Birne Binegar: Oh pretty good how you doing.
Birne Binegar: yeah I hope this is gonna go real smoothly i've got a practice exam i'm going to step you through what I want you to observe, of course, his my strategy which I approached Roberto that's most the exam home or work those out.
Birne Binegar: explicitly but one problem that people have sometimes with that first exam if they do row reduction but they don't have a consistent strategy, so they go around in circles.
Birne Binegar: And they never get to reduce relational and form it cetera so watch what I do.
Birne Binegar: What you'll see is that everything you have plenty of room on the exam, even though it looks like you don't have much room but it's just.
Birne Binegar: you'll have plenty of time, I think you can, if you knew what you're doing you could do the exam and 20 minutes so take 40 minutes to check your work as you're working along the other thing I want to point out is that i'm.
Birne Binegar: Sorry let's see this is.
Birne Binegar: Let me, let me bring this up here.
laptop.
Birne Binegar: Think six.
Birne Binegar: And um what what all you'll see as we step through the exam typically.
Birne Binegar: A big part of the exam is going to be solving linear system and that that takes several steps you have to.
Birne Binegar: win the augmented matrix you have to reduce relational and form, you have to further reduce it to reduce your Ashkelon porn.
Birne Binegar: And then you have to write the solution as a as a hyper plane well instead of giving you one problem with parts A, B and C I broken up, and in the for problems.
Birne Binegar: So that a mistake made in the first part doesn't carry over to a mistake made my second part because that's the one weakness of reduction is it's easy to make a small, you know, Miss assign or.
Birne Binegar: incorrectly add two numbers together, and then it screws up everything else that you do, subsequently, so I broken the reduction problems where you're solving linear system into.
Birne Binegar: Several independent problems it's just that I start you midway through the actual calculation in them all, but the first one.
Birne Binegar: yeah I mean i'll so i'll show you explicitly will go through this So the first problem problem three is, you have a linear system you to write down and augmented matrix and Rover to use it.
Birne Binegar: And for you, given them a matrix in row so on form, but it needs to be reduced to reduce your actual on forums and then in a third problem you're giving a matrix and reduce relational on form and you need to write down that fiber plane so.
Birne Binegar: yeah I like to break things up.
Birne Binegar: But um and.
Birne Binegar: In some sense and examine maybe for problems, except that I changed the matrix every time I.
Birne Binegar: start a new step anyway let's step through this.
Birne Binegar: till the first part of the course we introduced vectors and and matrices etc this this exam is mostly calculation, so the theory is not going to appear to explicitly.
Birne Binegar: Maybe a little bit in the second problem but let's do the first problem first all we need to do is multiply together to make your CS so what you have to know in order to do that is that.
Birne Binegar: If you have a in this case, we have a matrix with three rows so I get a B and C on exam so B and C, so we have a matrix that has three rows and two columns.
Birne Binegar: And it's multiplying a matrix with.
Birne Binegar: Two rows and three columns so before we even start calculating.
Birne Binegar: We noticed, we have the match that we need for that.
Birne Binegar: product to be defined, and that these outer dimensions are going to give us the result.
Birne Binegar: The result, the dimension to the matrix that can come from this product and we're know we're going to get a three by three matrix out of this product it's defined because the inner dimensions match and the outer dimensions we're going to give you.
Birne Binegar: The size of the eventual matrix what we use is the following formula to get the entry in the I throw and jade column of the product, we take the dot product at the ice row with the first factor.
Birne Binegar: With the jade column of the second factor.
Birne Binegar: and
Birne Binegar: For the given matrix we can do that, let me do it explicitly on the paper.
Birne Binegar: Here we go 102 minus 131.
Birne Binegar: And I have 1010 minus one one okay so what's my fingertips and that's how you can rapidly compute these.
Birne Binegar: dot products of rows of the first factor and call them to the second factor still i'm going to need one.
Birne Binegar: In this one one slot up in this corner i'll take the first row and first column and i'll take their product so that will be one times one plus zero times zero is one.
Birne Binegar: Then i'm going to have, I have a second column, I can work with, so I have one time zero Plus zero times, one is zero.
Birne Binegar: And finally, I have one times one plus zero times, one is one.
Birne Binegar: Alright, so that's give give me the first row of the product now i'm gonna you know I got all these elements, by using just the first row with the first factor.
Birne Binegar: Now I moved to the second row, and I do the same thing one two times one plus minus one time zero it's just going to be to two times zero Plus minus one minus one is one two times one plus minus one times one let's see two.
Birne Binegar: Oh sorry see it's written incorrectly here, this should have been zero there.
Birne Binegar: till I have two times one plus 02 and then the third row three times one plus zero is three.
Birne Binegar: zero times three.
Birne Binegar: minus one times, one is minus one.
Birne Binegar: Like not can't really do that, I think, maybe I can do is turn off the lights here, maybe that will help.
Birne Binegar: That a little better.
Birne Binegar: yeah.
Birne Binegar: try out of focus.
Birne Binegar: see if that straight into that a little bit.
Birne Binegar: looks better on the computer monitor, I hope that yep or the problem is my Pen, I think.
Birne Binegar: i'll do that last entry in the last one would be three times one plus one time zero or simply three right, so there it is three minus one one.
Birne Binegar: All right, okay so i'll work with this done all right.
Birne Binegar: So yeah matrix multiplication should be really easy Oh, by the way.
Birne Binegar: notice at the top of the exam no calculators I want you to work out all these problems by hand there are calculators out there even smartphones but which you can.
Birne Binegar: Put in the entries of the matrix and they'll tell you everything you want to know about that matrix it's real Echelon form, whether it's in verses etc.
Birne Binegar: I don't want you to do that because part of the point my point on emphasizing row reduction and we'll see that in the next problem is your ability to the interpret interpret the calculation as you're working it through.
Birne Binegar: So let me give you an example that with the second problem.
Birne Binegar: I give you three major sees it happened to be in relational on form and what I want you to do is tell me about the linear system from which these.
Birne Binegar: augmented matrix these came from, and to describe the solution set without doing any row reduction you just have to look at those things and and be able to detect.
Birne Binegar: What kind of linear system, it came from with number of equations and a number of variables, you ought to be able to tell whether or not this system even has a solution by looking at these matrices.
Birne Binegar: And you ought to be able to tell actually the dimension of the solution, set by looking at other major sees.
Birne Binegar: The way you do that let's look at problem be first in problem be.
Birne Binegar: it's got three rows so that means you we had three equations in this really and.
Birne Binegar: The left hand side of these augmented matrix corresponds to the coefficient matrix each column here corresponds to a particular variable be like X one X two X three x four.
Birne Binegar: So what I want you to look at and in this second example is that the equation correspond to the third row is zero times X one zero times X two zero times X three plus zero times X four is equal to negative one.
Birne Binegar: which basically that last row was telling me that zero is equal to minus one which is a contradiction it's not true that zero is equal to one and so.
Birne Binegar: Because one of the equations corresponding to this augmented matrix least to a contradiction there's not going to be any solution to the corresponding linear system, so the full answer to Part B would be three equations.
Birne Binegar: For four variables.
Birne Binegar: But no solution.
Birne Binegar: And you don't have to consider this third third item here because.
Birne Binegar: hypothesis if the literary system does have a solution, then I want to know the number of free variables, but there's no solution, so you can ignore that third question for Part B.
Birne Binegar: Now the other examples of the matrices you don't have that phenomenon anytime you have a pivot in the last column of an augmented matrix.
Birne Binegar: you're going to end up from from the equation that says basically zero is equal to some non zero number that's when you know you don't have a solution as long as you don't have a row like this, then you will have a solution.
Birne Binegar: All right, so let's look at now parts A and B they do have solutions because you don't have this phenomenon with a pivot in the last call.
Birne Binegar: So let in Part A, I have four rows so that's four equations.
Birne Binegar: On the left hand side, I have four columns so it was four equations and four variables.
Birne Binegar: Now, how many.
Birne Binegar: Solutions do we have well, it does have a solution.
Birne Binegar: Because this phenomena is not occurring, and then to get the free parameters what you do is you count.
Birne Binegar: The number of columns that do not have pivots.
Birne Binegar: The pivot columns contouring can corresponding to where pivots live they correspond the variables that you can solve for and the columns without pivots and here's only the third column here that don't have the pivot.
Birne Binegar: that's a pivot that to pivot that to pivot so it's just this third column that's without a pivot that means 1321 column.
Birne Binegar: Without a pivot.
Birne Binegar: means one free variable in the solution.
Birne Binegar: Alright well that's.
Birne Binegar: Part A, and part see it looks like we have.
Birne Binegar: an audience, even though this is a zero row I want you to imagine that this came from.
Birne Binegar: A system of.
Birne Binegar: equations and three unknowns but somehow during the course of row reduction zero appeared will still think of this augmented matrix corresponding to for equations because there for roads and three variables.
Birne Binegar: For equations.
Birne Binegar: In three variables.
Birne Binegar: And then it does have a solution.
Birne Binegar: Because this phenomenon here does not occur and, in fact, if you look on the left hand side.
Birne Binegar: Every column on the left has a pivot.
Birne Binegar: So there are no columns.
Birne Binegar: Without it.
Birne Binegar: And that means no free variables.
Birne Binegar: Which is not the same thing as saying there's no solution actually is saying that there's a unique solution that you don't have any degrees of freedom that you can adjust to get more solutions.
Birne Binegar: Okay, so you should probably you'll probably see a problem, like that, on the exam and that's how quickly you can solve it.
Birne Binegar: All right now we're actually getting the nuts and bolts of doing the calculation it's necessary to solve a linear system will do it in three steps.
Birne Binegar: In problem three I give you a set of linear equations and your to write down the augmented matrix and then reduce it.
Birne Binegar: to at least row Echelon form.
Birne Binegar: Right so let's do that first equation there's nothing multiplying X one.
Birne Binegar: i'll put a zero here minus one times X to two times X three and then on the right hand side of the equation, the number one.
Birne Binegar: Second equation coefficient matrix is 211.
Birne Binegar: On the right hand side we have zero and then minus 1113.
Birne Binegar: All right, that's half the problem they're just writing down that matrix now what we have to do is reduce it to row Echelon form.
Birne Binegar: that's getting it in the form.
Birne Binegar: Where.
Birne Binegar: The entries.
Birne Binegar: occur in sort of the zeros concentrated in this lower left hand corner, but the thing we're looking at actually mostly as these pivots we want them to occur in a downward staircase.
Birne Binegar: When we've got the matrix in row Echelon form and there's a particular way that we going to take our given matrix and put it in this form, we have to use elementary row operations.
Birne Binegar: So these are.
Birne Binegar: For example, interchanging the order of two weeks two rows or what we can do is, we can take a row and rip and replace it by a multiple of itself.
Birne Binegar: or the third loud operation is to replace a row by some.
Birne Binegar: For multiple have another row.
Birne Binegar: Just using these three operations, we want to convert this matrix into row Echelon form all right now here's a pivot and there's a entry below a pivot but.
Birne Binegar: more of a problem if we don't have the pivot up in this upper right hand corner that's where we want to start.
Birne Binegar: So the first thing i'm going to do is maybe take this third row interchange it with the first row, so I get a pivot up in this corner that will give me something I can work with.
Birne Binegar: So what we'll do is we'll take row one interchange it.
Birne Binegar: With row three.
Birne Binegar: now get minus 1113.
Birne Binegar: nothing's happening with the second row.
Birne Binegar: And third row first row is going down to the bottom.
Birne Binegar: One okay.
Birne Binegar: All right now.
Birne Binegar: i've had someone Rochelle unfortunate at least the second and third rows of the pattern, but here we have a pivot sitting directly below a pivot i'm going to want to clear out that guy.
Birne Binegar: and
Birne Binegar: I can do that easily by sending.
Birne Binegar: row to to row to two plus two times row one i'm trying to get i'm trying to use this element in row one to clear out this entry so that's why I multiply by two and then i'll have.
Birne Binegar: let's see first row, if not changing 1113 laphroaig not changing in this operation two one okay now we're going to.
Birne Binegar: add two times row one, two, the second row so minus two plus two is 02 plus one is three.
Birne Binegar: Two plus one is three and three plus zero is three.
Birne Binegar: Okay well and now.
Birne Binegar: let's simplify this a little bit let's send row 321 third of sorry row two one third row two.
Birne Binegar: minus 11130111 and zero minus one to one.
Birne Binegar: That just to make the second, the next step, a little bit easier, the real problem I have written the reason why this matrix is not in in row Echelon form is because I have a pivot below a pivot I want to clear out this guy but i'm going to do so without introducing fractions.
Birne Binegar: Ah yeah it should be shouldn't because i'm adding three to three yeah so that should have been sift.
Birne Binegar: Be six and then that would make this too.
Birne Binegar: Alright, so the last thing i'm gonna do is i'm going to send the third row to row three plus row two.
Birne Binegar: minus 11130112 i'm just rewriting the first two rows but here's where the action is going to go on 01 plus or minus one is 01 plus two is three and two plus one is three all right now i'm stopping here.
Birne Binegar: i'm only we asked to go to row Echelon form and i've got a matrix now where the pivots are in that Nice descending staircase and we're done alright so.
Birne Binegar: that's the first stage of production and having covered that in problem for what I do is I move on to the next stage of reduction, I give you a matrix it's already.
Birne Binegar: In row Echelon form.
Birne Binegar: Underlying the pivot here and the goal now is to.
Birne Binegar: transform this matrix into reduce your salon.
Birne Binegar: So now our strategy is different, instead of trying to clear out entries below pivot we worry about the entries above that and we worry about the value of the pivots themselves, they should be normalized to be equal to one.
Birne Binegar: So let's take care of that second problem first or second row one to one half rho one that will convert that two into one will send row two to one third row to that will change that 321 and in the third row what we'll do is we'll.
Birne Binegar: divide.
Birne Binegar: By minus two.
Birne Binegar: Then we get a new matrix which 11231100122100001 minus one and the last row nothing ever goes on there's darted off all zeros it's going to stay on zero.
Birne Binegar: All right, so good now i've got a matrix that's enroll Echelon form.
Birne Binegar: And the pivots are all equal to one last thing to do is to clear out these entries that are like these non zero entries it sit above tibbetts and lower rows so I need to get rid of that too, because pivot below it and the one in the to have to be removed, using the third row.
Birne Binegar: But we start from the bottom now and we'll send them, for example.
Birne Binegar: Real one that replace it by row one minus wrote three when i'm trying to do is get rid of this guy up here, so all subtract the third row from it to get that to cancel.
Birne Binegar: and similarly row to it's going to go to row to I have to hear so i'm going to need negative two times this guy to get that cancellation minus two.
Birne Binegar: row threes and we'll get 1123 I can write down those first four entries easily because.
Birne Binegar: As I add three multiples of three to these things, these first two four entries are not going to have any effect, but then here's what we will have an effect will get a 01 minus 101 minus negative one, is to.
Birne Binegar: write you subtracting the third row from the first so one minus negative one effectively one plus one, and then, when we do this, we get 0012 and then we're going to have to minus two is zero and one minus a negative two is going to produce three.
Birne Binegar: Last row it's not changing 00 sorry second natural one minus 1000000.
Birne Binegar: All right, last problems.
Birne Binegar: That.
Birne Binegar: entry in the first row third column is sitting above the pivot in the second row so we'll send row one two row on minus two times row to.
Birne Binegar: Then we'll get 110 let's see three minus four is negative one.
Birne Binegar: Zero minus two times zero is zero.
Birne Binegar: And two minus.
Birne Binegar: Two times three is going to be.
To.
Birne Binegar: Negative six it'll be minus four I guess.
Birne Binegar: And then we had 00120300001 minus 1000000 and then we're done here the pivots.
Birne Binegar: And recurring in the downward staircase they're all equal to one and sorry off screen.
Birne Binegar: and
Birne Binegar: Above and below pivots I have nothing but zeros So this is the answer to that problem.
Birne Binegar: Okay question so far.
Birne Binegar: Alright, so.
Birne Binegar: If we were solving linear system we're now at the final stage where we've got an augmented matrix and reduce your actual on form and the problem is going to be to.
Birne Binegar: write down the hyper planes expression for the solution well i'll start a new problems with the same basic principle at play here's a matrix in reduce racial on for, and I want you to take this matrix and tell me what the solution looks like as a hyper planes.
Birne Binegar: Alright, so.
Birne Binegar: let's first write down the equations corresponding this the augmented matrix in reducing national on form is the augmented matrix of the solution.
Birne Binegar: So when I go to the equation so i'll have zero times X one plus x two plus zero times X three minus two times X four plus x five.
Birne Binegar: is equal to one that's the first row needs to that equation second row is similar going to lead to X three plus x four minus X five is equal to two and the last two rows simply say zero is equal to zero and zero is equal to zero.
Birne Binegar: All right now.
Birne Binegar: These are supposed to be the equations of the solution, but I just had these guys look like another linear system.
Birne Binegar: It doesn't particularly look like the solution to anything.
Birne Binegar: Well what's missing here, and what will you recognizes the solution is you pick the variables that correspond the columns with pivot so that would be X two X three.
Birne Binegar: You think of these as a fixed variables of the of the solution, these are ones that you can express in terms of the remaining dairy both in each equation so X four X five.
Birne Binegar: are going to be the free variables in the solutions, and if that also X one don't forget about X one there's no coefficients there but it's it still was effectively part of the original linear system there's a column, for it there, so.
Birne Binegar: Aside from the equations that correspond to this augmented matrix there's also my interpretation of variables.
Birne Binegar: The columns that contain pivots that say X two and X three, these are going to be three variables or sorry, these are the fixed variables, because they correspond the columns with tidbits.
Birne Binegar: And the other variables X one for the first column X for for the fourth column X five for the fifth column that they don't have pivots in those columns So those are going to be three variables X one X four X five are free.
Birne Binegar: Right, so what we do the write down that hyper planes solution is first of all.
Birne Binegar: We try to see the solutions as vectors so what i'm going to do is i'm going to write down a generic vector but where I can i'm going to replace.
Birne Binegar: The fixed variables their expressions in terms of the free variables so here from the first equation, I see that X is equal to one plus two X four minus X i'm taking these guys to the right hand side of the equations.
Birne Binegar: So what this first equation says, is it X is equal to one plus two X four minus X five.
Birne Binegar: Sorry, and then second equation is saying X three is equal to two minus X four plus X, Y.
Birne Binegar: movie me spree variables, to the right hand side, I have to switch the science of the coefficient and now, when I do.
Birne Binegar: For the free variables i'll just write the corresponding symbol but for the fixed variable so i'll replace him by the formulas I get from solving the equations X two is one plus two X four minus X five X three is two minus X four plus x five X four is free, so I leave it alone.
Birne Binegar: Five is free I leave it alone.
Birne Binegar: And now I have an expression.
Birne Binegar: A vector expression for the solution of the original linear systems is not quite yet displaying it as a hyperlink.
Birne Binegar: To see that form.
Birne Binegar: What you do is you take this vector.
Birne Binegar: And you break it up.
Birne Binegar: There are some constant that appear in each of these components one in the second component to, and the third component no constant in the first fourth and fifth components.
Birne Binegar: And then we have terms that are proportional to the free variable X one.
Birne Binegar: Let me grab them from each component first component, I have a one.
Birne Binegar: And X one doesn't appear in any other components and then we have X for appearing let's see it does not occur in the first slot it occurs with coefficient to in the second slot.
Birne Binegar: coefficient minus one coefficient one and zero the last slot and finally we have X five the last three variable and its corresponding vector is zero minus one plus one zero doesn't occur in the fourth slot and one.
Birne Binegar: All right, and now i've got my hyper plane expression of the solution, this is your starting point, your so called P zero and then.
Birne Binegar: X one X four X five or free parameters in the solution, so they can be any number you want.
Birne Binegar: What you solution that you reach any point in the solution space by starting off at this with this vector and then heading arbitrary scale or multiples of these three doctor she just add them on.
Birne Binegar: so effectively my solutions space is now being displayed as a three dimensional hyper plane.
Birne Binegar: it's it's a vector or every solution.
Birne Binegar: it's gonna look like.
Birne Binegar: Anything that works here.
Birne Binegar: hyper planes.
Birne Binegar: are set to vectors.
Birne Binegar: Three of the three where we allow the parameters to one key to MP3 the very over the entire set of real numbers and what this does is it it generates a hyper plane yeah starting point and you're allowed to stray from that starting point in three distinct directions.
Birne Binegar: So i've got that this vector expressing that form this is P zero, this is the one B two and v3 and the three variables well they correspond to these.
Birne Binegar: parameters tier one tier two and three that are allowed to be any real number.
Birne Binegar: Okay, and so in those series a pre problems I have you sold linear linear systems.
Birne Binegar: All right, and then.
Birne Binegar: The last problem on the exam number six is to find the inverse of a three by three matrix so the way that's done is.
Birne Binegar: You form this augmented matrix where instead of.
Birne Binegar: Putting a call and after next to your coefficient matrix you actually put the identity matrix on the right hand side so i'll have 111221103 and two that I joined.
Birne Binegar: The three by three identity matrix.
Birne Binegar: Alright, so the strategy is going to be.
Birne Binegar: reduced to reduce row Echelon form.
Birne Binegar: And one of two things is going to happen.
Birne Binegar: either going to get.
Birne Binegar: This side gets converted into the identity matrix and if that happens.
Birne Binegar: The left, the right hand side it's going to be whatever happens to the right hand side if the left hand side go to identity matrix the right hand side automatically goes to the inverse and that's what you're trying to compute or you get something that's.
Birne Binegar: Not the identity matrix and it doesn't matter what happens here.
Birne Binegar: If you don't get the identity matrix on the left, when you converted this matrix to each unique reduce relational on form in this case you have no inverse.
Birne Binegar: Alright So what do we have to do well, we just have to reduce this matrix all the way to reduce your national on form stare at it and decide whether we have an inverse we don't have an inverse.
Birne Binegar: Alright, so.
Birne Binegar: let's see maybe the first thing i'll do is send row to.
Birne Binegar: row two minus two times row one and i'll work on the third row tubes and wrote three two row three minus row one.
Birne Binegar: What did I do those two operations, because I need to get rid of this entry below the pivot and this entry below that pivot in the first row so first thing i'll do is i'll start with the Tibetan the upper left hand corner and use it to clear out entries below it.
Birne Binegar: We do that.
Birne Binegar: let's see first row is not changing 111100 now the second row i'm going to subtract two times the first round let's do it in place two times two minus two is 02 minus two is 01 minus two is minus 101 minus zero is one.
Birne Binegar: Zero minus two is negative two and zero minus 00.
Birne Binegar: Third row we're going to subtract the first row so one minus one is 00 minus one negative one, three minus one is 200 minus one is minus one zero minus 001 minus zero is once.
Birne Binegar: Okay, so.
Birne Binegar: that's that clear out all the entries below this first pivot you look at where we're at now, and here we have a pivot alpha row and below it, we have a pivot off to the left.
Birne Binegar: Or the simplest thing to do here is just to interchange the order of the roads that will put a pivot up here and i'll move this tidbit down there.
Birne Binegar: So let's do that.
Birne Binegar: let's send.
Birne Binegar: row to interchange it riff row three and I get 111100 nothing happened there they don't have zero minus one two minus 10100 negative one one minus two zero good.
Birne Binegar: Now we're in row Echelon forms but we got to go all the way to reduce relational and form, so we have more work to do what we'll do next is will make all the pivots equal to one.
Birne Binegar: that's we can do, simply by sending row two to it's negative and wrote three gets sent to it's negative.
Birne Binegar: We end up with 11110001 minus 210 minus 1001 minus 120.
Birne Binegar: Good.
Birne Binegar: last step well it's going to take a couple calculations, but the last thing we need to do is clear out these non zero entries that are sitting above the pits.
Birne Binegar: So we can do that, we can clear out this guy by subtracting the third row from the first and we can clear out this guy by adding two times the third row to the second row.
Birne Binegar: So let's do that real one is going to go to row one minus row three.
Birne Binegar: row to it's going to go to row two plus two Times wrote three.
Birne Binegar: And what will get is 110 and let's see i'm subtracting the third row, so all get a.
Birne Binegar: Two minus a negative one is one minus negative one, is to zero minus two is negative, two zero minus zero is zero.
Birne Binegar: Now the second row will add two times the third row so zero there one two plus negative two is zero.
Birne Binegar: minus two plus one is.
Birne Binegar: one.
Birne Binegar: Oh sorry minus one I guess.
Birne Binegar: Two plus zero.
Birne Binegar: Today that's going to equal for.
Birne Binegar: Two times the third row, plus the second row and then we'll get minus one there and then 001 yeah.
Birne Binegar: Well, I may have.
These two variables.
Birne Binegar: Ah, OK.
Birne Binegar: let's see.
Birne Binegar: Let me.
Birne Binegar: Right, so I would have.
Birne Binegar: yeah to minus two.
Birne Binegar: Two minus 201 minus two.
Birne Binegar: is negative one, then I have.
Birne Binegar: Negative yeah I did I did I minus two and then one there one minus one minus two times zero sorry that should have been one so minus.
Birne Binegar: That went down here so.
Birne Binegar: minus one.
Birne Binegar: plus two.
Birne Binegar: shoot me shoot up.
Birne Binegar: yeah let's let's let's let's just fix it up on the blackboard and i'll work from the blackboard rasmus piece paper, by the way.
Birne Binegar: I hope you see that the exam is is really short and so these kind of mistakes that i'm making because i'm presenting it, rather than working out, you should just take your time and do the calculation slowly you'll have plenty of time.
Birne Binegar: So what we should have had ended up with.
Birne Binegar: Is.
Birne Binegar: To just throw them away let's see.
Birne Binegar: 111100 and then we had 00.
Birne Binegar: minus one.
Birne Binegar: minus 210000.
Birne Binegar: minus one two minus 101.
Birne Binegar: So that was not intermediate step but next thing to do would be to switch maybe 111100 let's get make the pivot equal to 112 minus 1001 minus 210 minus one now we'll switch these guys.
Birne Binegar: will get 11110001 minus 210 minus one and then 0012 minus one zero.
Birne Binegar: And then now we'll work on these guys.
Birne Binegar: Okay do that here.
Birne Binegar: i'll squeeze it in, I think.
Birne Binegar: One one and then i'm subtracting the third row from the first and then i'll get zero and then one minus two is negative one zero minus one is plus one and zero minus zero is zero.
Birne Binegar: And then here we're going to subtract will add two times the third row to the second row, and that will produce 01 and now zero and let's see four plus one is five.
Birne Binegar: Two times the third so this will be negative two.
Birne Binegar: And then zero Plus minus one minus one, and then we have 0010 sorry.
Birne Binegar: Two minus one zero all right, and then the last thing to get rid of is that guy there it's the last entry that that sitting in a column that's not a pivot that sitting in a column, with a ticket.
Birne Binegar: until the last thing we do is we send row one row one minus the second row and we'll end up with.
Birne Binegar: 100.
Birne Binegar: Then let's see i'm subtracting the second row from the first minus one minus five is negative six.
Birne Binegar: One minus a negative two is going to be three zero minus one it's going to be one.
Birne Binegar: somewhere.
Birne Binegar: Out of box there and then the rest of the matrix is unchanged live minus two minus 10012 minus one zero and now we've got the identity matrix here.
Birne Binegar: No problem is completed as soon as you point out that this right hand side is the inverse okay.
Birne Binegar: Now, if we had we had a zero here that would need their matrix does not have an inverse but to alignment we get the identity matrix on the left hand side, it means are in versus whatever occurred on the right hand sides.
Birne Binegar: Okay, well, I did make it to the exam.
Birne Binegar: The exam is can be really, really close to this practice exam so I understand how to do these calculations, in particular role reaction, this is a big part of the exam so just.
Birne Binegar: Just be able to do that not only efficiently, but also accurately and you'll do well on the exam I typically.
Birne Binegar: If I have a class it's doing their homework I typically get average scores on the first exam it's like 87 or 90% so you should.
Birne Binegar: This is the easiest exam and you should do well on it, but I make it easy because, in fact, these calculations this rolled up some calculation that's an underpinning the rest of the course.
Birne Binegar: yeah yeah it's gonna look it's going to change the numbers but it's gonna look at central like that.