, an exponential function assign function, or some or rational function that has singularities etc it's a very simple function, the complication the places where.
Birne Binegar: The inverse is not necessarily a polynomial function of the entries of the original matrix you have this one factor here, where you have to go one over the terminal pay this guy is also a polynomial and the entries of a but it's in the denominator, so, in fact.
Birne Binegar: The inverse of the matrix is, in fact.
Birne Binegar: Each of its entries is irrational function a ratio of two polynomials consisting of the original matrix a lot there's one circumstance.
Birne Binegar: That makes things particularly easy if you specialize two matrices who determinant is equal to one, then the inverse is a pure polynomial, and that makes.
Birne Binegar: Its analysis really simple because polynomials are linear functions are very easy, the next simplest function you consider might be a quadratic function or a cubic function.
Birne Binegar: etc they're relatively simple functions me easy to evaluate easy to differentiate easy to integrate etc, and so.
Birne Binegar: Hopefully, what you can glean from this if you ever had to, for example, integrate the entries other of a matrix well, the fact that they could could be polynomial functions, but make life very easy so frequently.
Birne Binegar: that's one of the main formal applications of this co factor expansion formula for the determine.
Birne Binegar: Okay, so.
Birne Binegar: What I want to do.
Birne Binegar: To finish this discussion yeah.
Birne Binegar: Actually, what I.
Birne Binegar: What i've gotten to is I calculated.
Birne Binegar: Basically, taking up a matrix with variable entries and i'm able to express it in burst in terms of the.
Birne Binegar: The variable entries of the original matrix and this next slide I just confirm that, in fact, I really do this matrix here computed by.
Birne Binegar: The co factor expand our co factor formula for the inverse of the matrix like a that it really is an inverse.
Birne Binegar: And you just do the matrix multiplication and what will happen is the off diagonal terms, end up canceling so you get zeros and then the entries along the diagonal or exactly cancelled like this scale or factor, a corresponding to the terminal pay.
Birne Binegar: Finish today's lecture i'm going to do.
Birne Binegar: Two more examples, but now not dealing with simple two by two matrix ease, but more may slightly larger three by three systems.
Birne Binegar: Okay, so here's a three by three linear system.
Birne Binegar: Instead of leaving my gerbils X one X two X three in this example, yes, use.
Birne Binegar: X, Y and Z SP as the underlying variables.
Birne Binegar: So we still have we can still grab the coefficient matrix just think of X, is the first variable, why is the second variable Z is the third variable grab the coefficient range him in a matrix and grab the numbers on the right hand side, create a column vector from that.
Birne Binegar: Now to solve the system i'm going to use kramer's rule, so what we have to do to apply that rule is first of all, we have to compute the determinant of a.
Birne Binegar: Well let's look at that.
Birne Binegar: Maybe we'll use a Co factor expansion along the third row to calculate determine if a.
Birne Binegar: To do that, we would have zero times the determinant of a minor that we get through ignore because it's being multiplied by zero, then we have the entry in the 131 slot.
Birne Binegar: So we'll pick up a minus sign, because this entry, if you some each row in column index it's gonna be it's gonna be.
Birne Binegar: A cheese third row first call second column it's going to be five but minus one raised to an odd powers mine is.
Birne Binegar: Negative one raised to an odd power of negative one, so that's The purpose of that minus sign it's coming from the fact that when you add the row index and column index of the corresponding this position.
Birne Binegar: You get an odd number and you should get a minus sign there and then for this entry here.
Birne Binegar: The miners obtain.
Birne Binegar: Like eliminate the entry we're looking at we eliminate the rowan column that contain that guy and we're left with the matrix 1110 so that's the term that we have to keep you.
Birne Binegar: But we also have an additional turman that third row we have this one in the three three slot so we're gonna have one we're gonna have a factor of.
Birne Binegar: Sorry.
Birne Binegar: We have gonna have a factor of minus one to the three plus three this here had a factor of minus one put the plus sign there, but we had a minus one to the.
Birne Binegar: three plus two because this entry was in the third row second call and that minus one to the odd power lead to the minus sign this minus one to an even power basically produces the factor one, and then we have that determinant to compute.
Birne Binegar: So it worked out to be for.
Birne Binegar: me to get.
Birne Binegar: To apply Kramer tool.
Birne Binegar: The components of the solution.
Birne Binegar: Are determinant given by determinants divided by determines where the ice component of the solution vector is going to be the determinant of a matrix I create from a by replacing it I call them with that vector be.
Birne Binegar: The specter be I use that to replace a columns of a here they are.
Birne Binegar: A up.
Birne Binegar: hey is one.
Birne Binegar: minus 111200101011 and B is equal to.
Birne Binegar: 602.
Birne Binegar: All right, let me keep that data up So you see what happened, the next slide.
Birne Binegar: I need to create these major sees the sub scripts here is going to tell you which call me replace So when I replaced the first column of a with this vector I get this matrix.
Birne Binegar: replace the second column of a with the same vector I get that and will replace the third column, with a with a vector be that appeared on the right hand side original in your system, I get that.
Birne Binegar: But let's go ahead and compute some determinants this is good practice for us so i'll step through this.
Birne Binegar: race and a little bit.
Birne Binegar: Alright, so let's look at this matrix will be a good way to compute the determinant of be one.
Birne Binegar: I would suggest, using a co-factor expansion along the second row, because it means i'm going to have only one or jewelry determined to compute.
Birne Binegar: So let's do that, we have zero and that minus sign, what did I took the minus sign here, because this.
Birne Binegar: Entry here is in the two one physician two plus one adds up to an odd number and that's what we take up the minus sign, then we have.
Birne Binegar: To that's explains that factor, we have a science factor, but this is in the two to slot so the corresponding sign factor of B minus one race, the power, for which we just be one, so I can ignore that.
Birne Binegar: And then I have the corresponding minor, so I eliminate the second column second row and i'm left with 6121 all right so there's the expression for the determinant of this matrix all we really need to do is calculate the determinant of this two by two matrix so that six minus two.
Birne Binegar: Six minus two, which would be for but or times two is equal to eight.
Birne Binegar: All right, and similarly.
Birne Binegar: The second matrix be do.
Birne Binegar: Here we again have.
Birne Binegar: A special row in the matrix that is going to make the Co factor formula for the determinant of be too easy to compute but just expand along the second row, we only have one.
Birne Binegar: determinant to compute until we get assigned factor.
Birne Binegar: This isn't the two months slot that two plus one adds up to three so that you get the some of the rolling column indexes and odd numbers, we pick up a minus sign.
Birne Binegar: Then we have the entry of the two in that to one slot and then we have the determinant of the minor we get by eliminating the ro n calm that contains this entry so we're left with 61126166121 and last one.
Birne Binegar: let's see in the last of these are jewelry matrices here we don't really have.
Birne Binegar: A special row that makes it trivial, I think the best I can do is do a Co factor expansion, on the second row or maybe along the third row, but because I have to non zero entries in those roads i'm gonna have to.
Birne Binegar: Unfortunately i'm gonna have to compete to determinants so here, I think it looks like what I did is I expanded along the third row, so we got zero for the first contribution.
Birne Binegar: This one is in a.
Birne Binegar: The sum of its column and row index it's five and so we'll pick up a minus sign, then we have the entry and then.
Birne Binegar: determine if the corresponding minor and then this last term corresponds to the Co factor associated with this entry here at co factor is essentially.
Birne Binegar: minus one to the three plus three, which is going to be a factor one times the determinant the minor, and then the multiply by two because that's the corresponding entry we get 12 okay good so.
Birne Binegar: relatively easy to set up the matrix ease or three by three matrix ease they're determined or not, that hard to compute and now we get the solution to the system by.
Birne Binegar: The first variable which we call X its value is going to be determine if the matrix we got by replacing.
Birne Binegar: The first column of a live the vector be on the right hand side of different of the linear system plug in the numbers, you get to similarly.
Birne Binegar: Why determinant the second auxiliary makers divide by determine a is going to equal one minus one and Z and the third variable that's going to be.
Birne Binegar: determined at that third matrix divided by determinant a which is for and so very quickly you get numerical answers just by applying that code back.
Birne Binegar: The Kramer rule for.
Birne Binegar: All right, so.
Birne Binegar: In fact.
Birne Binegar: When a computer a modern computer algebra system it's actually solving square linear systems in the background, the criminal rules, the most expedient way to do that and so.
Birne Binegar: If you just using mathematics or maple or something like this, you ask it that's all the linear system if it's a square linear system that's usually applying kramer's rule in the background, the Catholic and things because it has the formulas any respectable.
Birne Binegar: Mathematical software program is going to have the the the formula for the determinant of the matrix Egypt arbitrary size zeki can compute those really rapid.
Birne Binegar: Alright i'm going to do another example I showed you how to use the Co factor expressions for the inverse of the matrix to calculate the D, the inverse of a general two by two matrix.
Birne Binegar: And this example i'm going to start with the explicit three by three matrix or no instead of having variable entries I have particular numbers, but what we'll do in this examples will calculate the aggregate matrix and then we'll use that aggregate matrix to.
Birne Binegar: provide us with the inverse of the original nature today, so what you have to do.
Birne Binegar: Is begin, first of all, you have to begin computing all the code factors for this matrix so corresponding to the one one slot you have minus.
Birne Binegar: The co factor is minus one race of the some of the rolling call them the next, and then you have the terminal the corresponding minor minor is right there I you eliminate the row and column continue to one one entry you're left with that two by two matrix.
Birne Binegar: And when you calculate the determinant that two by two matrix multiply by sign you get minus one Similarly, the one to entry is readily calculated.
Birne Binegar: And the one three entry is readily calculated now.
Birne Binegar: The formula, I want to use is this.
Birne Binegar: And I thought I need i'm so sorry it's not that.
Birne Binegar: formula, I want to use is that.
Birne Binegar: The inverse of a is equal to one over that determine okay times.
Birne Binegar: See ads matrix obey.
Birne Binegar: Let me write it.
Birne Binegar: See transpose or C.
Birne Binegar: Yes.
Birne Binegar: And I J slot is minus one the I plus J times the determinant of the corresponding mind.
Birne Binegar: that's The co factor alright, so what i'm going to need in order to compute that inverse is i'm gonna.
Birne Binegar: need all these co factors to create this matrix but i'm also going to need a determinant of a but remember the terminal a can also be calculated using co factors.
Birne Binegar: So at this stage of the game why, although I haven't computed all the Co factors of my tricks i've only computed the ones that live in the first row I can anyway use those co factors to compute at.
Birne Binegar: The determinant of a.
Birne Binegar: let's do that real quickly in general, the determinant of a is I can pick a row let's pick the first row, and then I move along the first row by changing the column index and I pick up all these.
Birne Binegar: factors here, but each one of these.
Birne Binegar: it's just to see one J, so one of the co factors that we just computed on the preceding slide.
Birne Binegar: And so what we get.
Birne Binegar: is perfect, so this week we've got the entries of the original matrix those we can read off the.
Birne Binegar: off its definition and then.
Birne Binegar: We use the code factors that we just completed on the preceding slide and so.
Birne Binegar: We get one piece.
Birne Binegar: Of the formula already, but we still have.
Birne Binegar: To create the rest of this co-factor matrix so i'll have to compete The co factors for the second row in the third row.
Birne Binegar: So here's all those complications go there, go exactly the same as we did before, to one co factor is there's the entry I eliminate it.
Birne Binegar: And I need to take to determine this minor that's the equivalent to minor and there's a sign factor, corresponding to this to one position you can go ahead and quickly calculate those things as well.
Birne Binegar: and similarly.
Birne Binegar: Along the third row you have co factors as well, each computed in exactly the same way you pick an entry you get assigned factor and you get a determinant of a matrix obtained by eliminating the row and calm that you're looking at.
Birne Binegar: And so.
Birne Binegar: Once you've got all these numbers notice that all they're all indexed by two numbers, so you can think of this as a row index and the column index, and you can arrange those in a matrix.
Birne Binegar: All right now.
Birne Binegar: you've got the matrix see computed the formula we need requires you to use the transpose of that matrix so.
Birne Binegar: here's this matrix see I created now i'll take it transposed so i'll take this first column alright, so the first rub i'll take the second column right is the second room.
Birne Binegar: The third column writer that's the third row and now i've got everything computed add matrix as you get to a and the terminal a have now been computed and.
Birne Binegar: That determine it was just a factor of two and so when you multiply.
Birne Binegar: This matrix you divide it by two you get exactly that matrix and you're done.
Birne Binegar: Okay, so.
Birne Binegar: that's about as far as i'd hoped to get today, so I think i'll just stop here you'll get practice in applying kramer's rule and these.
Birne Binegar: calculating these ads to get makes your season there by calculating the inverse of matrices that those calculations are covered and problem set eight so do those and then.
Birne Binegar: What we'll do starting on Friday and it's going to occupy us for a couple weeks is the eigenvector I can down your problem.
Birne Binegar: And so i'll pick up with those topics on Friday.