 Are there any, so I don't know, I know at least one person tried to watch the video for the last time and noticed that there wasn't a lot of math in it because I had a little problem with the camera. So, you're welcome to watch it. It's a bunch of guys, it's two guys playing. Any questions or things that I should address before I go on? No? Everybody's good with everything? So, there's a homework assignment that's due Wednesday and you shouldn't have problems with it. You should collect them in an algorithm that has recitation, which is upstairs here if you weren't here last week. Okay, so let's pick up roughly where we were. So, this might be our last, that's not right. We were talking about if I have an matrix A, B, a vector V, two four. Then the product I get by taking this times this plus this times that, 16 is some number, I guess it's 10, 22. And more generally, this is just abstracting the idea of writing linear equations as products of the coefficients and the variables as a column vector. And then we can generalize if we have two matrices A and B and their product I think of taking the matrix A just here and thinking of B, so B consists of thinking of it as column vectors B1, B2, up to B, I don't know, K. And I just do this for each of the column vectors. So, and I get the answer. So that would mean that if I did something like 1, 0, 2, 4, 1, 3 times the matrix 1, 2, 3, 4, 5, 6, 0, 1, 0. And this would be, I take 1, 0, 2 and dot it with 1, 4, 0. So that would just be 1 plus 0 plus 0 here. And then similarly here, it'll be 1, so it'll be 2 plus 0 plus 2 here. And then here it'll be 3 plus 0 plus 0. And so I guess I might go just do the other one, right? 4 plus 4 plus 0, 8 plus 5 plus 3, 12 plus 6. It'll be that resulting matrix, whatever the heck it is, right? Any confusion on this? Everybody's good with this? So, notice that if we think of this guy as generalizing 3x plus 2y and 5x minus 4y is the same thing as saying 3, 2, 5, 8, 4 times x, y, then here we're doing a similar thing. We're saying here one of these, none of those and two of those is whatever the heck this is, this is, and so on, right? We're just taking linear, this is just saying linear combinations of these three vectors, of those three vectors as describing for me some kind of equations. So this is really, so a matrix is just a short way of writing linear combinations and this stuff is going to be like equations like this. I am writing 3x plus 2y without writing the x's and the y's. Now, linear combinations of stuff is another way of writing a function. The variables are linear, they have power 1 and the coefficients are numbers. It's a linear function on higher dimensions, higher is how many rows of columns tell me the number of variables in the number of variables count. Does this make sense? It's just stuff. But this is important because we're, I mean maybe you don't see this at this point but this will keep coming back in this place. We think of really this matrix not necessarily as being this calculational object but it really represents a transformation. This is telling me a way to combine three numbers to get to. The way of telling me that I take x plus no y's plus 2z's and that gives me my new x and 4x's plus 2y's plus 1y plus 3z's gives me my new y. So this is really a function from three, this matrix represents a kind of a function from three variables into two variables. Put that aside for a little while. Okay, so we know how to multiply matrices together. We know how to scale them by just multiplying everything by a constant. So we can do stuff with it. So for example, if I write a polynomial, so since I can multiply so for square matrices, n by n matrix, it makes sense saying a square is just a times a. And a cubed is just a times a times a. That's all. So I can talk about the square or the cube or the tenth power of a matrix. And similarly, 3a is either a plus a plus a, which is the same thing as the matrix consisting of three times each of the... Yeah? If you have like a cubed, that's going from left to right. This is not going to be a cube. Well, if you're multiplying a matrix, so one should check that the order doesn't matter here because it's the same a. So one has to check that this is a times a squared, which is the same as a squared times a. You have to check this, but it's true. So a matrix always commutes with its own powers. So one has to check that, but it's true. But if I'm doing a, b, that's not necessarily the same as b. And b are the same, but it's true. So this makes sense if I want to talk about 3 times the matrix 1, 2, 3, 4. And this is just the matrix 3, 6, 9, 12. So I can talk about 3a as well. And last time we also talked about a special matrix 0, or O, which is the matrix consisting of all zeros. It's aij where aij is always 0. And then we have the identity matrix, which is the matrix, again aij with the property that aij is 0 if i is not j and aij is 1 if i is j. Which is the matrix that has ones down the diagonals and zero everywhere else. If the row and column are the same, then it's 1. And if the row and column are different, it's 0. And last time, I believe last time, or maybe it's just obvious to see that if I take a times the identity matrix, I get a back. Because when I dot the first entry with the first column, the answer is the first column. When I dot the second entry, this guy with the second column, I get no first columns, the entire second column, and none of the other columns and so on. So if I take the product a times a, I get a back, and also this is the same as i times a. And similarly, a times the zero matrix is the zero matrix, which is zero times a. But also a plus the zero matrix is a. So now I want to specialize the square matrices. And if I, so something like 3a squared plus 2a minus the identity, this is something we can calculate. If I had a matrix, we can calculate it square. We can multiply it by 3. You can take twice it and add it on there. And we can subtract. So this makes sense. I don't know if it's any good, but I mean I do know. But it makes sense. So in general, if we have some polynomial, we can actually calculate, even though this doesn't seem to make sense, p of a matrix. We can define this to be whatever happens when I replace a by that matrix and do the calculation. So for any square matrix, the polynomial applied to that matrix makes sense. What it means, I don't know, but I'm not telling you. But it's a thing. We can do it. Yeah? You add like 3a squared plus 2x plus 7. Yes. So if I put a 7 here, that would be a problem on this. So we replace the 1 here with the identity. The generalization is the powers of x's become powers of a. And we think of this as x to the 0, which is the identity matrix. Yeah? Can we do cross-multiply? Say again? Can we do cross-multiply? What does cross-multiply mean in this context? We can foil it out. Is that what you mean? We can factor it? Yeah. So maybe we can factor it. We can. Certainly if I want to think of a plus the identity squared, I can do this just like a polynomial, where this will be a squared plus 2a plus the identity. So that, sure. And I can certainly make sense of a plus the identity and a minus the identity, and see that that is in fact a squared minus the identity. So those kinds of things work. But something like a plus the identity over a necessarily makes sense. I can multiply, but I haven't made sense of division. And in fact division won't always work. It will only sometimes work. Right? Just like you can't divide by 0, there are certain matrices that we can't divide by. So that's something we have to work on. Is whether we can make sense of doing division. So here I can say a plus i times a minus i equals a squared minus i. But maybe a squared minus i over a plus i isn't defined. Well maybe it is. Because I don't know what divide means. Right? So divide was easy to define. But we never defined dividing. So that's something. Does that exist in the matrix at all? Is there a matrix in it? Sure. For some of the matrices, but not for all of the matrices. So I want to put that aside for a minute, but I'm going to talk about it in a couple of minutes. So, and then here, so I want to, I can just stay here. So we can do that. Another thing that is not so useful in this class, but is extremely useful in differential equations, is not just for polynomials, but we can think of infinite series. So for example, we know that e to the x is 1 plus x plus x squared over 2 plus x cubed over 3 factorial plus that infinite series, which is the sum of x to the n over n factorial from n equals 0 to infinity. And we can define, if it makes sense, so this is actually the limit as big n goes to infinity of the sum i equals 0 to infinity of x to the little n over n factorial. So this is a limit of polynomials, a higher and higher degree, as we let the degree go to infinity, added up in this way. So I claim that this makes sense. For a matrix, I can take e to the matrix power, which I mean, certainly this is 1 plus a 1 half a squared plus 1 6 a cubed. Certainly, if I stop this somewhere, it's a well-defined thing. If this limit exists, which is maybe a little work to show, I'm not going to do it. This is not a 1, this is a 5. Yeah. Well, a has to be, well, here this is 0 because a is a constant, but I could think of a matrix which has variables in the entries. And I can make sense of, I'm not going to just now, but we can make sense of the derivative. But there's lots of derivatives. Because, I mean, certainly I could make sense of, and I'll just stay over here for a minute, I could define, for example, a of x is 2x squared 3 fourth. I could make that function. And then I could try and make sense of d dx of a of x. And not surprisingly, well, what should this be if this is going to be something? This should be the limit as h goes to 0 of a of x plus h minus a of x divided by the standard h. Well, this is going to be fine. This is going to be, well, all of these guys are just going to add to each other in just nice ways, divided by h. Nothing fancy there. So this is just going to be 4x1 4xq. So that makes sense. What we haven't covered is if I have x, y, c, w, and I want to talk about the derivative of that, with respect to what? What's my derivative? So I have lots of derivatives here. So it can get more complicated if we have more variables. And we haven't addressed that yet, but we will talk about that too. So let's certainly think of, this is really a function from the reals into the set of 2 by 2 matrices. And this function, it makes sense to push it a little bit and ask how each of the components in the matrix change when I move x a little bit, and this is the way they change by this matrix, 4x0 1 4 xq. It tells me how, when I move x, how much is it going to change? Well, it's going to be 4 times 0.06. Nothing in that one. One in that one. And, well, time is 0.06. So it's going to tell me how that's going to, the infinitesimal change of this matrix during the x away. It's a mess. It's not always a mess. Sometimes it's easy. But usually it's a mess. So let me just start computing what I'll go down here. So this is, I'm just pointing out that you can do these things and they actually mean something. So if I take the matrix, what, should I do the identity? That one's easy. So if a is the identity matrix, then the identity matrix plus the identity matrix plus what? This is 1 plus x x squared over 2. The next one is 1 half times the identity matrix square 1 over 3 factorial times the identity matrix cubed, and so on. But these are all identity matrixes, identity. So this is going to be the identity matrix. I can factor, I'll just leave them alone for a minute, which is 1 plus 1 plus 1 half plus 3 factorial times the identity matrix, which is just e, and the identity matrix, which is the matrix with e's down the diagonal. Well, that was easy. So it's just e times the identity matrix. That one was easy. But if I change this a little bit, that the powers do something funny, then I will get a different answer. So don't worry too much about e to a matrix power at this point. This is really important in differential equations. In differential equations, a lot of times the way you solve the linear equation, the solution turns out to be e to a matrix power. Yeah. So do you still hold the i pi plus 1 equals 0? Well, I'm not quite sure what a matrix pi means. I mean, to matrixify i pi maybe maybe. I can tell you that e to the i where i is the square root of negative 1 times pi times the identity is just a matrix with minus 1's down the diagonal. Sure. But that's stupid. It's an equation. That's just as stupid as this one. Saying e to the identity is all e's down. We didn't learn anything there. But there might be some way to make some sense of that. We could make sense of, for example, the sign of s i and e of a matrix. Again, using this kind of thing. I have no idea what that means. It's just a function. Here, the exponential has a nice relationship to derivatives. And so, that tells us something about differential equations. But what is the relevance of the sine function here? Well, it's fourth derivative is itself. And the second derivative is itself with the sine change. But, you know, so what? So, maybe? I just wanted to throw this out here. You do encounter e to a matrix power in other contexts. So, since we can do it, we'll do it. If you take 308, this will come back with a vengeance. We won't really use it much now. Or, for that matter, if you take even 303, it'll come back. Okay. So, I wanted to say some more about division stuff. So, again, I'm just focusing on square matrices so that I can multiply and get powers and so on. So, if you think about, you know, what is one third mean? So, it's the number so that three times x is one. This is, in fact, the number back to, what about fact fractions? Fourth grade? Something like that. Remember back to elementary school, people kind of knew what one half was and they kind of knew what maybe a third or even an eighth is. But if you talk about, you know, five-ninths, then fuses them. And you could say, okay, I got a thing and I chop it up into nine equal pieces and then I take five of them and have five-ninths. Okay, that makes sense. But it's another way of saying that if I take nine of these things together I'm not going to and add them up, then I'll have five objects. Right, so this is one way of thinking about it, but it's maybe better to think that nine times five-ninths equals five. That is, it solves nine x equals five. It's the unique real number so that nine x equals five. And we have a similar kind of relationship here except our multiplication can get a little more complicated. So for example, if we wanted to find if I'm given matrices well, let's just worry about the identity matrix. So I'm given some matrix A and I want to solve find some matrix B so that A B is the identity. Maybe I want B A. It doesn't matter. So that B A is the identity then I'm going to call this guy the inverse. B is A to the minus one power. I mean here I'm thinking that it's one over A but that doesn't necessarily make sense. But only, though for example if I take the matrix zero, zero, zero, zero it doesn't have an inverse or one, zero, zero, zero that has no inverse. So I'm going to find another matrix so I can't solve find some matrix A B C D equals one, zero, zero, one just not going to work. Because that would mean that A has to equal one but also zero has to equal one. If I take this product I will get A zero, zero, zero I have to have A zero, zero, zero equals one, zero, zero and that's kind of a problem. And of course I can work up more less obvious matrices that I don't have inverses for. Maybe not. But if I take a matrix like I don't know one, two three, seven then since I already did this so if I take that and I multiply it by seven, negative two, negative three, one then this will be the identity let's just check. So seven minus six is one what am I doing? Negative two plus two is zero 21 minus 21 is zero and negative six plus seven is one so I'm happy. This guy has an inverse this guy doesn't they're different and so we kind of need some way to figure out when a guy has when a matrix has an inverse and when it doesn't and of course we can just measure a matrix like one, two, three, four and I want to find its inverse well actually let's just check if A, B is the identity and I multiply this matrix this equation on both sides but on the right by the inverse matrix A then that tells me since this equation has to hold in other words B is A inverse so that's okay but now I can multiply on both sides on the whatever side that is the right by A in other words that means that B, A is also the identity so if A, B is the identity then B, A is also the identity in other words it doesn't matter the right or the left so a matrix and its inverse will commute in the sense of a matrix and its inverse and we don't care whether I want to solve this this equation or the other equation A inverse to the identity which is always the same as A now if the matrices aren't square then well we have generalized inverses right inverse and left inverse but if it's invertible it doesn't matter which order we do it so one way that we could find an inverse for one way that we could try and find an inverse so if you give me some matrix let's just try one 1, 2, 3, 4 and I want to find A, B, C, D so that equals equals 1, 1, 0, 0 well I can just write down the equations right this is the same thing I'm saying that A plus is 1 plus 4B is 0 C plus 3D is 0 I've lost track 2C plus 4D 4 equations and 4 unknowns so I can solve this for A, B, C and D ok but also we already saw that we can solve equations by adding multiples of rows to each other and doing all that sort of thing so I could another way that I could do this is I could start wait a minute another way I could do this is I could just say ok I'm going to solve the system where I want this to be 1, 0 but I also want it to be 0, 1 in the second column right I did this before when I was just solving equals 1 column vector but I can solve it equals 2 column vectors at the same time and so I could just start doing row reductions to transform this guy into something that looks like the identity matrix is it clear to people that this is ok that if I just start row reducing this thing that will tell me what A is well that will tell me this will tell me my answers in terms of A, B, C, D do you have a question? ok what does that line mean? it's just if you prefer you can write them side by side and just keep track of whatever you do here so I just put the line in the middle remember I'm going to mess on with this guy but everything I do with this guy I'm going to do that guy I'm going to do the same collection of row operations to both matrices at the same time in such a way that this one becomes the identity and this one is whatever it is so let me do that so I'm going to take 3 times this row and subtract it from this one so if I take 3 times this row let's just do that I'm going to take 3 times that and subtract it from that so this stays the same but this guy becomes a 0 and then this is a negative 6 that I'm adding to a 4 so that gives me a negative 2 and then I have to remember that that's what I did so I have a negative 3 here is that clear what I did? I just took 3 times let's just label this R1 and R2 and so this is label R1 and this is minus 3 times this is R2 minus 3 times R1 a really naive question but the 1001 on the right side was that just chosen as an example? no I want to solve that I want to see how to transform this guy so he becomes the identity matrix so I'm trying to find the inverse which is another way of saying I'm just rewriting this set of equations so that this is 1, 2, 3, 4 ok so let me let me say this again let me write it the other way so that it matches 1, 2, 3, 4 so I'm some matrix A, B, C, D I'm really thinking of this as a pair of column vectors and I'm saying I want to know find A, B, C and D that satisfies the equation this dot product is 1 this dot product is 0 this dot product is 0 and this dot product is 1 in other words it's the same as solving the equation 1, 2 x, y equals A, C 1, 0 and also 1, 2, 3, 4 B, D equals 0, 1 because that's what I want to do because I want it so that the product is the identity because I'm trying to find A times A inverse so that the product is the identity oh and I'm just thinking of that equation and solve that equation I want you to do it at the same time because the process is going to be the same so we're specifically trying to find the inverse yes, exactly if I wanted to find it to be something else I could do this with something else over there but once I have the inverse then I don't need to work anymore because now I don't need anything but specifically I'm looking for the inverse but I've just broken it up and it's going to be something else and we already saw I mean I had x's and y's here when we did it before we already saw that to solve something like this we take the augmented matrix 1, 2, 1, 3, 4, 0 and we mess with it to get rid of variables we mess with this to turn this into 1, 0, 0, 1 and whatever I get here I'm starting to do the same process here I'm just doing it with two rows at a time so I'm really solving this one this one at the same time but since whatever I do to turn this into 1, 0 is going to be the same for both of them when I just do them both at the same time I mean you're nodding but you're nodding like if you say so what is the second column what is that what is the second column so you're trying a 0 and on one side and on the other both ways so I want to solve the pair of equations I want to solve this one and I want to solve this one but the way I can solve them together is just doing them both again we're going to try and pull out of this the identity matrix over here and magically my answer will appear over here now there's a formula you can just probably already know it but it's good to know where these formulas come from but I've lost track of what I've done so I have that and I don't like that minus 2 so now I'm going to divide everything by minus 2 so this will become 1 actually I shouldn't just change it 1, 2, 0, 1 1, 0 minus 2 so this is 3 halves and this is minus 1 half and now it's easy to kill this 2 so that'll become just subtract twice this from that so 1, 0, 0, 1 so twice this is 3 so that's a negative 2 twice this is so it's 1 because I was correct did I make a mistake? 3 halves so my inverse matrix should be that so it should be true that this matrix and this matrix is the identity I guess I should check it I don't know what I'm going to do I messed up the board so I could check that 1, 2, 3, 4 negative 2, 1 3 halves I probably just need something else because I always do negative 2 plus 3 is in fact 1 and 1 minus 1 is in fact 0 and negative 6 plus 6 is in fact 0 and 3 minus 2 is in fact 1 so it even worked I'm really proud of doing this I mean this works and it's fine I have to look at it because I always forget where it goes so in my notes I was going to do this with a general matrix ABCD but ED and we do the augment all of these will have an AD minus BC so you start with this and you do all of the tedious stuff to get to the identity matrix here this is what you'll get don't memorize this I mean you feel free memorize whatever you want you'll get something like this maybe I messed it up already but something like this there's a formula the important here is to realize any time you transpose my life D negative B negative C A so those signs are wrong for a 2 by 2 it's pretty easy you just swap these two and you change those signs and that's the inverse and then you divide by the determinant you swap the diagonal elements here you close along the change of the signs and there you go you want to memorize that if you don't that's also fine so we can invert a 2 by 2 matrix easily and this process that I gave over there works just fine with a 12 by 12 matrix as well just is a long trick to do that there's something else I need to say you know sometimes you won't be able to solve this equation I want to solve the equation matrix times vector equals the zero matrix then sometimes I'll get many solutions and sometimes I will get only the zero solution and so without this this is the same suppose A so if then this equation is still true if I multiply both sides by A inverse on the right but that's the same as saying that this vector is zero this vector x or matrix x is the zero A inverse is the identity so I'm left with x and A inverse times zero is the zero matrix so A is invertible then the only solution to this is the zero vector and it works the other way too so in other words is invertible and only if the only solution I only proved one way with to Ax zero is x being the zero vector but we talked about that a little bit too this is the same as saying the columns linearly independent which is also the same thing the rows are the same so in other words the matrix A consists of a bunch of linearly independent vectors well if you think about this statement this is the same as saying when I mix and match the rows it can eventually arrive at the identity because what linearly independent means I'm taking combinations together I see some confused faces you can actually say what does that mean? somebody want to say that? the only solution I didn't want to write the whole word the only solution to this equation is the zero vector but that's the same as saying the columns are linearly independent vectors the columns of A and then this is the same I didn't prove this but the same as saying the rows so all of these things are the same which this is the same I ran out of room the same as saying that I can solve well I can reduce to the identity by elementary operations I can do this jump that I was doing over there they're all the same so invertible matrices are exactly the same as matrices where the rows and columns can't be you can't kill off one of the rows by adding multiples of the other ones and for 2 by 2s all the same stuff and so we also just already saw except they didn't do the work so for a 2 by 2 matrix that's the same so a 2 by 2 matrix A, B, C, D that's the same as saying that A D minus B C is minus 0 because if A D minus B C were 0 I would have to divide by 0 in the form for the inverse so one of the answers would have to be infinity now I can arrive at this equation by just manipulating and trying to solve where I can arrive at it in other ways but let's say I do this it's there at my nose I just don't want to do it because I'll screw it up for sure so this is an important thing it's no accident so you've probably seen this before and this guy is the determinant I'm not a determinant yet but this is the determinant of the matrix A, B, C, D which we just take this product and subtract that from that product now a slightly we can generalize this to higher dimensions so this is also the area if I take the vector A, C and the vector B, D it's the area of the parallelogram between them if this is the vector then this area except somehow my picture if I put them in the right order but we can crank up this notion of determinant to higher dimensions to deal with 3x3s or 4x4s or 5x5s in an inducted way so if I have and I did this in an earlier class and the standard notation actually is you write the matrix with straight lines instead of curved lines and this means the determinant and if you write straight lines on the side of the matrix it means take a determinant of this absolute value which is the size of okay? so if I wanted to do a higher determinant like this I'd have to define it and so this is actually can be defined of sub matrices so I'm going to take ABC, DEF, GHJ I wrote J so you don't think it's the square root of negative one it's going to be A times the matrix A times the determinant EF HJ plus B minus B times the determinant DEH in other words I take I pick an element and I cross out his row and column and I take the determinant of what's left over except that this one should have been green oh well green means minus here if I cross out his row and column I subtract the green and then I add and I do the same here and I add that back so I alternate plus minus plus plus minus and I can expand by any row and any column so one needs to so this will be A times EJ FH minus B times DJ minus FG plus C DH minus BG and in general we can define this inductively degree matrix any size we can play the same game so that if I have 0 2 1 3 4 I'm just going to write the first step what I do let's just start with the top row although it would be easier to use the bottom is I take 1 times the determinant 4 5 6 2 3 4 0 1 0 subtract off 0 so I don't even have to do that but I will write it minus A times the determinant 3 5 6 1 3 4 0 1 0 plus 2 times the determinant 3 4 6 1 2 4 0 0 0 so that would be 0 minus 1 times the determinant 3 4 5 1 2 3 0 0 1 pick a row, picking the top row you could also pick a column and expand along that getting the little sub matrices so now to do this I have to do it again right so doing a 10 by 10 can be a long process there are more efficient ways right so here so like this one is 0 this one is easy 0 and this one is A times so let me just point out rather than prove it now I can actually expand by any row in any column so this one is 0 because I choose to expand by this bottom row which is 0 times that plus 0 times that that was easy this one I can expand by this row because that would be fairly easy too so this will be I have to figure out whether it's plus or minus so it's plus minus plus minus so it's going to be minus so this pattern starting from the corner I go plus and I alternate pluses and minuses or another way of saying it is if the sum of the R and J yeah if I plus J is odd it's minus and if I plus J is even it's plus yeah so why is that pattern a thing why is that pattern a thing though it has to do with the fact that the wedge product is anti... that doesn't help let's just accept it for now I'm trying to think of a good reason that I can does anyone know it's because the wedge product is anti-symmetric but that doesn't help me what's the wedge product it's because the cross product is anti-symmetric but is that what we're wondering about yeah well I'm not doing a cross product here but it's related to the cross product it's the same process it's actually very clever yeah so let's just say cause a damn I don't have a good answer I should have a good answer and so that would be so this guy this gives me a minus so that was already a minus A there and then I take here the determinant 3, 4, 6, 1 which is going to be 3 minus 24 so huh so I take the row where except I wrote the wrong thing 3, 1, 6, 4 3, 1, 6, 4 that makes more sense yeah okay so 12 minus 6 which is 6 so this will be 6A here and here I can do the same thing here to get the determinant 4, 2, 6, 4 which will be 16 minus 12 which is F4 and here this is 0 and here this will be minus the determinant 3, 4, 1, 2 plus traffic plus so which is 6 minus 4 which is 2 and there's a minus so you can do that sort of thing and I'm writing along with Convary so let me just tell you some stuff about the determinants so one thing that you can check there's a proof in the book so if I if I so here's a fact which one can calculate and check the multiple to another or a column to another or a column it does not change the determinant almost row reduction except that remember in row reduction you often will not just add them but you'll scale them and here I can't scale because the scale will definitely change the determinant A is the same as I can multiply any row by r so multiplying a row or a column by a number r changes the determinant by that factor you said keep it in mind so you could row reuse it but of course if you could just row if you could do the whole row reduction process then every determinant is either 1 or 0 so that's kind of stupid so we have to keep track that when you're solving and putting this here in a second you often do this scaling by r we have to keep track of those scales and the other thing is if you interchange two adjacent rows you change the sum interchange if I exchange this with that well I'm stupid if I exchange this with that you can add a row to a column that doesn't mean you can add a row to a column no you can add a column to a column or a row to a row add a row to a column so you can check all of these facts I think they're done in the book we don't really have enough time to do them now but you can do this stuff and so that helps a lot in calculating the determinants because you can sort of make it convenient to have a row of all 0's and 1's by or at least some 0's and 1's and make your life a little easier when calculating things of course computers are really good at calculating determinants so you can just type it into maple and there it comes or Mathematica or any of those kind of guys and it just falls out okay what else do you want to say here so you should also notice that the determinant of the inverse is 1 over if you have a matrix A and you know it's determinant this is the same as 1 over the determinant of A inverse the determinant of A inverse that sort of just falls out from those observations and if I multiply two things together this is determinant A the determinant is multiplicative in this way here we can add to this which is not well it is sort of obvious here because the rows and columns are linearly independent if they're linearly dependent that means I can mush together rows and columns to get 1 being all 0's which means the determinant will be 0 so this statement is the same that the determinant of A is not 0 and this is a really critical observation this is often a lot of times how you check whether a system has a unique solution or not you take the determinant and if the determinant is not 0 then you know there's a solution and if the determinant is 0 then there's not going to be well there's going to be yeah because these are numbers so this is equal to determinant of A which has to be because these are real numbers and real numbers alright so and go through it so I think I won't so there's a thing there's a formula called Kramer's Rule that invert matrices relating determinants and stuffy stuffy stuff feel free to know Kramer's Rule it's just a little thing that you know you can work out and I can never remember it and you know so it expresses the inverse of a matrix in terms of the determinant and sub matrices and all sorts of complicated but about the end of the section so we have these tools of linear algebra and the reason for covering these tools of linear algebra is to be able to talk about matrices and indices and vectors and things like that and I'm going to return starting next time to doing multivariable calculus because the derivative from Rn to Rm or a vector if one of the things is one dimensional so when we when we think about functions from one set of variables to another set of variables the derivative is a linear map that approximates that function and so derivatives will be matrices because matrices are general linear maps from Rn Rn to Rm so we have to deal with those kinds of things so the reason for this little excursion here into linear algebra that some of which you may have seen in high school is precisely so we have the tools to talk about the calculus we need to do so in some sense this course is multivariable calculus with linear algebra rather than they're not really quite on the equal part except we can't do multivariable calculus very well unless we know the linear algebra the other the other version of this course down plays the linear algebra one but anyway there it is this is probably a good place to stop when you can all go upstairs see what's another our model