WEBVTT 00:00:12.380 --> 00:00:16.800 Okay, this is linear algebra, lecture four. 00:00:16.800 --> 00:00:24.800 And, the first thing I have to do is something that was on the list for last time, but here 00:00:24.810 --> 00:00:26.169 it is now. 00:00:26.169 --> 00:00:29.369 What's the inverse of a product? 00:00:29.369 --> 00:00:36.300 If I multiply two matrices together and I know their inverses, how do I get the inverse 00:00:36.300 --> 00:00:39.559 of A times B? 00:00:39.559 --> 00:00:46.760 So I know what inverses mean for a single matrix A and for a matrix B. 00:00:46.760 --> 00:00:53.229 What matrix do I multiply by to get the identity if I have A B? 00:00:53.229 --> 00:00:57.100 Okay, that'll be simple but so basic. 00:00:57.100 --> 00:01:06.120 Then I'm going to use that to -- I will have a product of matrices and the product that 00:01:06.120 --> 00:01:13.659 we'll meet will be these elimination matrices and the net result of today's lectures is 00:01:13.660 --> 00:01:22.960 the big formula for elimination, so the net result of today's lecture is this great way 00:01:22.960 --> 00:01:27.700 to look at Gaussian elimination. 00:01:27.710 --> 00:01:31.400 We know that we get from A to U by elimination. 00:01:31.400 --> 00:01:38.060 We know the steps -- but now we get the right way to look at it, A equals L U. 00:01:38.060 --> 00:01:40.800 So that's the high point for today. 00:01:40.800 --> 00:01:41.800 Okay. 00:01:41.800 --> 00:01:47.320 Can I take the easy part, the first step first? 00:01:47.320 --> 00:01:54.560 So, suppose A is invertible -- and of course it's going to be a big question, when is the 00:01:54.560 --> 00:01:55.860 matrix invertible? 00:01:55.860 --> 00:02:04.920 But let's say A is invertible and B is invertible, then what matrix gives me the inverse of A B? 00:02:04.920 --> 00:02:07.460 So that's the direct question. 00:02:07.470 --> 00:02:08.869 What's the inverse of A B? 00:02:08.869 --> 00:02:12.120 Do I multiply those separate inverses? 00:02:12.120 --> 00:02:21.220 Yes. I multiply the two matrices A inverse and B inverse, but what order do I multiply? 00:02:21.230 --> 00:02:22.290 In reverse order. 00:02:22.290 --> 00:02:24.020 And you see why. 00:02:24.020 --> 00:02:29.780 So the right thing to put here is B inverse A inverse. 00:02:29.780 --> 00:02:31.820 That's the inverse I'm after. 00:02:31.820 --> 00:02:38.940 We can just check that A B times that matrix gives the identity. 00:02:38.950 --> 00:02:39.950 Okay. 00:02:39.950 --> 00:02:46.480 So why -- once again, it's this fact that I can move parentheses around. 00:02:46.480 --> 00:02:52.320 I can just erase them all and do the multiplications any way I want to. 00:02:52.329 --> 00:02:56.209 So what's the right multiplication to do first? 00:02:56.209 --> 00:02:58.220 B times B inverse. 00:02:58.220 --> 00:03:02.400 This product here I is the identity. 00:03:02.400 --> 00:03:08.100 Then A times the identity is the identity and then finally A times A inverse gives the 00:03:08.100 --> 00:03:10.060 identity. 00:03:10.060 --> 00:03:15.680 So forgive the dumb example in the book. 00:03:15.680 --> 00:03:21.180 Why do you, do the inverse things in reverse order? 00:03:21.189 --> 00:03:27.019 It's just like -- you take off your shoes, you take off your socks, then the good way 00:03:27.020 --> 00:03:35.060 to invert that process is socks back on first, then shoes. 00:03:35.060 --> 00:03:36.660 Sorry, okay. 00:03:36.660 --> 00:03:41.320 I'm sorry that's on the tape. 00:03:41.320 --> 00:03:46.460 And, of course, on the other side we should really just check -- on the other side I have 00:03:46.460 --> 00:03:47.480 B inverse, 00:03:47.480 --> 00:03:54.099 A inverse. That does multiply A B, and this time it's these guys that give the identity, 00:03:54.099 --> 00:03:58.040 squeeze down, they give the identity, we're in shape. 00:03:58.040 --> 00:04:01.700 Okay. So there's the inverse. 00:04:01.700 --> 00:04:08.480 Good. While we're at it, let me do a transpose, because the next lecture has got a lot to 00:04:08.480 --> 00:04:09.779 -- involves transposes. 00:04:09.780 --> 00:04:16.860 So how do I -- if I transpose a matrix, I'm talking about square, invertible matrices 00:04:16.860 --> 00:04:18.298 right now. 00:04:18.300 --> 00:04:23.500 If I transpose one, what's its inverse? 00:04:23.500 --> 00:04:27.640 Well, the nice formula is -- let's see. 00:04:27.640 --> 00:04:34.760 Let me start from A, A inverse equal the identity. 00:04:34.760 --> 00:04:39.780 And let me transpose both sides. 00:04:39.790 --> 00:04:43.199 That will bring a transpose into the picture. 00:04:43.199 --> 00:04:48.300 So if I transpose the identity matrix, what do I have? 00:04:48.300 --> 00:04:49.600 The identity, right? 00:04:49.600 --> 00:04:54.990 If I exchange rows and columns, the identity is a symmetric matrix. 00:04:54.990 --> 00:04:56.940 It doesn't know the difference. 00:04:56.940 --> 00:05:07.780 If I transpose these guys, that product, then again it turns out that I have to reverse 00:05:07.780 --> 00:05:09.160 the order. 00:05:09.160 --> 00:05:15.260 I can transpose them separately, but when I multiply, those transposes come in the opposite 00:05:15.260 --> 00:05:16.220 order. 00:05:16.220 --> 00:05:23.600 So it's A inverse transpose times A transpose giving the identity. 00:05:23.600 --> 00:05:26.700 So that's -- this equation is -- just comes directly from that 00:05:26.700 --> 00:05:32.690 one. But this equation tells me what I wanted to know, namely what is the inverse of this 00:05:32.690 --> 00:05:35.700 guy A transpose? 00:05:35.700 --> 00:05:43.180 What's the inverse of that -- if I transpose a matrix, what'ss the inverse of the result? 00:05:43.180 --> 00:05:47.000 And this equation tells me that here it is. 00:05:47.000 --> 00:05:56.659 This is the inverse of A transpose. 00:05:56.659 --> 00:06:02.220 Inverse of A transpose. 00:06:02.220 --> 00:06:08.100 Of A transpose. 00:06:08.100 --> 00:06:13.480 So I'll put a big circle around that, because that's the answer, that's the best answer 00:06:13.480 --> 00:06:16.560 we could hope for. 00:06:16.560 --> 00:06:24.640 That if you want to know the inverse of A transpose and you know the inverse of A, then 00:06:24.640 --> 00:06:26.640 you just transpose that. 00:06:26.640 --> 00:06:33.790 So in a -- to put it another way, transposing and inversing you can do in either order for 00:06:33.790 --> 00:06:35.390 a single matrix. 00:06:35.390 --> 00:06:42.770 Okay. So these are like basic facts that we can now use, all right -- so now I put it 00:06:42.770 --> 00:06:44.920 to use. 00:06:44.920 --> 00:06:52.980 I put it to use by thinking -- we're really completing, the subject of elimination. 00:06:52.980 --> 00:07:05.920 Actually, -- the thing about elimination is it's the right way to understand what the 00:07:05.930 --> 00:07:06.930 matrix has got. 00:07:06.930 --> 00:07:15.550 This A equal L U is the most basic factorization of a matrix. 00:07:15.550 --> 00:07:23.960 I always worry that you will think this course is all elimination. 00:07:23.960 --> 00:07:27.100 It's just row operations. 00:07:27.100 --> 00:07:29.420 And please don't. 00:07:29.420 --> 00:07:41.120 We'll be beyond that, but it's the right algebra to do first. 00:07:41.120 --> 00:07:48.630 Okay. So, now I'm coming near the end of it, but I want to get it in a decent form. 00:07:48.630 --> 00:07:52.270 So my decent form is matrix form. 00:07:52.270 --> 00:07:59.360 I have a matrix A, let's suppose it's a good matrix, I can do elimination, no row exchanges 00:07:59.360 --> 00:08:03.220 -- So no row exchanges for now. 00:08:03.220 --> 00:08:07.640 Pivots all fine, nothing zero in the pivot position. 00:08:07.640 --> 00:08:10.240 I get to the very end, which is U. 00:08:10.240 --> 00:08:12.000 So I get from A to U. 00:08:12.000 --> 00:08:15.220 And I want to know what's the connection? 00:08:15.220 --> 00:08:17.680 How is A related to U? 00:08:17.690 --> 00:08:21.991 And this is going to tell me that there's a matrix L that connects them. 00:08:21.991 --> 00:08:22.991 Okay. 00:08:23.000 --> 00:08:28.700 Can I do it for a two by two first? 00:08:28.700 --> 00:08:32.679 Okay. Two by two, elimination. 00:08:32.679 --> 00:08:37.179 Okay, so I'll do it under here. 00:08:37.186 --> 00:08:46.680 Okay. So let my matrix A be -- We'll keep it simple, say two and an eight, so we know 00:08:46.680 --> 00:08:54.029 that the first pivot is a two, and the multiplier's going to be a four and then let me put a one 00:08:54.029 --> 00:08:58.720 here and what number do I not want to put there? 00:08:58.720 --> 00:09:05.780 Four. I don't want a four there, because in that case, the second pivot would not -- we 00:09:05.780 --> 00:09:07.642 wouldn't have a second pivot. 00:09:07.642 --> 00:09:11.360 The matrix would be singular, general screw-up. Okay. 00:09:11.360 --> 00:09:15.090 So let me put some other number here like seven. 00:09:15.090 --> 00:09:16.580 Okay. 00:09:16.580 --> 00:09:24.680 Okay. Now I want to operate on that with my elementary matrix. 00:09:24.680 --> 00:09:28.340 So what's the elementary matrix? 00:09:28.340 --> 00:09:34.560 Strictly speaking, it's E21, because it's the guy that's going to produce a zero in 00:09:34.560 --> 00:09:35.560 that position. 00:09:35.560 --> 00:09:43.310 And it's going to produce U in one shot, because it's just a two by two matrix. 00:09:43.310 --> 00:09:50.150 So two one and I'm going to take four of those away from those, produce that zero and leave 00:09:50.150 --> 00:09:51.580 a three there. 00:09:51.580 --> 00:09:53.080 And that's U. 00:09:53.080 --> 00:09:55.140 And what's the matrix that did it? 00:09:55.140 --> 00:09:56.900 Quick review, then. 00:09:56.900 --> 00:10:03.060 What's the elimination elementary matrix E21 -- it's one zero, thanks. 00:10:03.060 --> 00:10:05.760 And -- negative four one, 00:10:05.760 --> 00:10:07.880 right. Good. 00:10:07.880 --> 00:10:14.100 Okay. So that -- you see the difference between this and what I'm shooting for. 00:10:14.100 --> 00:10:21.980 I'm shooting for A on one side and the other matrices on the other side of the equation. 00:10:21.990 --> 00:10:22.990 Okay. 00:10:22.990 --> 00:10:25.940 So I can do that right away. 00:10:25.940 --> 00:10:29.500 Now here's going to be my A equals L U. 00:10:29.500 --> 00:10:38.160 And you won't have any trouble telling me what -- so A is still two one eight seven. 00:10:38.160 --> 00:10:44.940 L is what you're going to tell me and U is still two one zero three. 00:10:44.940 --> 00:10:49.800 Okay. So what's L in this case? 00:10:49.800 --> 00:10:57.220 Well, first -- so how is L related to this E guy? 00:10:57.220 --> 00:11:02.650 It's the inverse, because I want to multiply through by the inverse of this, which will 00:11:02.650 --> 00:11:08.570 put the identity here, and the inverse will show up there and I'll call it L. 00:11:08.570 --> 00:11:11.360 So what is the inverse of this? 00:11:11.360 --> 00:11:17.640 Remember those elimination matrices are easy to invert. 00:11:17.650 --> 00:11:30.130 The inverse matrix for this one is 1 0 4 1, it has the plus sign because it adds back 00:11:30.130 --> 00:11:32.200 what this removes. 00:11:32.200 --> 00:11:39.860 Okay. Do you want -- if we did the numbers right, we must -- this should be correct. 00:11:39.860 --> 00:11:42.380 Okay. And of course it is. 00:11:42.390 --> 00:11:47.730 That says the first row's right, four times the first row plus the second row is eight 00:11:47.730 --> 00:11:49.700 seven. Good. Okay. 00:11:49.720 --> 00:11:51.920 That's simple, two by two. 00:11:51.920 --> 00:11:57.380 But it already shows the form that we're headed for. 00:11:57.390 --> 00:12:01.190 It shows -- so what's the L stand for? 00:12:01.190 --> 00:12:02.640 Why the letter L? 00:12:02.640 --> 00:12:09.860 If U stood for upper triangular, then of course L stands for lower triangular. 00:12:09.860 --> 00:12:16.920 And actually, it has ones on the diagonal, where this thing has the pivots on the diagonal. 00:12:16.920 --> 00:12:27.080 Oh, sometimes we may want to separate out the pivots, so can I just mention that sometimes 00:12:27.089 --> 00:12:34.339 we could also write this as -- we could have this one zero four one -- I'll just show you 00:12:34.340 --> 00:12:39.160 how I would divide out this matrix of pivots -- two three. 00:12:39.160 --> 00:12:41.880 There's a diagonal matrix. 00:12:41.880 --> 00:12:45.980 And I just -- whatever is left is here. 00:12:45.980 --> 00:12:49.620 Now what's left? 00:12:49.630 --> 00:12:57.140 If I divide this first row by two to pull out the two, then I have a one and a one half. 00:12:57.140 --> 00:13:02.580 And if I divide the second row by three to pull out the three, then I have a one. 00:13:02.580 --> 00:13:11.180 So if this is L U, this is maybe called L D or pivot U. 00:13:11.180 --> 00:13:21.100 And now it's a little more balanced, because we have ones on the diagonal here and here. 00:13:21.100 --> 00:13:22.920 And the diagonal matrix in the middle. 00:13:22.920 --> 00:13:24.380 So both of those... 00:13:24.380 --> 00:13:29.400 Matlab would produce either one. 00:13:29.400 --> 00:13:32.860 I'll basically stay with L U. 00:13:32.860 --> 00:13:39.960 Okay. Now I have to think about bigger than two by two. 00:13:39.960 --> 00:13:43.780 But right now, this was just like easy exercise. 00:13:43.780 --> 00:13:50.920 And, to tell the truth, this one was a minus sign and this one was a plus sign. 00:13:50.920 --> 00:13:54.740 I mean, that's the only difference. 00:13:54.740 --> 00:14:02.700 But, with three by three, there's a more significant difference. 00:14:02.700 --> 00:14:05.740 Let me show you how that works. 00:14:05.740 --> 00:14:17.760 Let me move up to a three by three, let's say some matrix A, okay? 00:14:17.760 --> 00:14:20.120 Let's imagine it's three by three. 00:14:20.130 --> 00:14:23.050 I won't write numbers down for now. 00:14:23.050 --> 00:14:29.351 So what's the first elimination step that I do, the first matrix I multiply it by, what 00:14:29.351 --> 00:14:30.870 letter will I use for 00:14:30.870 --> 00:14:41.490 that? It'll be E two one, because it's -- the first step will be to get a zero in that two 00:14:41.490 --> 00:14:43.520 one position. right? 00:14:43.520 --> 00:14:48.260 And then the next step will be to get a zero in the three one position. 00:14:48.260 --> 00:14:53.560 And the final step will be to get a zero in the three two 00:14:53.560 --> 00:14:57.089 That's what elimination is, and it produced U. position. 00:14:57.089 --> 00:15:06.820 And again, no row exchanges. 00:15:06.820 --> 00:15:13.180 I'm taking the nice case, now, the typical case, too -- when I don't have to do any row 00:15:13.180 --> 00:15:15.920 exchange, all I do is these elimination steps. 00:15:15.920 --> 00:15:16.900 Okay. 00:15:16.900 --> 00:15:22.880 Now, suppose I want that stuff over on the right-hand side, as I really do. 00:15:22.880 --> 00:15:25.470 That's, like, my point here. 00:15:25.470 --> 00:15:30.300 I can multiply these together to get a matrix E, but I want it over on the right. 00:15:30.300 --> 00:15:33.670 I want its inverse over there. 00:15:33.670 --> 00:15:39.160 So what's the right expression now? 00:15:39.160 --> 00:15:46.440 If I write A and U, what goes there? 00:15:46.440 --> 00:15:47.060 Okay. 00:15:47.060 --> 00:15:50.040 So I've got the inverse of this, I've got three matrices in 00:15:50.040 --> 00:15:52.020 a row now. 00:15:52.020 --> 00:15:58.300 And it's their inverses that are going to show up, because each one is easy to invert. 00:15:58.300 --> 00:16:01.460 Question is, what about the whole bunch? 00:16:01.460 --> 00:16:04.780 How easy is it to invert the whole bunch? 00:16:04.780 --> 00:16:08.630 So, that's what we know how to do. 00:16:08.630 --> 00:16:12.840 We know how to invert, we should take the separate inverses, but they go in the opposite 00:16:12.840 --> 00:16:13.840 order. 00:16:13.840 --> 00:16:15.800 So what goes here? 00:16:15.800 --> 00:16:21.640 E three two inverse, right, because I'll multiply from the left by E three two inverse, then 00:16:21.640 --> 00:16:26.600 I'll pop it up next to U. 00:16:26.600 --> 00:16:30.880 And then will come E three one inverse. 00:16:30.880 --> 00:16:40.900 And then this'll be the only guy left standing and that's gone when I do an E two one inverse. 00:16:40.910 --> 00:16:43.590 So there is L. 00:16:43.590 --> 00:16:47.300 That's L U. 00:16:47.300 --> 00:16:50.860 L is product of inverses. 00:16:50.860 --> 00:16:57.680 Now you still can ask why is this guy preferring inverses? 00:16:57.680 --> 00:16:59.570 And let me explain why. 00:16:59.570 --> 00:17:05.619 Let me explain why is this product nicer than this one? 00:17:05.619 --> 00:17:10.380 This product turns out to be better than this one. 00:17:10.380 --> 00:17:15.060 Let me take a typical case here. 00:17:15.060 --> 00:17:17.250 Let me take a typical case. 00:17:17.250 --> 00:17:23.000 So let me -- I have to do three by three for you to see the improvement. 00:17:23.000 --> 00:17:27.400 Two by two, it was just one E, no problem. 00:17:27.400 --> 00:17:29.960 But let me go up to this case. 00:17:29.960 --> 00:17:41.460 Suppose my matrices E21 -- suppose E21 has a minus two in there. 00:17:41.460 --> 00:17:49.840 Suppose that -- and now suppose -- oh, I'll even suppose E31 is the identity. 00:17:49.840 --> 00:17:54.360 I'm going to make the point with just a couple of these. 00:17:54.360 --> 00:17:54.900 Okay. 00:17:54.900 --> 00:18:02.720 Now this guy will have -- do something -- now let's suppose minus five one. 00:18:02.720 --> 00:18:07.560 Okay. There's typical. 00:18:07.560 --> 00:18:12.760 That's a typical case in which we didn't need an E31. Maybe we already had a zero in that 00:18:12.760 --> 00:18:14.660 three one position. 00:18:14.660 --> 00:18:17.560 Okay. 00:18:17.560 --> 00:18:24.700 Let me see -- is that going to be enough to, show my point? 00:18:24.700 --> 00:18:29.500 Let me do that multiplication. 00:18:29.500 --> 00:18:32.640 So if I do that multiplication it's like good practice to 00:18:32.640 --> 00:18:35.000 multiply these matrices. 00:18:35.000 --> 00:18:39.900 Tell me what's above the diagonal when I do this multiplication? 00:18:39.900 --> 00:18:44.500 All zeroes. When I do this multiplication, I'm going to get ones on the diagonal and 00:18:44.500 --> 00:18:48.280 zeroes above. 00:18:48.280 --> 00:18:50.140 Because -- what does that say? 00:18:50.140 --> 00:18:54.540 That says that I'm subtracting rows from lower rows. 00:18:54.549 --> 00:19:00.970 So nothing is moving upwards as it did last time in Gauss-Jordan. Okay. 00:19:00.970 --> 00:19:10.280 Now -- so really, what I have to do is check this minus two one zero, now this is -- what's 00:19:10.280 --> 00:19:11.240 that number? 00:19:11.240 --> 00:19:15.740 This is the number that I'm really have in mind. 00:19:15.740 --> 00:19:19.900 That number is ten. 00:19:19.900 --> 00:19:25.620 And this one is -- what goes here? 00:19:25.620 --> 00:19:28.960 Row three against column two, it looks like the minus five. 00:19:32.000 --> 00:19:35.100 What – it's that ten. 00:19:35.100 --> 00:19:37.020 How did that ten get in there? 00:19:37.020 --> 00:19:39.040 I don't like that ten. 00:19:39.040 --> 00:19:42.220 I mean -- of course, I don't want to erase it, because it's right. 00:19:42.220 --> 00:19:45.840 But I don't want it there. 00:19:45.840 --> 00:19:53.800 It's because -- the ten got in there because I subtracted two of row one from row two, 00:19:53.800 --> 00:19:58.760 and then I subtracted five of that new row two from row three. 00:19:58.760 --> 00:20:04.920 So doing it in that order, how did row one effect row three? 00:20:04.920 --> 00:20:10.480 Well, it did, because two of it got removed from row two and then five of those got removed 00:20:10.480 --> 00:20:11.590 from row three. 00:20:11.590 --> 00:20:17.800 So altogether ten of row one got thrown into row three. 00:20:17.800 --> 00:20:27.540 Now my point is in the reverse direction -- so now I can do it -- below it I'll do the inverses. 00:20:27.540 --> 00:20:28.100 Okay. 00:20:28.100 --> 00:20:29.820 And, of course, opposite order. 00:20:29.820 --> 00:20:31.660 Reverse order. 00:20:31.660 --> 00:20:35.960 Reverse order. 00:20:35.960 --> 00:20:44.020 Okay. So now this is going to -- this is the E that goes on the left side. 00:20:44.020 --> 00:20:47.790 Left of A. 00:20:47.790 --> 00:20:54.340 Now I'm going to do the inverses in the opposite order, so what's the -- So the opposite order 00:20:54.340 --> 00:20:57.530 means I put this inverse first. 00:20:57.530 --> 00:20:58.890 And what is its inverse? 00:20:58.890 --> 00:21:04.660 What's the inverse of E21? Same thing with a plus sign, right? 00:21:04.660 --> 00:21:12.730 For the individual matrices, instead of taking away two I add back two of row one to row 00:21:12.730 --> 00:21:16.620 two, so no problem. 00:21:16.620 --> 00:21:21.400 And now, in reverse order, I want to invert that. 00:21:21.400 --> 00:21:22.840 Just right? 00:21:22.840 --> 00:21:25.540 I'm doing just this, this. 00:21:25.540 --> 00:21:36.320 So now the inverse is again the same thing, but add in the five. 00:21:36.320 --> 00:21:45.460 And now I'll do that multiplication and I'll get a happy result. 00:21:45.460 --> 00:21:47.800 I hope. 00:21:47.800 --> 00:21:50.220 Did I do it right so far? 00:21:50.220 --> 00:21:50.960 Yes, okay. 00:21:50.960 --> 00:21:51.640 Let me do the multiplication. 00:21:51.650 --> 00:21:53.350 I believe this comes out. 00:21:53.350 --> 00:21:56.660 So row one of the answer is one zero zero. 00:21:56.660 --> 00:21:59.300 Oh, I know that all this is going to be left, 00:21:59.300 --> 00:22:04.080 right? Then I have two one zero. 00:22:04.090 --> 00:22:07.559 So I get two one zero there, right? 00:22:07.560 --> 00:22:10.060 And what's the third row? 00:22:10.060 --> 00:22:15.160 What's the third row in this product? 00:22:15.160 --> 00:22:18.620 Just read it out to me, the third row? 00:22:18.640 --> 00:22:22.980 0 5 1 00:22:22.980 --> 00:22:27.240 Because one way to say is – this is saying take one of the 00:22:27.240 --> 00:22:29.640 last row and there it is. 00:22:29.640 --> 00:22:32.580 And this is my matrix L. 00:22:32.590 --> 00:22:38.450 And it's the one that goes on the left of U. 00:22:38.450 --> 00:22:45.190 It goes into -- what do I mean here? 00:22:45.190 --> 00:22:52.040 Maybe rather than saying left of A, left of U, let me right down again what I mean. 00:22:52.040 --> 00:22:58.600 E A is U, whereas A is L U. 00:22:58.600 --> 00:23:01.560 Okay. 00:23:01.560 --> 00:23:06.280 Let me make the point now in words. 00:23:06.300 --> 00:23:11.090 The order that the matrices come for L is the right order. 00:23:11.090 --> 00:23:21.780 The two and the five don't sort of interfere to produce this ten one. In the 00:23:21.800 --> 00:23:26.919 right order, the multipliers just sit in the matrix L. 00:23:26.919 --> 00:23:35.530 That's the point -- that if I want to know L, I have no work to do. 00:23:35.530 --> 00:23:41.370 I just keep a record of what those multipliers were, and that gives me L. 00:23:41.370 --> 00:23:52.610 So I'll draw the -- let me say it. 00:23:52.610 --> 00:23:56.320 So this is the A=L U. 00:23:56.320 --> 00:24:14.010 So if no row exchanges, the multipliers that those numbers that we multiplied rows by and 00:24:14.010 --> 00:24:27.550 subtracted, when we did an elimination step -- the multipliers go directly into L. 00:24:27.550 --> 00:24:29.950 Okay. 00:24:29.950 --> 00:24:42.770 So L is -- this is the way, to look at elimination. 00:24:42.770 --> 00:24:51.510 You go through the elimination steps, and actually if you do it right, you can throw 00:24:51.510 --> 00:24:57.120 away A as you create L U. 00:24:57.120 --> 00:25:08.520 If you think about it, those steps of elimination, as when you've finished with row two of A, 00:25:08.520 --> 00:25:16.540 you've created a new row two of U, which you have to save, and you've created the multipliers 00:25:16.540 --> 00:25:22.980 that you used -- which you have to save, and then you can forget A. 00:25:22.980 --> 00:25:25.919 So because it's all there in L and U. 00:25:25.919 --> 00:25:41.580 So that's -- this moment is maybe the new insight in elimination that comes from matrix 00:25:41.580 --> 00:25:44.070 -- doing it in matrix form. 00:25:44.070 --> 00:25:52.350 So it was -- the product of Es is -- we can't see what that product of Es is. 00:25:52.350 --> 00:25:56.350 The matrix E is not a particularly attractive one. 00:25:56.350 --> 00:26:02.130 What's great is when we put them on the other side -- their inverses in the opposite order, 00:26:02.130 --> 00:26:06.210 there the L comes out just right. Okay. 00:26:06.210 --> 00:26:09.580 Now -- oh gosh, so today's a sort of, 00:26:09.580 --> 00:26:14.160 like, practical day. 00:26:14.160 --> 00:26:20.670 Can we think together how expensive is elimination? 00:26:20.670 --> 00:26:24.570 How many operations do we do? 00:26:24.570 --> 00:26:32.520 So this is now a kind of new topic which I didn't list as -- on the program, but here 00:26:32.540 --> 00:26:34.980 it came. Here it comes. 00:26:34.980 --> 00:26:55.780 How many operations on an n by n matrix A. 00:26:55.780 --> 00:26:58.020 I mean, it's a very practical question. 00:26:58.020 --> 00:27:08.280 Can we solve systems of order a thousand, in a second or a minute or a week? 00:27:08.280 --> 00:27:16.460 Can we solve systems of order a million in a second or an hour or a week? 00:27:16.460 --> 00:27:25.059 I mean, what's the -- if it's n by n, we often want to take n bigger. 00:27:25.059 --> 00:27:28.700 I mean, we've put in more information. 00:27:28.700 --> 00:27:33.880 We make the whole thing is more accurate for the bigger matrix. 00:27:33.880 --> 00:27:39.000 But it's more expensive, too, and the question is how much more expensive? 00:27:39.000 --> 00:27:41.679 If I have matrices of order a hundred. 00:27:41.679 --> 00:27:43.150 Let's say a hundred by a hundred. 00:27:43.150 --> 00:27:47.660 Let me take n to be a hundred. 00:27:47.660 --> 00:27:50.840 Say n equal a hundred. 00:27:50.840 --> 00:27:54.860 How many steps are we doing? 00:27:54.860 --> 00:28:02.770 How many operations are we actually doing that we -- And let's suppose there aren't 00:28:02.770 --> 00:28:07.799 any zeroes, because of course if a matrix has got a lot of zeroes in good places, we 00:28:07.800 --> 00:28:09.870 don't have to do those operations, and, 00:28:09.870 --> 00:28:13.500 it'll be much faster. 00:28:13.500 --> 00:28:21.940 But -- so just think for a moment about the first step. 00:28:21.940 --> 00:28:27.179 So here's our matrix A, hundred by a hundred. 00:28:27.180 --> 00:28:34.880 And the first step will be -- that column, is got zeroes down 00:28:34.880 --> 00:28:41.080 here. So it's down to 99 by 99, right? 00:28:41.080 --> 00:28:45.920 That's really like the first stage of elimination, 00:28:45.920 --> 00:28:48.480 to get from this hundred 00:28:48.490 --> 00:28:55.600 by hundred non-zero matrix to this stage where the first pivot is sitting up here and the 00:28:55.600 --> 00:28:59.440 first row's okay the first column is okay. 00:28:59.440 --> 00:29:05.120 So, eventually -- how many steps did that take? 00:29:05.120 --> 00:29:07.400 You see, I'm trying to get an idea. 00:29:07.409 --> 00:29:11.860 Is the answer proportional to n? 00:29:11.860 --> 00:29:17.260 Is the total number of steps in elimination, the total number, is it proportional to n 00:29:17.260 --> 00:29:22.520 -- in which case if I double n from a hundred to two hundred -- does it take me twice as 00:29:22.520 --> 00:29:24.140 long? 00:29:24.140 --> 00:29:27.340 Does it square, so it would take me four times as long? 00:29:27.350 --> 00:29:30.600 Does it cube so it would take me eight times as long? 00:29:30.600 --> 00:29:37.120 Or is it n factorial, so it would take me a hundred times as long? 00:29:37.120 --> 00:29:41.560 I think, you know, from a practical point of view, we have to have some idea of the 00:29:41.560 --> 00:29:44.280 cost, here. 00:29:44.280 --> 00:29:48.720 So these are the questions that I'm -- let me ask those questions again. 00:29:48.720 --> 00:29:54.900 Is it proportional -- does it go like n, like n squared, like n cubed -- or some higher 00:29:54.900 --> 00:29:56.420 power of n? 00:29:56.420 --> 00:30:04.580 Like n factorial where every step up multiplies by a hundred and then by a hundred and one 00:30:04.580 --> 00:30:07.980 and then by a hundred and two -- which is it? 00:30:07.980 --> 00:30:14.320 Okay, so that's the only way I know to answer that is to think through what we actually 00:30:14.320 --> 00:30:16.100 had to do. 00:30:16.100 --> 00:30:17.100 Okay. 00:30:17.100 --> 00:30:22.680 So what was the cost here? 00:30:22.680 --> 00:30:23.300 Well, let's see. 00:30:23.309 --> 00:30:25.700 What do I mean by an operation? 00:30:25.700 --> 00:30:32.549 I guess I mean, well an addition or -- yeah. 00:30:32.549 --> 00:30:33.549 No big deal. 00:30:33.549 --> 00:30:38.980 I guess I mean an addition or a subtraction or a multiplication or a division. 00:30:38.980 --> 00:30:41.540 Okay. 00:30:41.540 --> 00:30:48.980 And actually, what operation I doing all the time? 00:30:48.980 --> 00:30:58.870 When I multiply row one by multiplier L and I subtract from row six. 00:30:58.870 --> 00:31:01.060 What's happening there individually? 00:31:01.060 --> 00:31:03.460 What's going on? 00:31:03.460 --> 00:31:08.490 If I multiply -- I do a multiplication by L and then a subtraction. 00:31:08.490 --> 00:31:15.610 So I guess operation -- Can I count that for the moment as, like, one operation? 00:31:15.610 --> 00:31:18.520 Or you may want to count them separately. 00:31:18.520 --> 00:31:26.740 The typical operation is multiply plus a subtract. 00:31:26.740 --> 00:31:31.420 So if I count those together, my answer's going to come out 00:31:31.420 --> 00:31:36.540 half as many as if -- I mean, if I count them separately, I'd have a certain 00:31:36.540 --> 00:31:39.000 number of multiplies, certain number of subtracts. 00:31:39.000 --> 00:31:40.580 That's really want to do. 00:31:40.580 --> 00:31:43.360 Okay. How many have I got here? 00:31:43.360 --> 00:31:49.500 So, I think -- let's see. 00:31:49.500 --> 00:31:56.293 It's about -- well, how many, roughly? 00:31:56.293 --> 00:31:59.352 How many operations to get from here to here? 00:31:59.360 --> 00:32:05.800 Well, maybe one way to look at it is all these numbers had to get changed. 00:32:05.800 --> 00:32:11.900 The first row didn't get changed, but all the other rows got changed at this step. 00:32:11.900 --> 00:32:25.880 So this step -- well, I guess maybe -- shall I say it cost about a hundred squared. 00:32:25.880 --> 00:32:32.100 I mean, if I had changed the first row, then it would have been exactly hundred squared, 00:32:32.110 --> 00:32:35.549 because -- because that's how many numbers are here. 00:32:35.549 --> 00:32:42.279 A hundred squared numbers is the total count of the entry, and all but this insignificant 00:32:42.279 --> 00:32:44.299 first row got changed. 00:32:44.300 --> 00:32:47.660 So I would say about a hundred squared. 00:32:47.660 --> 00:32:48.620 Okay. 00:32:48.620 --> 00:32:54.150 Now, what about the next step? 00:32:54.150 --> 00:32:56.960 So now the first row is fine. 00:32:56.960 --> 00:33:00.020 The second row is fine. 00:33:00.020 --> 00:33:06.700 And I'm changing these zeroes are all fine, so what's up with the second step? 00:33:06.700 --> 00:33:08.620 And then you're with me. 00:33:08.620 --> 00:33:10.380 Roughly, what's the cost? 00:33:10.390 --> 00:33:17.330 If this first step cost a hundred squared, about, operations then this one, which is 00:33:17.330 --> 00:33:27.490 really working on this guy to produce this, costs about what? 00:33:27.490 --> 00:33:31.250 How many operations to fix? 00:33:31.250 --> 00:33:36.610 About ninety-nine squared, or ninety-nine times ninety-eight. But less, right? 00:33:36.610 --> 00:33:39.590 Less, because our problem's getting smaller. 00:33:39.590 --> 00:33:42.280 About ninety-nine squared. 00:33:42.280 --> 00:33:46.160 And then I go down and down and the next one will be ninety-eight squared, the next ninety-seven 00:33:46.160 --> 00:33:53.750 squared and finally I'm down around one squared or -- where it's just like the little numbers. 00:33:53.750 --> 00:33:55.890 The big numbers are here. 00:33:55.890 --> 00:34:06.190 So the number of operations is about n squared plus that was n, right? n was a hundred? 00:34:06.190 --> 00:34:13.589 n squared for the first step, then n minus one squared, then n minus two squared, finally 00:34:13.589 --> 00:34:23.040 down to three squared and two squared and even one squared. 00:34:23.040 --> 00:34:27.929 No way I should have written that -- squeezed that in. 00:34:27.929 --> 00:34:37.570 Let me try it so the count is n squared plus n minus one squared plus -- all the way down 00:34:37.570 --> 00:34:41.980 to one squared. 00:34:41.980 --> 00:34:44.899 That's a pretty decent count. 00:34:44.899 --> 00:34:56.040 Admittedly, we didn't catch every single tiny operation, but we got the right leading term 00:34:56.040 --> 00:34:57.040 here. 00:34:57.040 --> 00:35:01.180 And what do those add up to? 00:35:01.180 --> 00:35:10.030 Okay, so now we're coming to the punch of this, question, this operation count. 00:35:10.030 --> 00:35:19.820 So the operations on the left side, on the matrix A to finally get to U. 00:35:19.820 --> 00:35:28.540 And anybody -- so which of these quantities is the right ballpark for that count? 00:35:28.540 --> 00:35:34.080 If I add a hundred squared to ninety nine squared to ninety eight squared -- ninety 00:35:34.080 --> 00:35:44.020 seven squared, all the way down to two squared then one squared, what have I got, about? 00:35:44.020 --> 00:35:48.100 It's just one of these -- let's identify it first. 00:35:48.100 --> 00:35:49.020 Is it n? 00:35:49.020 --> 00:35:52.080 Certainly not. 00:35:52.080 --> 00:35:55.700 Is it n factorial? 00:35:55.700 --> 00:35:56.460 No. 00:35:56.460 --> 00:36:02.020 If it was n factorial, we would -- with determinants, it is n factorial. 00:36:02.020 --> 00:36:12.400 I'll put in a bad mark against determinants, because that -- okay, so what is it? 00:36:12.400 --> 00:36:18.580 It's n -- well, this is the answer. 00:36:18.580 --> 00:36:21.240 It's this order -- n cubed. 00:36:21.240 --> 00:36:26.380 It's like I have n terms, right? 00:36:26.380 --> 00:36:28.680 I've got n terms in this sum. 00:36:28.690 --> 00:36:31.140 And the biggest one is n squared. 00:36:31.140 --> 00:36:40.750 So the worst it could be would be n cubed, but it's not as bad as -- it's n cubed times 00:36:40.750 --> 00:36:45.270 -- it's about one third of n cubed. 00:36:45.270 --> 00:36:53.690 That's the magic operation count. 00:36:53.690 --> 00:37:02.339 Somehow that one third takes account of the fact that the numbers are getting smaller. 00:37:02.339 --> 00:37:06.900 If they weren't getting smaller, we would have n terms times n squared, but it would 00:37:06.900 --> 00:37:08.240 be exactly n cubed. 00:37:08.240 --> 00:37:10.640 But our numbers are getting smaller -- actually, row two and row one moves down to row three. 00:37:10.640 --> 00:37:19.400 do you remember where does one third come in this -- I'll even allow a mention of calculus. 00:37:19.400 --> 00:37:25.940 So calculus can be mentioned, integration can be mentioned now in the next minute and 00:37:25.940 --> 00:37:28.360 not again for weeks. 00:37:28.360 --> 00:37:33.440 It's not that I don't like 18.01, but18.06 is better. 00:37:33.440 --> 00:37:42.340 Okay. So, -- so what's -- what's the calculus formula that looks like? 00:37:42.350 --> 00:37:50.589 It looks like -- if we were in calculus instead of summing stuff, we would integrate. 00:37:50.589 --> 00:37:56.800 So I would integrate x squared and I would get one third x 00:37:56.800 --> 00:38:07.430 cubed. So if that was like an integral from one to n, of x squared b x, if the answer 00:38:07.430 --> 00:38:13.690 would be one third n cubed -- and it's correct for the sum also, because that's, like, the 00:38:13.690 --> 00:38:14.900 whole point of calculus. 00:38:14.900 --> 00:38:19.315 The whole point of calculus is -- oh, I don't want to tell you the whole -- I mean, you 00:38:19.315 --> 00:38:21.280 know the whole point of calculus. 00:38:21.280 --> 00:38:27.990 Calculus is like sums except it's continuous. 00:38:27.990 --> 00:38:31.950 Okay. And algebra is discreet. 00:38:31.950 --> 00:38:32.950 Okay. 00:38:32.950 --> 00:38:35.010 So the answer is one third n cubed. 00:38:35.010 --> 00:38:40.080 Now I'll just -- let me say one more thing about operations. 00:38:40.080 --> 00:38:41.920 What about the right-hand side? 00:38:41.920 --> 00:38:45.250 This was what it cost on the left side. 00:38:45.250 --> 00:38:50.380 This is on A. 00:38:50.380 --> 00:38:52.180 Because this is A that we're working with. 00:38:52.180 --> 00:39:00.660 But what's the cost on the extra column vector b that we're hanging around here? 00:39:00.660 --> 00:39:07.570 So b costs a lot less, obviously, because it's just one column. 00:39:07.570 --> 00:39:14.260 We carry it through elimination and then actually we do back substitution. 00:39:14.260 --> 00:39:16.160 Let me just tell you the answer there. 00:39:16.160 --> 00:39:18.420 It's n squared. 00:39:18.420 --> 00:39:23.220 So the cost for every right hand side is n squared. 00:39:23.220 --> 00:39:37.000 So let me -- I'll just fit that in here -- for the cost of b turns out to be n squared. 00:39:37.000 --> 00:39:49.580 So you see if we have, as we often have, a a matrix A and several right-hand sides, then 00:39:49.580 --> 00:39:57.670 we pay the price on A, the higher price on A to get it split up into L and U to do elimination 00:39:57.670 --> 00:40:02.810 on A, but then we can process every right-hand side at low cost. 00:40:02.810 --> 00:40:03.859 Okay. 00:40:03.860 --> 00:40:16.560 So the -- We really have discussed the most fundamental algorithm for a system of equations. 00:40:16.560 --> 00:40:19.700 Okay. 00:40:19.700 --> 00:40:29.500 So, I'm ready to allow row exchanges. 00:40:29.500 --> 00:40:34.220 I'm ready to allow -- now what happens to this whole -- today's lecture if there are 00:40:34.220 --> 00:40:38.400 row exchanges? 00:40:38.400 --> 00:40:42.240 When would there be row exchanges? 00:40:42.240 --> 00:40:47.840 There are row -- we need to do row exchanges if a zero shows up in the pivot position. 00:40:47.840 --> 00:40:55.820 So moving then into the final section of this chapter, which is about transposes -- well, 00:40:55.820 --> 00:41:08.640 we've already seen some transposes, and -- the title of this section is, 00:41:08.640 --> 00:41:13.700 "Transposes and Permutations." 00:41:13.700 --> 00:41:21.160 Okay. So can I say, now, where does a permutation come in? 00:41:21.160 --> 00:41:23.300 Let me talk a little about permutations. 00:41:23.300 --> 00:41:34.620 So that'll be up here, permutations. 00:41:34.620 --> 00:41:41.420 So these are the matrices that I need to do row exchanges. 00:41:41.420 --> 00:41:44.560 And I may have to do two row exchanges. 00:41:44.560 --> 00:41:52.320 Can you invent a matrix where I would have to do two row exchanges and then would come 00:41:52.320 --> 00:41:54.180 out fine? 00:41:54.180 --> 00:42:01.460 Yeah let's just, for the heck of it -- so I'll put it here. 00:42:01.460 --> 00:42:04.120 Let me do three by threes. 00:42:04.120 --> 00:42:10.480 Actually, why don't I just plain list all the three by three permutation matrices. 00:42:10.480 --> 00:42:13.000 There're a nice little group of them. 00:42:13.010 --> 00:42:20.880 What are all the matrices that exchange no rows at all? 00:42:20.880 --> 00:42:26.070 Well, I'll include the identity. 00:42:26.070 --> 00:42:29.710 So that's a permutation matrix that doesn't do anything. 00:42:29.710 --> 00:42:38.680 Now what's the permutation matrix that exchanges -- what is P12? The permutation matrix that 00:42:38.680 --> 00:42:48.540 exchanges rows one and two would be -- 0 1 0 -- 1 0 0, right. 00:42:48.540 --> 00:42:53.360 I just exchanged those rows of the identity and I've got it. 00:42:53.360 --> 00:42:56.080 Okay. Actually, I'll -- yes. 00:42:56.080 --> 00:43:01.840 Let me clutter this up. 00:43:01.840 --> 00:43:06.340 Okay. Give me a complete list of all the row exchange matrices. 00:43:06.340 --> 00:43:07.540 So what are they? 00:43:07.540 --> 00:43:13.780 They're all the ways I can take the identity matrix and rearrange its rows. 00:43:13.780 --> 00:43:16.620 How many will there be? 00:43:16.620 --> 00:43:21.820 How many three by three permutation matrices? 00:43:21.820 --> 00:43:24.100 Shall we keep going and get the answer? 00:43:24.100 --> 00:43:27.500 So tell me some more. 00:43:27.500 --> 00:43:28.580 STUDENT: Zero one – 00:43:28.580 --> 00:43:31.000 STRANG: Zero – What one are you going to do now? 00:43:31.000 --> 00:43:32.840 STUDENT: I'm going to switch the – 00:43:32.840 --> 00:43:43.520 STRANG: Switch rows one and -- One and three, okay. One and three, leaving two alone. 00:43:43.520 --> 00:43:44.460 Okay. 00:43:44.460 --> 00:43:50.780 Now what else? Switch -- what would be the next easy one -- is switch two and 00:43:50.780 --> 00:43:57.240 three, good. So I'll leave one zero zero alone and I'll switch -- I'll move number three 00:43:57.250 --> 00:44:00.329 up and number two down. 00:44:00.329 --> 00:44:05.380 Okay. Those are the ones that just exchange single -- a pair of 00:44:05.380 --> 00:44:13.339 rows. This guy, this guy and this guy exchanges a pair of rows, but now there are more possibilities. 00:44:13.340 --> 00:44:16.080 What's left? 00:44:16.080 --> 00:44:19.640 So tell -- there is another one here. 00:44:19.640 --> 00:44:22.260 What's that? 00:44:22.260 --> 00:44:26.440 It's going to move -- it's going to change all rows, right? 00:44:26.440 --> 00:44:28.160 Where shall we put them? 00:44:28.160 --> 00:44:29.780 So -- give me a first row. 00:44:29.780 --> 00:44:30.800 STUDENT: Zero one zero? 00:44:30.800 --> 00:44:32.740 STRANG: Zero one zero. 00:44:32.740 --> 00:44:38.000 Okay, now a second row -- say zero zero one and the third guy 00:44:38.000 --> 00:44:41.960 One zero zero. 00:44:41.960 --> 00:44:44.420 So that is like a cycle. 00:44:44.420 --> 00:44:49.800 That puts row two moves up to row one, row three moves up to 00:44:49.800 --> 00:44:53.420 row two and row one moves down to row three. 00:44:53.420 --> 00:45:00.180 And there's one more, which is -- let's see. 00:45:00.180 --> 00:45:03.280 What's left? 00:45:03.280 --> 00:45:04.160 I'm lost. 00:45:04.160 --> 00:45:05.240 STUDENT: Is it zero zero one? 00:45:05.240 --> 00:45:06.780 STRANG: Is it zero zero one? Okay. 00:45:06.780 --> 00:45:08.120 STUDENT: One zero zero. 00:45:08.120 --> 00:45:11.260 STRANG: One zero zero, okay. 00:45:11.260 --> 00:45:16.460 Zero one zero, okay. 00:45:16.460 --> 00:45:17.980 Great. 00:45:17.980 --> 00:45:21.780 Six. Six of them. 00:45:21.780 --> 00:45:35.580 Six P. And they're sort of nice, because what happens if I write, multiply two of them together? 00:45:35.580 --> 00:45:42.100 If I multiply two of these matrices together, what can you tell me about the answer? 00:45:42.100 --> 00:45:44.360 It's on the list. 00:45:44.360 --> 00:45:49.220 If I do some row exchanges and then I do some more row exchanges, then all together I've 00:45:49.220 --> 00:45:50.520 done row exchanges. 00:45:50.520 --> 00:45:54.520 So if I multiply -- but, I don't know. 00:45:54.520 --> 00:46:00.340 And if I invert, then I'm just doing row exchanges to get back again. 00:46:00.349 --> 00:46:01.869 So the inverses are all there. 00:46:01.869 --> 00:46:12.790 It's a little family of matrices that -- they've got their own -- if I multiply, I'm still 00:46:12.790 --> 00:46:14.400 inside this group. 00:46:14.400 --> 00:46:18.920 If I invert I'm inside this group -- actually, group is the right name for this subject. 00:46:18.920 --> 00:46:24.320 It's a group of six matrices, and what about the inverses? 00:46:24.320 --> 00:46:28.160 What's the inverse of this guy, for example? 00:46:28.160 --> 00:46:34.700 What's the inverse -- if I exchange rows one and two, what's the inverse matrix? 00:46:34.700 --> 00:46:36.760 Just tell me fast. 00:46:36.760 --> 00:46:46.910 The inverse of that matrix is -- if I exchange rows one and two, then what I should do to 00:46:46.910 --> 00:46:51.150 get back to where I started is the same thing. 00:46:51.150 --> 00:46:54.905 So this thing is its own inverse. 00:46:54.905 --> 00:46:55.905 That's probably its own inverse. 00:46:55.905 --> 00:47:00.380 This is probably not -- actually, I think these are inverses of each other. 00:47:00.380 --> 00:47:06.320 Oh, yeah, actually -- the inverse is the transpose. 00:47:06.320 --> 00:47:16.020 There's a curious fact about permutations matrices, that the inverses are the transposes. 00:47:16.020 --> 00:47:21.820 And final moment -- how many are there if I -- how many four by four permutations? 00:47:21.820 --> 00:47:29.880 So let me take four by four -- how many Ps? 00:47:29.880 --> 00:47:32.720 Well, okay. 00:47:32.720 --> 00:47:34.940 Make a good guess. 00:47:34.940 --> 00:47:38.020 Twenty four, right. Twenty four Ps. 00:47:38.020 --> 00:47:48.820 Okay. So, we've got these permutation matrices, and in the next lecture, we'll use them. 00:47:48.820 --> 00:47:51.660 So the next lecture, finishes Chapter 2 00:47:51.660 --> 00:47:55.160 and moves to Chapter 3. 00:47:55.160 --> 00:47:57.480 Thank you.