1 00:00:12,380 --> 00:00:16,800 Okay, this is linear algebra, lecture four. 2 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 3 00:00:24,810 --> 00:00:26,169 it is now. 4 00:00:26,169 --> 00:00:29,369 What's the inverse of a product? 5 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 6 00:00:36,300 --> 00:00:39,559 of A times B? 7 00:00:39,559 --> 00:00:46,760 So I know what inverses mean for a single matrix A and for a matrix B. 8 00:00:46,760 --> 00:00:53,229 What matrix do I multiply by to get the identity if I have A B? 9 00:00:53,229 --> 00:00:57,100 Okay, that'll be simple but so basic. 10 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 11 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 12 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 13 00:01:22,960 --> 00:01:27,700 to look at Gaussian elimination. 14 00:01:27,710 --> 00:01:31,400 We know that we get from A to U by elimination. 15 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. 16 00:01:38,060 --> 00:01:40,800 So that's the high point for today. 17 00:01:40,800 --> 00:01:41,800 Okay. 18 00:01:41,800 --> 00:01:47,320 Can I take the easy part, the first step first? 19 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 20 00:01:54,560 --> 00:01:55,860 matrix invertible? 21 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? 22 00:02:04,920 --> 00:02:07,460 So that's the direct question. 23 00:02:07,470 --> 00:02:08,869 What's the inverse of A B? 24 00:02:08,869 --> 00:02:12,120 Do I multiply those separate inverses? 25 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? 26 00:02:21,230 --> 00:02:22,290 In reverse order. 27 00:02:22,290 --> 00:02:24,020 And you see why. 28 00:02:24,020 --> 00:02:29,780 So the right thing to put here is B inverse A inverse. 29 00:02:29,780 --> 00:02:31,820 That's the inverse I'm after. 30 00:02:31,820 --> 00:02:38,940 We can just check that A B times that matrix gives the identity. 31 00:02:38,950 --> 00:02:39,950 Okay. 32 00:02:39,950 --> 00:02:46,480 So why -- once again, it's this fact that I can move parentheses around. 33 00:02:46,480 --> 00:02:52,320 I can just erase them all and do the multiplications any way I want to. 34 00:02:52,329 --> 00:02:56,209 So what's the right multiplication to do first? 35 00:02:56,209 --> 00:02:58,220 B times B inverse. 36 00:02:58,220 --> 00:03:02,400 This product here I is the identity. 37 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 38 00:03:08,100 --> 00:03:10,060 identity. 39 00:03:10,060 --> 00:03:15,680 So forgive the dumb example in the book. 40 00:03:15,680 --> 00:03:21,180 Why do you, do the inverse things in reverse order? 41 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 42 00:03:27,020 --> 00:03:35,060 to invert that process is socks back on first, then shoes. 43 00:03:35,060 --> 00:03:36,660 Sorry, okay. 44 00:03:36,660 --> 00:03:41,320 I'm sorry that's on the tape. 45 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 46 00:03:46,460 --> 00:03:47,480 B inverse, 47 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, 48 00:03:54,099 --> 00:03:58,040 squeeze down, they give the identity, we're in shape. 49 00:03:58,040 --> 00:04:01,700 Okay. So there's the inverse. 50 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 51 00:04:08,480 --> 00:04:09,779 -- involves transposes. 52 00:04:09,780 --> 00:04:16,860 So how do I -- if I transpose a matrix, I'm talking about square, invertible matrices 53 00:04:16,860 --> 00:04:18,298 right now. 54 00:04:18,300 --> 00:04:23,500 If I transpose one, what's its inverse? 55 00:04:23,500 --> 00:04:27,640 Well, the nice formula is -- let's see. 56 00:04:27,640 --> 00:04:34,760 Let me start from A, A inverse equal the identity. 57 00:04:34,760 --> 00:04:39,780 And let me transpose both sides. 58 00:04:39,790 --> 00:04:43,199 That will bring a transpose into the picture. 59 00:04:43,199 --> 00:04:48,300 So if I transpose the identity matrix, what do I have? 60 00:04:48,300 --> 00:04:49,600 The identity, right? 61 00:04:49,600 --> 00:04:54,990 If I exchange rows and columns, the identity is a symmetric matrix. 62 00:04:54,990 --> 00:04:56,940 It doesn't know the difference. 63 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 64 00:05:07,780 --> 00:05:09,160 the order. 65 00:05:09,160 --> 00:05:15,260 I can transpose them separately, but when I multiply, those transposes come in the opposite 66 00:05:15,260 --> 00:05:16,220 order. 67 00:05:16,220 --> 00:05:23,600 So it's A inverse transpose times A transpose giving the identity. 68 00:05:23,600 --> 00:05:26,700 So that's -- this equation is -- just comes directly from that 69 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 70 00:05:32,690 --> 00:05:35,700 guy A transpose? 71 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? 72 00:05:43,180 --> 00:05:47,000 And this equation tells me that here it is. 73 00:05:47,000 --> 00:05:56,659 This is the inverse of A transpose. 74 00:05:56,659 --> 00:06:02,220 Inverse of A transpose. 75 00:06:02,220 --> 00:06:08,100 Of A transpose. 76 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 77 00:06:13,480 --> 00:06:16,560 we could hope for. 78 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 79 00:06:24,640 --> 00:06:26,640 you just transpose that. 80 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 81 00:06:33,790 --> 00:06:35,390 a single matrix. 82 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 83 00:06:42,770 --> 00:06:44,920 to use. 84 00:06:44,920 --> 00:06:52,980 I put it to use by thinking -- we're really completing, the subject of elimination. 85 00:06:52,980 --> 00:07:05,920 Actually, -- the thing about elimination is it's the right way to understand what the 86 00:07:05,930 --> 00:07:06,930 matrix has got. 87 00:07:06,930 --> 00:07:15,550 This A equal L U is the most basic factorization of a matrix. 88 00:07:15,550 --> 00:07:23,960 I always worry that you will think this course is all elimination. 89 00:07:23,960 --> 00:07:27,100 It's just row operations. 90 00:07:27,100 --> 00:07:29,420 And please don't. 91 00:07:29,420 --> 00:07:41,120 We'll be beyond that, but it's the right algebra to do first. 92 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. 93 00:07:48,630 --> 00:07:52,270 So my decent form is matrix form. 94 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 95 00:07:59,360 --> 00:08:03,220 -- So no row exchanges for now. 96 00:08:03,220 --> 00:08:07,640 Pivots all fine, nothing zero in the pivot position. 97 00:08:07,640 --> 00:08:10,240 I get to the very end, which is U. 98 00:08:10,240 --> 00:08:12,000 So I get from A to U. 99 00:08:12,000 --> 00:08:15,220 And I want to know what's the connection? 100 00:08:15,220 --> 00:08:17,680 How is A related to U? 101 00:08:17,690 --> 00:08:21,991 And this is going to tell me that there's a matrix L that connects them. 102 00:08:21,991 --> 00:08:22,991 Okay. 103 00:08:23,000 --> 00:08:28,700 Can I do it for a two by two first? 104 00:08:28,700 --> 00:08:32,679 Okay. Two by two, elimination. 105 00:08:32,679 --> 00:08:37,179 Okay, so I'll do it under here. 106 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 107 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 108 00:08:54,029 --> 00:08:58,720 here and what number do I not want to put there? 109 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 110 00:09:05,780 --> 00:09:07,642 wouldn't have a second pivot. 111 00:09:07,642 --> 00:09:11,360 The matrix would be singular, general screw-up. Okay. 112 00:09:11,360 --> 00:09:15,090 So let me put some other number here like seven. 113 00:09:15,090 --> 00:09:16,580 Okay. 114 00:09:16,580 --> 00:09:24,680 Okay. Now I want to operate on that with my elementary matrix. 115 00:09:24,680 --> 00:09:28,340 So what's the elementary matrix? 116 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 117 00:09:34,560 --> 00:09:35,560 that position. 118 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. 119 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 120 00:09:50,150 --> 00:09:51,580 a three there. 121 00:09:51,580 --> 00:09:53,080 And that's U. 122 00:09:53,080 --> 00:09:55,140 And what's the matrix that did it? 123 00:09:55,140 --> 00:09:56,900 Quick review, then. 124 00:09:56,900 --> 00:10:03,060 What's the elimination elementary matrix E21 -- it's one zero, thanks. 125 00:10:03,060 --> 00:10:05,760 And -- negative four one, 126 00:10:05,760 --> 00:10:07,880 right. Good. 127 00:10:07,880 --> 00:10:14,100 Okay. So that -- you see the difference between this and what I'm shooting for. 128 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. 129 00:10:21,990 --> 00:10:22,990 Okay. 130 00:10:22,990 --> 00:10:25,940 So I can do that right away. 131 00:10:25,940 --> 00:10:29,500 Now here's going to be my A equals L U. 132 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. 133 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. 134 00:10:44,940 --> 00:10:49,800 Okay. So what's L in this case? 135 00:10:49,800 --> 00:10:57,220 Well, first -- so how is L related to this E guy? 136 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 137 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. 138 00:11:08,570 --> 00:11:11,360 So what is the inverse of this? 139 00:11:11,360 --> 00:11:17,640 Remember those elimination matrices are easy to invert. 140 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 141 00:11:30,130 --> 00:11:32,200 what this removes. 142 00:11:32,200 --> 00:11:39,860 Okay. Do you want -- if we did the numbers right, we must -- this should be correct. 143 00:11:39,860 --> 00:11:42,380 Okay. And of course it is. 144 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 145 00:11:47,730 --> 00:11:49,700 seven. Good. Okay. 146 00:11:49,720 --> 00:11:51,920 That's simple, two by two. 147 00:11:51,920 --> 00:11:57,380 But it already shows the form that we're headed for. 148 00:11:57,390 --> 00:12:01,190 It shows -- so what's the L stand for? 149 00:12:01,190 --> 00:12:02,640 Why the letter L? 150 00:12:02,640 --> 00:12:09,860 If U stood for upper triangular, then of course L stands for lower triangular. 151 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. 152 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 153 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 154 00:12:34,340 --> 00:12:39,160 how I would divide out this matrix of pivots -- two three. 155 00:12:39,160 --> 00:12:41,880 There's a diagonal matrix. 156 00:12:41,880 --> 00:12:45,980 And I just -- whatever is left is here. 157 00:12:45,980 --> 00:12:49,620 Now what's left? 158 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. 159 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. 160 00:13:02,580 --> 00:13:11,180 So if this is L U, this is maybe called L D or pivot U. 161 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. 162 00:13:21,100 --> 00:13:22,920 And the diagonal matrix in the middle. 163 00:13:22,920 --> 00:13:24,380 So both of those... 164 00:13:24,380 --> 00:13:29,400 Matlab would produce either one. 165 00:13:29,400 --> 00:13:32,860 I'll basically stay with L U. 166 00:13:32,860 --> 00:13:39,960 Okay. Now I have to think about bigger than two by two. 167 00:13:39,960 --> 00:13:43,780 But right now, this was just like easy exercise. 168 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. 169 00:13:50,920 --> 00:13:54,740 I mean, that's the only difference. 170 00:13:54,740 --> 00:14:02,700 But, with three by three, there's a more significant difference. 171 00:14:02,700 --> 00:14:05,740 Let me show you how that works. 172 00:14:05,740 --> 00:14:17,760 Let me move up to a three by three, let's say some matrix A, okay? 173 00:14:17,760 --> 00:14:20,120 Let's imagine it's three by three. 174 00:14:20,130 --> 00:14:23,050 I won't write numbers down for now. 175 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 176 00:14:29,351 --> 00:14:30,870 letter will I use for 177 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 178 00:14:41,490 --> 00:14:43,520 one position. right? 179 00:14:43,520 --> 00:14:48,260 And then the next step will be to get a zero in the three one position. 180 00:14:48,260 --> 00:14:53,560 And the final step will be to get a zero in the three two 181 00:14:53,560 --> 00:14:57,089 That's what elimination is, and it produced U. position. 182 00:14:57,089 --> 00:15:06,820 And again, no row exchanges. 183 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 184 00:15:13,180 --> 00:15:15,920 exchange, all I do is these elimination steps. 185 00:15:15,920 --> 00:15:16,900 Okay. 186 00:15:16,900 --> 00:15:22,880 Now, suppose I want that stuff over on the right-hand side, as I really do. 187 00:15:22,880 --> 00:15:25,470 That's, like, my point here. 188 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. 189 00:15:30,300 --> 00:15:33,670 I want its inverse over there. 190 00:15:33,670 --> 00:15:39,160 So what's the right expression now? 191 00:15:39,160 --> 00:15:46,440 If I write A and U, what goes there? 192 00:15:46,440 --> 00:15:47,060 Okay. 193 00:15:47,060 --> 00:15:50,040 So I've got the inverse of this, I've got three matrices in 194 00:15:50,040 --> 00:15:52,020 a row now. 195 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. 196 00:15:58,300 --> 00:16:01,460 Question is, what about the whole bunch? 197 00:16:01,460 --> 00:16:04,780 How easy is it to invert the whole bunch? 198 00:16:04,780 --> 00:16:08,630 So, that's what we know how to do. 199 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 200 00:16:12,840 --> 00:16:13,840 order. 201 00:16:13,840 --> 00:16:15,800 So what goes here? 202 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 203 00:16:21,640 --> 00:16:26,600 I'll pop it up next to U. 204 00:16:26,600 --> 00:16:30,880 And then will come E three one inverse. 205 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. 206 00:16:40,910 --> 00:16:43,590 So there is L. 207 00:16:43,590 --> 00:16:47,300 That's L U. 208 00:16:47,300 --> 00:16:50,860 L is product of inverses. 209 00:16:50,860 --> 00:16:57,680 Now you still can ask why is this guy preferring inverses? 210 00:16:57,680 --> 00:16:59,570 And let me explain why. 211 00:16:59,570 --> 00:17:05,619 Let me explain why is this product nicer than this one? 212 00:17:05,619 --> 00:17:10,380 This product turns out to be better than this one. 213 00:17:10,380 --> 00:17:15,060 Let me take a typical case here. 214 00:17:15,060 --> 00:17:17,250 Let me take a typical case. 215 00:17:17,250 --> 00:17:23,000 So let me -- I have to do three by three for you to see the improvement. 216 00:17:23,000 --> 00:17:27,400 Two by two, it was just one E, no problem. 217 00:17:27,400 --> 00:17:29,960 But let me go up to this case. 218 00:17:29,960 --> 00:17:41,460 Suppose my matrices E21 -- suppose E21 has a minus two in there. 219 00:17:41,460 --> 00:17:49,840 Suppose that -- and now suppose -- oh, I'll even suppose E31 is the identity. 220 00:17:49,840 --> 00:17:54,360 I'm going to make the point with just a couple of these. 221 00:17:54,360 --> 00:17:54,900 Okay. 222 00:17:54,900 --> 00:18:02,720 Now this guy will have -- do something -- now let's suppose minus five one. 223 00:18:02,720 --> 00:18:07,560 Okay. There's typical. 224 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 225 00:18:12,760 --> 00:18:14,660 three one position. 226 00:18:14,660 --> 00:18:17,560 Okay. 227 00:18:17,560 --> 00:18:24,700 Let me see -- is that going to be enough to, show my point? 228 00:18:24,700 --> 00:18:29,500 Let me do that multiplication. 229 00:18:29,500 --> 00:18:32,640 So if I do that multiplication it's like good practice to 230 00:18:32,640 --> 00:18:35,000 multiply these matrices. 231 00:18:35,000 --> 00:18:39,900 Tell me what's above the diagonal when I do this multiplication? 232 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 233 00:18:44,500 --> 00:18:48,280 zeroes above. 234 00:18:48,280 --> 00:18:50,140 Because -- what does that say? 235 00:18:50,140 --> 00:18:54,540 That says that I'm subtracting rows from lower rows. 236 00:18:54,549 --> 00:19:00,970 So nothing is moving upwards as it did last time in Gauss-Jordan. Okay. 237 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 238 00:19:10,280 --> 00:19:11,240 that number? 239 00:19:11,240 --> 00:19:15,740 This is the number that I'm really have in mind. 240 00:19:15,740 --> 00:19:19,900 That number is ten. 241 00:19:19,900 --> 00:19:25,620 And this one is -- what goes here? 242 00:19:25,620 --> 00:19:28,960 Row three against column two, it looks like the minus five. 243 00:19:32,000 --> 00:19:35,100 What – it's that ten. 244 00:19:35,100 --> 00:19:37,020 How did that ten get in there? 245 00:19:37,020 --> 00:19:39,040 I don't like that ten. 246 00:19:39,040 --> 00:19:42,220 I mean -- of course, I don't want to erase it, because it's right. 247 00:19:42,220 --> 00:19:45,840 But I don't want it there. 248 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, 249 00:19:53,800 --> 00:19:58,760 and then I subtracted five of that new row two from row three. 250 00:19:58,760 --> 00:20:04,920 So doing it in that order, how did row one effect row three? 251 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 252 00:20:10,480 --> 00:20:11,590 from row three. 253 00:20:11,590 --> 00:20:17,800 So altogether ten of row one got thrown into row three. 254 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. 255 00:20:27,540 --> 00:20:28,100 Okay. 256 00:20:28,100 --> 00:20:29,820 And, of course, opposite order. 257 00:20:29,820 --> 00:20:31,660 Reverse order. 258 00:20:31,660 --> 00:20:35,960 Reverse order. 259 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. 260 00:20:44,020 --> 00:20:47,790 Left of A. 261 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 262 00:20:54,340 --> 00:20:57,530 means I put this inverse first. 263 00:20:57,530 --> 00:20:58,890 And what is its inverse? 264 00:20:58,890 --> 00:21:04,660 What's the inverse of E21? Same thing with a plus sign, right? 265 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 266 00:21:12,730 --> 00:21:16,620 two, so no problem. 267 00:21:16,620 --> 00:21:21,400 And now, in reverse order, I want to invert that. 268 00:21:21,400 --> 00:21:22,840 Just right? 269 00:21:22,840 --> 00:21:25,540 I'm doing just this, this. 270 00:21:25,540 --> 00:21:36,320 So now the inverse is again the same thing, but add in the five. 271 00:21:36,320 --> 00:21:45,460 And now I'll do that multiplication and I'll get a happy result. 272 00:21:45,460 --> 00:21:47,800 I hope. 273 00:21:47,800 --> 00:21:50,220 Did I do it right so far? 274 00:21:50,220 --> 00:21:50,960 Yes, okay. 275 00:21:50,960 --> 00:21:51,640 Let me do the multiplication. 276 00:21:51,650 --> 00:21:53,350 I believe this comes out. 277 00:21:53,350 --> 00:21:56,660 So row one of the answer is one zero zero. 278 00:21:56,660 --> 00:21:59,300 Oh, I know that all this is going to be left, 279 00:21:59,300 --> 00:22:04,080 right? Then I have two one zero. 280 00:22:04,090 --> 00:22:07,559 So I get two one zero there, right? 281 00:22:07,560 --> 00:22:10,060 And what's the third row? 282 00:22:10,060 --> 00:22:15,160 What's the third row in this product? 283 00:22:15,160 --> 00:22:18,620 Just read it out to me, the third row? 284 00:22:18,640 --> 00:22:22,980 0 5 1 285 00:22:22,980 --> 00:22:27,240 Because one way to say is – this is saying take one of the 286 00:22:27,240 --> 00:22:29,640 last row and there it is. 287 00:22:29,640 --> 00:22:32,580 And this is my matrix L. 288 00:22:32,590 --> 00:22:38,450 And it's the one that goes on the left of U. 289 00:22:38,450 --> 00:22:45,190 It goes into -- what do I mean here? 290 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. 291 00:22:52,040 --> 00:22:58,600 E A is U, whereas A is L U. 292 00:22:58,600 --> 00:23:01,560 Okay. 293 00:23:01,560 --> 00:23:06,280 Let me make the point now in words. 294 00:23:06,300 --> 00:23:11,090 The order that the matrices come for L is the right order. 295 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 296 00:23:21,800 --> 00:23:26,919 right order, the multipliers just sit in the matrix L. 297 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. 298 00:23:35,530 --> 00:23:41,370 I just keep a record of what those multipliers were, and that gives me L. 299 00:23:41,370 --> 00:23:52,610 So I'll draw the -- let me say it. 300 00:23:52,610 --> 00:23:56,320 So this is the A=L U. 301 00:23:56,320 --> 00:24:14,010 So if no row exchanges, the multipliers that those numbers that we multiplied rows by and 302 00:24:14,010 --> 00:24:27,550 subtracted, when we did an elimination step -- the multipliers go directly into L. 303 00:24:27,550 --> 00:24:29,950 Okay. 304 00:24:29,950 --> 00:24:42,770 So L is -- this is the way, to look at elimination. 305 00:24:42,770 --> 00:24:51,510 You go through the elimination steps, and actually if you do it right, you can throw 306 00:24:51,510 --> 00:24:57,120 away A as you create L U. 307 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, 308 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 309 00:25:16,540 --> 00:25:22,980 that you used -- which you have to save, and then you can forget A. 310 00:25:22,980 --> 00:25:25,919 So because it's all there in L and U. 311 00:25:25,919 --> 00:25:41,580 So that's -- this moment is maybe the new insight in elimination that comes from matrix 312 00:25:41,580 --> 00:25:44,070 -- doing it in matrix form. 313 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. 314 00:25:52,350 --> 00:25:56,350 The matrix E is not a particularly attractive one. 315 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, 316 00:26:02,130 --> 00:26:06,210 there the L comes out just right. Okay. 317 00:26:06,210 --> 00:26:09,580 Now -- oh gosh, so today's a sort of, 318 00:26:09,580 --> 00:26:14,160 like, practical day. 319 00:26:14,160 --> 00:26:20,670 Can we think together how expensive is elimination? 320 00:26:20,670 --> 00:26:24,570 How many operations do we do? 321 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 322 00:26:32,540 --> 00:26:34,980 it came. Here it comes. 323 00:26:34,980 --> 00:26:55,780 How many operations on an n by n matrix A. 324 00:26:55,780 --> 00:26:58,020 I mean, it's a very practical question. 325 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? 326 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? 327 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. 328 00:27:25,059 --> 00:27:28,700 I mean, we've put in more information. 329 00:27:28,700 --> 00:27:33,880 We make the whole thing is more accurate for the bigger matrix. 330 00:27:33,880 --> 00:27:39,000 But it's more expensive, too, and the question is how much more expensive? 331 00:27:39,000 --> 00:27:41,679 If I have matrices of order a hundred. 332 00:27:41,679 --> 00:27:43,150 Let's say a hundred by a hundred. 333 00:27:43,150 --> 00:27:47,660 Let me take n to be a hundred. 334 00:27:47,660 --> 00:27:50,840 Say n equal a hundred. 335 00:27:50,840 --> 00:27:54,860 How many steps are we doing? 336 00:27:54,860 --> 00:28:02,770 How many operations are we actually doing that we -- And let's suppose there aren't 337 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 338 00:28:07,800 --> 00:28:09,870 don't have to do those operations, and, 339 00:28:09,870 --> 00:28:13,500 it'll be much faster. 340 00:28:13,500 --> 00:28:21,940 But -- so just think for a moment about the first step. 341 00:28:21,940 --> 00:28:27,179 So here's our matrix A, hundred by a hundred. 342 00:28:27,180 --> 00:28:34,880 And the first step will be -- that column, is got zeroes down 343 00:28:34,880 --> 00:28:41,080 here. So it's down to 99 by 99, right? 344 00:28:41,080 --> 00:28:45,920 That's really like the first stage of elimination, 345 00:28:45,920 --> 00:28:48,480 to get from this hundred 346 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 347 00:28:55,600 --> 00:28:59,440 first row's okay the first column is okay. 348 00:28:59,440 --> 00:29:05,120 So, eventually -- how many steps did that take? 349 00:29:05,120 --> 00:29:07,400 You see, I'm trying to get an idea. 350 00:29:07,409 --> 00:29:11,860 Is the answer proportional to n? 351 00:29:11,860 --> 00:29:17,260 Is the total number of steps in elimination, the total number, is it proportional to n 352 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 353 00:29:22,520 --> 00:29:24,140 long? 354 00:29:24,140 --> 00:29:27,340 Does it square, so it would take me four times as long? 355 00:29:27,350 --> 00:29:30,600 Does it cube so it would take me eight times as long? 356 00:29:30,600 --> 00:29:37,120 Or is it n factorial, so it would take me a hundred times as long? 357 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 358 00:29:41,560 --> 00:29:44,280 cost, here. 359 00:29:44,280 --> 00:29:48,720 So these are the questions that I'm -- let me ask those questions again. 360 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 361 00:29:54,900 --> 00:29:56,420 power of n? 362 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 363 00:30:04,580 --> 00:30:07,980 and then by a hundred and two -- which is it? 364 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 365 00:30:14,320 --> 00:30:16,100 had to do. 366 00:30:16,100 --> 00:30:17,100 Okay. 367 00:30:17,100 --> 00:30:22,680 So what was the cost here? 368 00:30:22,680 --> 00:30:23,300 Well, let's see. 369 00:30:23,309 --> 00:30:25,700 What do I mean by an operation? 370 00:30:25,700 --> 00:30:32,549 I guess I mean, well an addition or -- yeah. 371 00:30:32,549 --> 00:30:33,549 No big deal. 372 00:30:33,549 --> 00:30:38,980 I guess I mean an addition or a subtraction or a multiplication or a division. 373 00:30:38,980 --> 00:30:41,540 Okay. 374 00:30:41,540 --> 00:30:48,980 And actually, what operation I doing all the time? 375 00:30:48,980 --> 00:30:58,870 When I multiply row one by multiplier L and I subtract from row six. 376 00:30:58,870 --> 00:31:01,060 What's happening there individually? 377 00:31:01,060 --> 00:31:03,460 What's going on? 378 00:31:03,460 --> 00:31:08,490 If I multiply -- I do a multiplication by L and then a subtraction. 379 00:31:08,490 --> 00:31:15,610 So I guess operation -- Can I count that for the moment as, like, one operation? 380 00:31:15,610 --> 00:31:18,520 Or you may want to count them separately. 381 00:31:18,520 --> 00:31:26,740 The typical operation is multiply plus a subtract. 382 00:31:26,740 --> 00:31:31,420 So if I count those together, my answer's going to come out 383 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 384 00:31:36,540 --> 00:31:39,000 number of multiplies, certain number of subtracts. 385 00:31:39,000 --> 00:31:40,580 That's really want to do. 386 00:31:40,580 --> 00:31:43,360 Okay. How many have I got here? 387 00:31:43,360 --> 00:31:49,500 So, I think -- let's see. 388 00:31:49,500 --> 00:31:56,293 It's about -- well, how many, roughly? 389 00:31:56,293 --> 00:31:59,352 How many operations to get from here to here? 390 00:31:59,360 --> 00:32:05,800 Well, maybe one way to look at it is all these numbers had to get changed. 391 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. 392 00:32:11,900 --> 00:32:25,880 So this step -- well, I guess maybe -- shall I say it cost about a hundred squared. 393 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, 394 00:32:32,110 --> 00:32:35,549 because -- because that's how many numbers are here. 395 00:32:35,549 --> 00:32:42,279 A hundred squared numbers is the total count of the entry, and all but this insignificant 396 00:32:42,279 --> 00:32:44,299 first row got changed. 397 00:32:44,300 --> 00:32:47,660 So I would say about a hundred squared. 398 00:32:47,660 --> 00:32:48,620 Okay. 399 00:32:48,620 --> 00:32:54,150 Now, what about the next step? 400 00:32:54,150 --> 00:32:56,960 So now the first row is fine. 401 00:32:56,960 --> 00:33:00,020 The second row is fine. 402 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? 403 00:33:06,700 --> 00:33:08,620 And then you're with me. 404 00:33:08,620 --> 00:33:10,380 Roughly, what's the cost? 405 00:33:10,390 --> 00:33:17,330 If this first step cost a hundred squared, about, operations then this one, which is 406 00:33:17,330 --> 00:33:27,490 really working on this guy to produce this, costs about what? 407 00:33:27,490 --> 00:33:31,250 How many operations to fix? 408 00:33:31,250 --> 00:33:36,610 About ninety-nine squared, or ninety-nine times ninety-eight. But less, right? 409 00:33:36,610 --> 00:33:39,590 Less, because our problem's getting smaller. 410 00:33:39,590 --> 00:33:42,280 About ninety-nine squared. 411 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 412 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. 413 00:33:53,750 --> 00:33:55,890 The big numbers are here. 414 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? 415 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 416 00:34:13,589 --> 00:34:23,040 down to three squared and two squared and even one squared. 417 00:34:23,040 --> 00:34:27,929 No way I should have written that -- squeezed that in. 418 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 419 00:34:37,570 --> 00:34:41,980 to one squared. 420 00:34:41,980 --> 00:34:44,899 That's a pretty decent count. 421 00:34:44,899 --> 00:34:56,040 Admittedly, we didn't catch every single tiny operation, but we got the right leading term 422 00:34:56,040 --> 00:34:57,040 here. 423 00:34:57,040 --> 00:35:01,180 And what do those add up to? 424 00:35:01,180 --> 00:35:10,030 Okay, so now we're coming to the punch of this, question, this operation count. 425 00:35:10,030 --> 00:35:19,820 So the operations on the left side, on the matrix A to finally get to U. 426 00:35:19,820 --> 00:35:28,540 And anybody -- so which of these quantities is the right ballpark for that count? 427 00:35:28,540 --> 00:35:34,080 If I add a hundred squared to ninety nine squared to ninety eight squared -- ninety 428 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? 429 00:35:44,020 --> 00:35:48,100 It's just one of these -- let's identify it first. 430 00:35:48,100 --> 00:35:49,020 Is it n? 431 00:35:49,020 --> 00:35:52,080 Certainly not. 432 00:35:52,080 --> 00:35:55,700 Is it n factorial? 433 00:35:55,700 --> 00:35:56,460 No. 434 00:35:56,460 --> 00:36:02,020 If it was n factorial, we would -- with determinants, it is n factorial. 435 00:36:02,020 --> 00:36:12,400 I'll put in a bad mark against determinants, because that -- okay, so what is it? 436 00:36:12,400 --> 00:36:18,580 It's n -- well, this is the answer. 437 00:36:18,580 --> 00:36:21,240 It's this order -- n cubed. 438 00:36:21,240 --> 00:36:26,380 It's like I have n terms, right? 439 00:36:26,380 --> 00:36:28,680 I've got n terms in this sum. 440 00:36:28,690 --> 00:36:31,140 And the biggest one is n squared. 441 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 442 00:36:40,750 --> 00:36:45,270 -- it's about one third of n cubed. 443 00:36:45,270 --> 00:36:53,690 That's the magic operation count. 444 00:36:53,690 --> 00:37:02,339 Somehow that one third takes account of the fact that the numbers are getting smaller. 445 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 446 00:37:06,900 --> 00:37:08,240 be exactly n cubed. 447 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. 448 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. 449 00:37:19,400 --> 00:37:25,940 So calculus can be mentioned, integration can be mentioned now in the next minute and 450 00:37:25,940 --> 00:37:28,360 not again for weeks. 451 00:37:28,360 --> 00:37:33,440 It's not that I don't like 18.01, but18.06 is better. 452 00:37:33,440 --> 00:37:42,340 Okay. So, -- so what's -- what's the calculus formula that looks like? 453 00:37:42,350 --> 00:37:50,589 It looks like -- if we were in calculus instead of summing stuff, we would integrate. 454 00:37:50,589 --> 00:37:56,800 So I would integrate x squared and I would get one third x 455 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 456 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 457 00:38:13,690 --> 00:38:14,900 whole point of calculus. 458 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 459 00:38:19,315 --> 00:38:21,280 know the whole point of calculus. 460 00:38:21,280 --> 00:38:27,990 Calculus is like sums except it's continuous. 461 00:38:27,990 --> 00:38:31,950 Okay. And algebra is discreet. 462 00:38:31,950 --> 00:38:32,950 Okay. 463 00:38:32,950 --> 00:38:35,010 So the answer is one third n cubed. 464 00:38:35,010 --> 00:38:40,080 Now I'll just -- let me say one more thing about operations. 465 00:38:40,080 --> 00:38:41,920 What about the right-hand side? 466 00:38:41,920 --> 00:38:45,250 This was what it cost on the left side. 467 00:38:45,250 --> 00:38:50,380 This is on A. 468 00:38:50,380 --> 00:38:52,180 Because this is A that we're working with. 469 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? 470 00:39:00,660 --> 00:39:07,570 So b costs a lot less, obviously, because it's just one column. 471 00:39:07,570 --> 00:39:14,260 We carry it through elimination and then actually we do back substitution. 472 00:39:14,260 --> 00:39:16,160 Let me just tell you the answer there. 473 00:39:16,160 --> 00:39:18,420 It's n squared. 474 00:39:18,420 --> 00:39:23,220 So the cost for every right hand side is n squared. 475 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. 476 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 477 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 478 00:39:57,670 --> 00:40:02,810 on A, but then we can process every right-hand side at low cost. 479 00:40:02,810 --> 00:40:03,859 Okay. 480 00:40:03,860 --> 00:40:16,560 So the -- We really have discussed the most fundamental algorithm for a system of equations. 481 00:40:16,560 --> 00:40:19,700 Okay. 482 00:40:19,700 --> 00:40:29,500 So, I'm ready to allow row exchanges. 483 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 484 00:40:34,220 --> 00:40:38,400 row exchanges? 485 00:40:38,400 --> 00:40:42,240 When would there be row exchanges? 486 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. 487 00:40:47,840 --> 00:40:55,820 So moving then into the final section of this chapter, which is about transposes -- well, 488 00:40:55,820 --> 00:41:08,640 we've already seen some transposes, and -- the title of this section is, 489 00:41:08,640 --> 00:41:13,700 "Transposes and Permutations." 490 00:41:13,700 --> 00:41:21,160 Okay. So can I say, now, where does a permutation come in? 491 00:41:21,160 --> 00:41:23,300 Let me talk a little about permutations. 492 00:41:23,300 --> 00:41:34,620 So that'll be up here, permutations. 493 00:41:34,620 --> 00:41:41,420 So these are the matrices that I need to do row exchanges. 494 00:41:41,420 --> 00:41:44,560 And I may have to do two row exchanges. 495 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 496 00:41:52,320 --> 00:41:54,180 out fine? 497 00:41:54,180 --> 00:42:01,460 Yeah let's just, for the heck of it -- so I'll put it here. 498 00:42:01,460 --> 00:42:04,120 Let me do three by threes. 499 00:42:04,120 --> 00:42:10,480 Actually, why don't I just plain list all the three by three permutation matrices. 500 00:42:10,480 --> 00:42:13,000 There're a nice little group of them. 501 00:42:13,010 --> 00:42:20,880 What are all the matrices that exchange no rows at all? 502 00:42:20,880 --> 00:42:26,070 Well, I'll include the identity. 503 00:42:26,070 --> 00:42:29,710 So that's a permutation matrix that doesn't do anything. 504 00:42:29,710 --> 00:42:38,680 Now what's the permutation matrix that exchanges -- what is P12? The permutation matrix that 505 00:42:38,680 --> 00:42:48,540 exchanges rows one and two would be -- 0 1 0 -- 1 0 0, right. 506 00:42:48,540 --> 00:42:53,360 I just exchanged those rows of the identity and I've got it. 507 00:42:53,360 --> 00:42:56,080 Okay. Actually, I'll -- yes. 508 00:42:56,080 --> 00:43:01,840 Let me clutter this up. 509 00:43:01,840 --> 00:43:06,340 Okay. Give me a complete list of all the row exchange matrices. 510 00:43:06,340 --> 00:43:07,540 So what are they? 511 00:43:07,540 --> 00:43:13,780 They're all the ways I can take the identity matrix and rearrange its rows. 512 00:43:13,780 --> 00:43:16,620 How many will there be? 513 00:43:16,620 --> 00:43:21,820 How many three by three permutation matrices? 514 00:43:21,820 --> 00:43:24,100 Shall we keep going and get the answer? 515 00:43:24,100 --> 00:43:27,500 So tell me some more. 516 00:43:27,500 --> 00:43:28,580 STUDENT: Zero one – 517 00:43:28,580 --> 00:43:31,000 STRANG: Zero – What one are you going to do now? 518 00:43:31,000 --> 00:43:32,840 STUDENT: I'm going to switch the – 519 00:43:32,840 --> 00:43:43,520 STRANG: Switch rows one and -- One and three, okay. One and three, leaving two alone. 520 00:43:43,520 --> 00:43:44,460 Okay. 521 00:43:44,460 --> 00:43:50,780 Now what else? Switch -- what would be the next easy one -- is switch two and 522 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 523 00:43:57,250 --> 00:44:00,329 up and number two down. 524 00:44:00,329 --> 00:44:05,380 Okay. Those are the ones that just exchange single -- a pair of 525 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. 526 00:44:13,340 --> 00:44:16,080 What's left? 527 00:44:16,080 --> 00:44:19,640 So tell -- there is another one here. 528 00:44:19,640 --> 00:44:22,260 What's that? 529 00:44:22,260 --> 00:44:26,440 It's going to move -- it's going to change all rows, right? 530 00:44:26,440 --> 00:44:28,160 Where shall we put them? 531 00:44:28,160 --> 00:44:29,780 So -- give me a first row. 532 00:44:29,780 --> 00:44:30,800 STUDENT: Zero one zero? 533 00:44:30,800 --> 00:44:32,740 STRANG: Zero one zero. 534 00:44:32,740 --> 00:44:38,000 Okay, now a second row -- say zero zero one and the third guy 535 00:44:38,000 --> 00:44:41,960 One zero zero. 536 00:44:41,960 --> 00:44:44,420 So that is like a cycle. 537 00:44:44,420 --> 00:44:49,800 That puts row two moves up to row one, row three moves up to 538 00:44:49,800 --> 00:44:53,420 row two and row one moves down to row three. 539 00:44:53,420 --> 00:45:00,180 And there's one more, which is -- let's see. 540 00:45:00,180 --> 00:45:03,280 What's left? 541 00:45:03,280 --> 00:45:04,160 I'm lost. 542 00:45:04,160 --> 00:45:05,240 STUDENT: Is it zero zero one? 543 00:45:05,240 --> 00:45:06,780 STRANG: Is it zero zero one? Okay. 544 00:45:06,780 --> 00:45:08,120 STUDENT: One zero zero. 545 00:45:08,120 --> 00:45:11,260 STRANG: One zero zero, okay. 546 00:45:11,260 --> 00:45:16,460 Zero one zero, okay. 547 00:45:16,460 --> 00:45:17,980 Great. 548 00:45:17,980 --> 00:45:21,780 Six. Six of them. 549 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? 550 00:45:35,580 --> 00:45:42,100 If I multiply two of these matrices together, what can you tell me about the answer? 551 00:45:42,100 --> 00:45:44,360 It's on the list. 552 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 553 00:45:49,220 --> 00:45:50,520 done row exchanges. 554 00:45:50,520 --> 00:45:54,520 So if I multiply -- but, I don't know. 555 00:45:54,520 --> 00:46:00,340 And if I invert, then I'm just doing row exchanges to get back again. 556 00:46:00,349 --> 00:46:01,869 So the inverses are all there. 557 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 558 00:46:12,790 --> 00:46:14,400 inside this group. 559 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. 560 00:46:18,920 --> 00:46:24,320 It's a group of six matrices, and what about the inverses? 561 00:46:24,320 --> 00:46:28,160 What's the inverse of this guy, for example? 562 00:46:28,160 --> 00:46:34,700 What's the inverse -- if I exchange rows one and two, what's the inverse matrix? 563 00:46:34,700 --> 00:46:36,760 Just tell me fast. 564 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 565 00:46:46,910 --> 00:46:51,150 get back to where I started is the same thing. 566 00:46:51,150 --> 00:46:54,905 So this thing is its own inverse. 567 00:46:54,905 --> 00:46:55,905 That's probably its own inverse. 568 00:46:55,905 --> 00:47:00,380 This is probably not -- actually, I think these are inverses of each other. 569 00:47:00,380 --> 00:47:06,320 Oh, yeah, actually -- the inverse is the transpose. 570 00:47:06,320 --> 00:47:16,020 There's a curious fact about permutations matrices, that the inverses are the transposes. 571 00:47:16,020 --> 00:47:21,820 And final moment -- how many are there if I -- how many four by four permutations? 572 00:47:21,820 --> 00:47:29,880 So let me take four by four -- how many Ps? 573 00:47:29,880 --> 00:47:32,720 Well, okay. 574 00:47:32,720 --> 00:47:34,940 Make a good guess. 575 00:47:34,940 --> 00:47:38,020 Twenty four, right. Twenty four Ps. 576 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. 577 00:47:48,820 --> 00:47:51,660 So the next lecture, finishes Chapter 2 578 00:47:51,660 --> 00:47:55,160 and moves to Chapter 3. 579 00:47:55,160 --> 00:47:57,480 Thank you.