1 00:00:07,730 --> 00:00:11,436 OK, this lecture is like the beginning 2 00:00:11,436 --> 00:00:13,060 of the second half of this is to prove. 3 00:00:13,060 --> 00:00:18,590 this course because up to now we paid a lot of attention 4 00:00:18,590 --> 00:00:22,240 to rectangular matrices. 5 00:00:22,240 --> 00:00:27,020 Now, concentrating on square matrices, 6 00:00:27,020 --> 00:00:29,150 so we're at two big topics. 7 00:00:29,150 --> 00:00:32,210 The determinant of a square matrix, 8 00:00:32,210 --> 00:00:35,040 so this is the first lecture in that new chapter 9 00:00:35,040 --> 00:00:38,940 on determinants, and the reason, the big reason 10 00:00:38,940 --> 00:00:43,030 we need the determinants is for the Eigen values. 11 00:00:43,030 --> 00:00:46,420 So this is really determinants and Eigen values, 12 00:00:46,420 --> 00:00:50,330 the next big, big chunk of 18.06. 13 00:00:50,330 --> 00:00:57,100 OK, so the determinant is a number associated 14 00:00:57,100 --> 00:01:01,360 with every square matrix, so every square matrix 15 00:01:01,360 --> 00:01:07,160 has this number associated with called the, its determinant. 16 00:01:07,160 --> 00:01:15,720 I'll often write it as D E T A or often also I'll write it as, 17 00:01:15,720 --> 00:01:19,440 A with vertical bars, so that's going to mean 18 00:01:19,440 --> 00:01:21,030 the determinant of the matrix. 19 00:01:21,030 --> 00:01:26,400 That's going to mean this one, like, magic number. 20 00:01:26,400 --> 00:01:32,420 Well, one number can't tell you what the whole matrix was. 21 00:01:32,420 --> 00:01:36,960 But this one number, just packs in as much information 22 00:01:36,960 --> 00:01:39,310 as possible into a single number, 23 00:01:39,310 --> 00:01:43,250 and of course the one fact that you've seen before 24 00:01:43,250 --> 00:01:48,280 and we have to see it again is the matrix 25 00:01:48,280 --> 00:01:53,680 is invertible when the determinant is not zero. 26 00:01:53,680 --> 00:01:58,260 The matrix is singular when the determinant is zero. 27 00:01:58,260 --> 00:02:03,590 So the determinant will be a test for invertibility, 28 00:02:03,590 --> 00:02:07,430 but the determinant's got a lot more to it than that, 29 00:02:07,430 --> 00:02:08,889 so let me start. 30 00:02:08,889 --> 00:02:12,260 OK, now the question is how to start. 31 00:02:12,260 --> 00:02:14,850 Do I give you a big formula for the determinant, 32 00:02:14,850 --> 00:02:16,600 all in one gulp? 33 00:02:16,600 --> 00:02:18,190 I don't think so! 34 00:02:18,190 --> 00:02:21,410 That big formula has got too much packed in it. 35 00:02:21,410 --> 00:02:28,900 I would rather start with three properties of the determinant, 36 00:02:28,900 --> 00:02:31,150 three properties that it has. 37 00:02:31,150 --> 00:02:33,990 And let me tell you property one. 38 00:02:33,990 --> 00:02:39,380 The determinant of the identity is one. 39 00:02:39,380 --> 00:02:40,670 OK. 40 00:02:40,670 --> 00:02:41,710 I... 41 00:02:41,710 --> 00:02:43,540 I wish the other two properties were 42 00:02:43,540 --> 00:02:46,810 as easy to tell you as that. 43 00:02:46,810 --> 00:02:51,490 But actually the second property is pretty straightforward too, 44 00:02:51,490 --> 00:02:56,050 and then once we get the third we will actually 45 00:02:56,050 --> 00:02:57,780 have the determinant. 46 00:02:57,780 --> 00:03:02,240 Those three properties define the determinant and we can -- 47 00:03:02,240 --> 00:03:13,610 then we can figure out, well, what is the determinant? 48 00:03:13,610 --> 00:03:16,880 What's a formula for it? 49 00:03:16,880 --> 00:03:22,000 Now, the second property is what happens if you 50 00:03:22,000 --> 00:03:24,620 exchange two rows of a matrix. 51 00:03:24,620 --> 00:03:27,040 What happens to the determinant? 52 00:03:27,040 --> 00:03:31,000 So, property two is exchange rows, 53 00:03:31,000 --> 00:03:46,390 reverse the sign of the determinant. 54 00:03:51,470 --> 00:03:53,360 A lot of plus and minus signs. 55 00:03:53,360 --> 00:03:56,150 I even wrote here, "plus and minus signs," 56 00:03:56,150 --> 00:03:57,880 because this is, like, that's what 57 00:03:57,880 --> 00:04:00,570 you have to pay attention to in the formulas 58 00:04:00,570 --> 00:04:03,080 and properties of determinants. 59 00:04:03,080 --> 00:04:08,200 So that -- you see what I mean by a property here? 60 00:04:08,200 --> 00:04:10,890 I haven't yet told you what the determinant is, 61 00:04:10,890 --> 00:04:13,890 but whatever it is, if I exchange 62 00:04:13,890 --> 00:04:17,910 two rows to get a different matrix that reverses 63 00:04:17,910 --> 00:04:20,950 the sign of the determinant. 64 00:04:20,950 --> 00:04:25,180 And so now, actually, what matrices 65 00:04:25,180 --> 00:04:28,140 do we now know the determinant of? 66 00:04:28,140 --> 00:04:32,260 From one and two, I now know the determinant. 67 00:04:32,260 --> 00:04:34,640 Well, I certainly know the determinant of the identity 68 00:04:34,640 --> 00:04:37,210 matrix and now I know the determinant 69 00:04:37,210 --> 00:04:41,970 of every other matrix that comes from row exchanges 70 00:04:41,970 --> 00:04:44,540 from the identities still. 71 00:04:44,540 --> 00:04:48,460 So what matrices have I gotten at this point? 72 00:04:48,460 --> 00:04:50,500 The permutations, right. 73 00:04:50,500 --> 00:04:55,020 At this point I know every permutation matrix, 74 00:04:55,020 --> 00:04:58,730 so now I'm saying the determinant of a permutation 75 00:04:58,730 --> 00:05:03,690 matrix is one or minus one. 76 00:05:03,690 --> 00:05:09,320 One or minus one, depending whether the number of exchanges 77 00:05:09,320 --> 00:05:16,720 was even or the number of exchanges was odd. 78 00:05:16,720 --> 00:05:19,100 So this is the determinant of a permutation. 79 00:05:19,100 --> 00:05:22,720 Now, P is back to standing for permutation. 80 00:05:22,720 --> 00:05:23,560 OK. 81 00:05:23,560 --> 00:05:28,180 if I could carry on this board, I could, like, 82 00:05:28,180 --> 00:05:29,940 do the two-by-two's. 83 00:05:29,940 --> 00:05:34,790 So, property one tells me that this two-by-two matrix. 84 00:05:34,790 --> 00:05:37,560 Oh, I better write absolute -- 85 00:05:37,560 --> 00:05:41,150 I mean, I'd better write vertical bars, not brackets, 86 00:05:41,150 --> 00:05:43,770 for that determinant. 87 00:05:43,770 --> 00:05:46,890 Property one said, in the two-by-two case, 88 00:05:46,890 --> 00:05:51,700 that this matrix has determinant one. 89 00:05:51,700 --> 00:05:59,130 Property two tells me that this matrix has determinant -- 90 00:05:59,130 --> 00:06:00,500 what? 91 00:06:00,500 --> 00:06:02,740 Negative one. 92 00:06:02,740 --> 00:06:04,530 This is, like, two-by-twos. 93 00:06:04,530 --> 00:06:08,090 Now, I finally want to get -- 94 00:06:08,090 --> 00:06:10,210 well, ultimately I want to get to, 95 00:06:10,210 --> 00:06:13,130 the formula that we all know. 96 00:06:13,130 --> 00:06:16,600 Let me put that way over here, that the determinant 97 00:06:16,600 --> 00:06:23,190 of a general two-by-two is ad-bc. 98 00:06:23,190 --> 00:06:23,690 OK. 99 00:06:23,690 --> 00:06:24,070 I'm going to leave that up, like, as the two by two case 100 00:06:24,070 --> 00:06:24,190 I'm down to the product of the diagonal and if I transpose, 101 00:06:24,190 --> 00:06:26,065 that we already know, so that every property, 102 00:06:26,065 --> 00:06:44,980 I can, like, check that it's correct for two-by-twos. 103 00:06:44,980 --> 00:06:47,960 But the whole point of these properties 104 00:06:47,960 --> 00:06:51,850 is that they're going to give me a formula for n-by-n. 105 00:06:51,850 --> 00:06:53,760 That's the whole point. 106 00:06:53,760 --> 00:06:57,060 They're going to give me this number that's 107 00:06:57,060 --> 00:07:00,460 a test for invertibility and other great properties 108 00:07:00,460 --> 00:07:02,240 for any size matrix. 109 00:07:02,240 --> 00:07:08,500 OK, now you see I'm like, slowing down because property 110 00:07:08,500 --> 00:07:13,220 three is the key property. 111 00:07:13,220 --> 00:07:17,570 Property three is the key property and can I somehow 112 00:07:17,570 --> 00:07:22,500 describe it -- maybe I'll separate it into 3A I said that 113 00:07:22,500 --> 00:07:26,610 if you do a row exchange, the determinant and 3B. 114 00:07:26,610 --> 00:07:30,750 Property 3A says that if I multiply one of the rows, 115 00:07:30,750 --> 00:07:40,020 say the first row, by a number T -- 116 00:07:40,020 --> 00:07:41,590 I'm going to erase that. 117 00:07:41,590 --> 00:07:46,150 That's, like, what I'm headed for but I'm not there yet. 118 00:07:46,150 --> 00:07:49,890 It's the one we know and you'll see that it's 119 00:07:49,890 --> 00:07:54,990 checked out by each property. 120 00:07:54,990 --> 00:08:00,220 OK, so this is for any matrix. 121 00:08:00,220 --> 00:08:03,990 For any matrix, if I multiply one row by T 122 00:08:03,990 --> 00:08:08,740 and leave the other row or other n-1 rows alone, 123 00:08:08,740 --> 00:08:11,220 what happens to the determinant? 124 00:08:11,220 --> 00:08:15,610 The factor T comes out. 125 00:08:15,610 --> 00:08:17,260 It's T times this determinant. 126 00:08:22,200 --> 00:08:23,230 That's not hard. 127 00:08:23,230 --> 00:08:25,560 I shouldn't have made a big deal out of property 3A, 128 00:08:25,560 --> 00:08:29,630 and 3B is that, if is, is if I keep -- 129 00:08:29,630 --> 00:08:33,640 I'm always keeping this second row the same, 130 00:08:33,640 --> 00:08:37,929 or that last n-1 rows are all staying the same. 131 00:08:37,929 --> 00:08:39,460 I'm just working -- 132 00:08:39,460 --> 00:08:43,679 I'm just looking inside the first row and if I have 133 00:08:43,679 --> 00:08:53,770 an a+a' there and a b+b' there -- 134 00:08:53,770 --> 00:08:54,480 sorry, I didn't. 135 00:08:54,480 --> 00:08:55,150 Ahh. 136 00:08:55,150 --> 00:08:59,690 Why don't -- I'll use an eraser, do it right. 137 00:08:59,690 --> 00:09:00,190 b+b' there. 138 00:09:00,190 --> 00:09:02,390 You see what I'm doing? 139 00:09:02,390 --> 00:09:06,730 This property and this property are about linear combinations, 140 00:09:06,730 --> 00:09:13,720 of the first row only, leaving the other rows unchanged. 141 00:09:13,720 --> 00:09:14,680 They'll copy along. 142 00:09:14,680 --> 00:09:20,320 Then, then I get the sum -- this breaks up into the sum of this 143 00:09:20,320 --> 00:09:23,500 determinant and this one. 144 00:09:32,740 --> 00:09:34,120 I'm putting up formulas. 145 00:09:34,120 --> 00:09:36,850 Maybe I can try to say it in words. 146 00:09:36,850 --> 00:09:41,110 The determinant is a linear function. 147 00:09:41,110 --> 00:09:48,170 It behaves like a linear function of first row 148 00:09:48,170 --> 00:09:51,540 if all the other rows stay the same. 149 00:09:51,540 --> 00:09:54,340 I not saying that -- 150 00:09:54,340 --> 00:09:55,640 let me emphasize. 151 00:09:55,640 --> 00:10:00,860 I not saying that the determinant of A plus B 152 00:10:00,860 --> 00:10:07,870 is determinant of A plus determinant of B. 153 00:10:07,870 --> 00:10:09,180 I not saying that. 154 00:10:09,180 --> 00:10:11,490 I'd better -- can I -- 155 00:10:11,490 --> 00:10:15,620 how do I get it onto tape that I'm not saying that? 156 00:10:15,620 --> 00:10:22,350 You see, this would allow all the rows -- you know, 157 00:10:22,350 --> 00:10:24,830 A to have a bunch of rows, B to have a bunch of rows. 158 00:10:24,830 --> 00:10:29,510 That's not the linearity I'm after. 159 00:10:29,510 --> 00:10:32,840 I'm only after linearity in each row. 160 00:10:32,840 --> 00:10:38,030 Linear for each row. 161 00:10:40,840 --> 00:10:44,990 Well, you may say I only talked about the first row, 162 00:10:44,990 --> 00:10:48,610 but I claim it's also linear in the second row, 163 00:10:48,610 --> 00:10:53,520 if I had this -- but not, I can't, I can't have 164 00:10:53,520 --> 00:10:56,320 a combination in both first and second. 165 00:10:56,320 --> 00:10:58,830 If I had a combination in the second row, 166 00:10:58,830 --> 00:11:04,230 then I could use rule two to put it up in the first row, 167 00:11:04,230 --> 00:11:09,560 use my property and then use rule two again to put it back, 168 00:11:09,560 --> 00:11:14,340 so each row is OK, not only the first row, 169 00:11:14,340 --> 00:11:17,370 but each row separately. 170 00:11:17,370 --> 00:11:19,760 OK, those are the three properties, 171 00:11:19,760 --> 00:11:23,030 and from those properties, so that's 172 00:11:23,030 --> 00:11:26,660 properties one, two, three. 173 00:11:26,660 --> 00:11:31,584 From those, I want to get all -- 174 00:11:31,584 --> 00:11:33,750 I'm going to learn a lot more about the determinant. 175 00:11:36,490 --> 00:11:38,030 Let me take an example. 176 00:11:38,030 --> 00:11:39,990 What would I like to learn? 177 00:11:39,990 --> 00:11:43,280 I would like to learn that -- so here's our property four. 178 00:11:43,280 --> 00:11:47,310 Let me use the same numbering as here. 179 00:11:47,310 --> 00:11:58,720 Property four is if two rows are equal, the determinant is zero. 180 00:11:58,720 --> 00:12:02,720 OK, so property four. 181 00:12:02,720 --> 00:12:13,570 Two equal rows lead to determinant equals zero. 182 00:12:13,570 --> 00:12:14,280 Right. 183 00:12:14,280 --> 00:12:19,360 Now, of course I can -- in the two-by-two case I can check, 184 00:12:19,360 --> 00:12:23,980 sure, the determinant of ab ab comes out zero. 185 00:12:23,980 --> 00:12:28,950 But I want to see why it's true for n-by-n. 186 00:12:28,950 --> 00:12:36,100 Suppose row one equals row three for a seven-by-seven matrix. 187 00:12:36,100 --> 00:12:39,070 So two rows are the same in a big matrix. 188 00:12:39,070 --> 00:12:43,350 And all I have to work with is these properties. 189 00:12:43,350 --> 00:12:47,540 The exchange property, which flips the sign, 190 00:12:47,540 --> 00:12:53,580 and the linearity property which works in each row separately. 191 00:12:53,580 --> 00:12:57,170 OK, can you see the reason? 192 00:12:57,170 --> 00:13:02,200 How do I get this one out of properties one, two, three? 193 00:13:02,200 --> 00:13:04,680 Because -- that's all I have to work with. 194 00:13:04,680 --> 00:13:07,770 Everything has to come from properties one, two, three. 195 00:13:07,770 --> 00:13:12,040 OK, so suppose I have a matrix, and two rows are even. 196 00:13:14,770 --> 00:13:16,310 How do I see that its determinant 197 00:13:16,310 --> 00:13:22,120 has to be zero from these properties? 198 00:13:22,120 --> 00:13:24,490 I do an exchange. 199 00:13:24,490 --> 00:13:27,190 Property two is just set up for this. 200 00:13:27,190 --> 00:13:28,610 Use property two. 201 00:13:28,610 --> 00:13:33,090 Use exchange -- exchange rows. 202 00:13:33,090 --> 00:13:40,720 Exchange those rows, and I get the same matrix. 203 00:13:40,720 --> 00:13:42,700 Of course, because those rows were equal. 204 00:13:47,870 --> 00:13:50,580 So the determinant didn't change. 205 00:13:50,580 --> 00:13:52,360 But on the other hand, property two 206 00:13:52,360 --> 00:13:56,170 says that the sign did change. 207 00:13:56,170 --> 00:13:59,350 So the -- so I, I have a determinant whose sign 208 00:13:59,350 --> 00:14:03,390 doesn't change and does change, and the only possibility then 209 00:14:03,390 --> 00:14:06,520 is that the determinant is zero. 210 00:14:06,520 --> 00:14:08,780 You see the reasoning there? 211 00:14:08,780 --> 00:14:09,550 Straightforward. 212 00:14:09,550 --> 00:14:15,250 Property two just told us, hey, if we've got two equal rows we. 213 00:14:15,250 --> 00:14:19,210 we've got a zero determinant. 214 00:14:19,210 --> 00:14:22,140 And of course in our minds, that matches the fact 215 00:14:22,140 --> 00:14:26,550 that if I have two equal rows the matrix isn't invertible. 216 00:14:26,550 --> 00:14:29,430 If I have two equal rows, I know that the rank 217 00:14:29,430 --> 00:14:31,190 is less changes sign. than n. 218 00:14:31,190 --> 00:14:34,540 OK, ready for property five. 219 00:14:34,540 --> 00:14:38,900 Now, property five you'll recognize as P. 220 00:14:38,900 --> 00:14:45,710 It says that the elimination step that I'm always doing, 221 00:14:45,710 --> 00:14:51,850 or U and U transposed, when they're triangular,4 subtract 222 00:14:51,850 --> 00:15:00,560 a multiple, subtract some multiple l times row one from 223 00:15:00,560 --> 00:15:04,800 another row, row k, let's say. 224 00:15:07,950 --> 00:15:10,760 You remember why I did that. 225 00:15:10,760 --> 00:15:14,620 In elimination I'm always choosing this multiplier so as 226 00:15:14,620 --> 00:15:17,190 to produce zero in that position. 227 00:15:17,190 --> 00:15:21,050 What I -- way, way back in property two,4 228 00:15:21,050 --> 00:15:24,230 Or row I from row k, maybe I should just 229 00:15:24,230 --> 00:15:28,250 make very clear that there's nothing special about row one 230 00:15:28,250 --> 00:15:30,530 here. 231 00:15:30,530 --> 00:15:34,400 OK, so that, you can see why I want that who cares? 232 00:15:34,400 --> 00:15:37,350 one, because that will allow me to start 233 00:15:37,350 --> 00:15:40,890 with this full matrix whose determinant I don't know, 234 00:15:40,890 --> 00:15:45,590 and I can do elimination and clean it out. 235 00:15:45,590 --> 00:15:47,830 I can get zeroes below the diagonal 236 00:15:47,830 --> 00:15:50,970 by these elimination steps and the point 237 00:15:50,970 --> 00:15:58,420 is that the determinant, the determinant doesn't change. 238 00:16:10,210 --> 00:16:12,450 So all those steps of elimination 239 00:16:12,450 --> 00:16:15,070 are OK not changing the determinant. 240 00:16:15,070 --> 00:16:18,580 In our language in the early chapter the determinant of A is 241 00:16:18,580 --> 00:16:20,900 So if I do seven row exchanges, the determinant 242 00:16:20,900 --> 00:16:23,989 changes sign, going to be the same as the determinant of U, 243 00:16:23,989 --> 00:16:25,030 the upper triangular one. 244 00:16:25,030 --> 00:16:27,090 It just has the pivots on the diagonal. 245 00:16:27,090 --> 00:16:29,100 That's why we'll want this property. 246 00:16:29,100 --> 00:16:31,250 OK, do you see where that property's coming from? 247 00:16:34,180 --> 00:16:37,110 Let me do the two-by-two case. 248 00:16:37,110 --> 00:16:39,490 Let me do the two-by-two case here. 249 00:16:39,490 --> 00:16:44,340 So, I'll keep property five going along. 250 00:16:44,340 --> 00:16:45,300 So what I doing? 251 00:16:45,300 --> 00:16:46,890 I'm going to keep -- 252 00:16:46,890 --> 00:16:52,910 I'm going to have ab cd, but I'm going 253 00:16:52,910 --> 00:16:57,460 to subtract l times the first row from the second row. 254 00:16:57,460 --> 00:17:03,182 And that's the determinant and of 255 00:17:03,182 --> 00:17:03,890 OK, that's not -- 256 00:17:03,890 --> 00:17:06,260 I didn't put in every comma and, course 257 00:17:06,260 --> 00:17:10,900 I can multiply that out and figure out, sure enough, ad-bc 258 00:17:10,900 --> 00:17:17,849 is there and this minus ALB plus ALB cancels out, 259 00:17:17,849 --> 00:17:19,569 but I just cheated, 260 00:17:19,569 --> 00:17:20,190 right? 261 00:17:20,190 --> 00:17:21,760 I've got to use the properties. 262 00:17:21,760 --> 00:17:22,510 So what property? 263 00:17:22,510 --> 00:17:24,599 Well, of course, this is a com -- 264 00:17:24,599 --> 00:17:28,420 I'm keeping the first row the same and the second row 265 00:17:28,420 --> 00:17:31,690 has a c and a d, and then there's 266 00:17:31,690 --> 00:17:36,190 the determinant of the A and the B, and the minus LA, 267 00:17:36,190 --> 00:17:37,240 and the minus LB. 268 00:17:41,840 --> 00:17:44,150 So what property was that? 269 00:17:44,150 --> 00:17:46,420 3B. 270 00:17:46,420 --> 00:17:49,340 I kept one row the same and I had 271 00:17:49,340 --> 00:17:52,690 a combination in the second, in the other row, 272 00:17:52,690 --> 00:17:56,520 and I just separated it out. 273 00:17:56,520 --> 00:17:59,300 OK, so that's property 3. 274 00:17:59,300 --> 00:18:03,350 That's by number 3, 3B if you like. 275 00:18:03,350 --> 00:18:04,980 OK, now use 3A. 276 00:18:04,980 --> 00:18:10,480 How do you use 3A, which says I can factor out an l, 277 00:18:10,480 --> 00:18:13,110 I can factor out a minus l here. 278 00:18:13,110 --> 00:18:17,140 I can factor a minus l out from this row, no problem. 279 00:18:17,140 --> 00:18:19,000 That was 3A. 280 00:18:19,000 --> 00:18:25,390 So now I've used property three and now I'm ready for the kill. 281 00:18:25,390 --> 00:18:32,070 Property four says that this determinant is zero, 282 00:18:32,070 --> 00:18:34,810 has two equal rows. 283 00:18:34,810 --> 00:18:37,100 You see how that would always work? 284 00:18:37,100 --> 00:18:40,280 I subtract a multiple of one row from another one. 285 00:18:40,280 --> 00:18:46,860 It gives me a combination in row k of the old row and l times 286 00:18:46,860 --> 00:18:51,310 this copy of the higher row, and then if -- 287 00:18:51,310 --> 00:18:53,550 since I have two equal rows, that's zero, 288 00:18:53,550 --> 00:18:56,990 so the determinant after elimination is the same 289 00:18:56,990 --> 00:18:58,280 as before. 290 00:18:58,280 --> 00:18:59,430 Good. 291 00:18:59,430 --> 00:19:00,310 OK. 292 00:19:00,310 --> 00:19:04,170 Now, let's see -- if I rescue my glasses, 293 00:19:04,170 --> 00:19:07,140 I can see what's property six. 294 00:19:07,140 --> 00:19:11,700 Oh, six is easy, plenty of space. 295 00:19:11,700 --> 00:19:22,450 Row of zeroes leads to determinant of A equals zero. 296 00:19:26,840 --> 00:19:28,380 A complete row of zeroes. 297 00:19:28,380 --> 00:19:32,550 So I'm again, this is like, something 298 00:19:32,550 --> 00:19:34,880 I'll use in the singular case. 299 00:19:34,880 --> 00:19:39,380 Actually, you can look ahead to why I need these properties. 300 00:19:39,380 --> 00:19:42,220 So I'm going to use property five, the elimination, 301 00:19:42,220 --> 00:19:45,980 use this stuff to say that this determinant is 302 00:19:45,980 --> 00:19:49,600 the same as that determinant and I'll produce a zero there. 303 00:19:49,600 --> 00:19:51,910 But what if I also produce a zero there? 304 00:19:51,910 --> 00:19:54,620 What if elimination gives a row of zeroes? 305 00:19:54,620 --> 00:19:59,120 That, that used to be my test for, mmm, singular, 306 00:19:59,120 --> 00:20:03,220 not invertible, rank two -- rank less than N, 307 00:20:03,220 --> 00:20:07,120 and now I'm seeing it's also gives determinant zero. 308 00:20:07,120 --> 00:20:12,600 How do I get that one from the previous properties? 309 00:20:12,600 --> 00:20:15,006 'Cause I -- this is not a new law, 310 00:20:15,006 --> 00:20:16,630 this has got to come from the old ones. 311 00:20:16,630 --> 00:20:20,635 So what shall I do? 312 00:20:23,210 --> 00:20:24,690 Yeah, use -- that's brilliant. 313 00:20:24,690 --> 00:20:26,430 If you use 3A with T equals zero. 314 00:20:26,430 --> 00:20:27,350 Right. 315 00:20:27,350 --> 00:20:32,760 So I have this zero zero cd, and I'm 316 00:20:32,760 --> 00:20:35,850 trying to show that that determinant is zero. triangular 317 00:20:35,850 --> 00:20:37,560 matrices, l and l transposed, 318 00:20:37,560 --> 00:20:41,000 OK, so the zero is the same is -- five, 319 00:20:41,000 --> 00:20:45,900 can I take T equals five, just to, like, pin it down? 320 00:20:45,900 --> 00:20:48,530 I'll multiply this row by five. 321 00:20:48,530 --> 00:20:52,780 Five, well then, five of this should -- if I, 322 00:20:52,780 --> 00:21:01,300 if there's a factor five in that, you see what -- 323 00:21:01,300 --> 00:21:05,500 so this is property 3A, with taking T as five. 324 00:21:05,500 --> 00:21:08,660 If I multiply a row by five, out comes a five. 325 00:21:08,660 --> 00:21:14,450 So why I doing this? 326 00:21:14,450 --> 00:21:19,320 Oh, because that's still zero zero, right? 327 00:21:19,320 --> 00:21:21,000 So that's this determinant equals 328 00:21:21,000 --> 00:21:28,780 five times this determinant, and the determinant has to be zero. 329 00:21:28,780 --> 00:21:34,100 I think I didn't do that the very best way. 330 00:21:34,100 --> 00:21:36,670 You were, yeah, you had the idea better. 331 00:21:36,670 --> 00:21:40,620 Multiply, yeah, take T equals zero. 332 00:21:40,620 --> 00:21:44,170 Was that your idea? 333 00:21:44,170 --> 00:21:46,840 Take T equals zero in rule 3B. 334 00:21:46,840 --> 00:21:51,990 If T is zero in rule 3B, and I bring the camera back to rule 335 00:21:51,990 --> 00:21:52,720 3B -- 336 00:21:52,720 --> 00:21:55,260 sorry. 337 00:21:55,260 --> 00:22:01,300 If T is zero, then I have a zero zero there 338 00:22:01,300 --> 00:22:03,330 and the determinant is zero. 339 00:22:03,330 --> 00:22:08,710 OK, one way or another, a row of zeroes means zero determinant. 340 00:22:08,710 --> 00:22:14,930 OK, now I have to get serious. 341 00:22:14,930 --> 00:22:20,500 The next properties are the ones that we're building up to. 342 00:22:20,500 --> 00:22:23,750 OK, so I can do elimination. 343 00:22:23,750 --> 00:22:26,540 I can reduce to a triangular matrix 344 00:22:26,540 --> 00:22:30,120 and now what's the determinant of that triangular matrix? 345 00:22:30,120 --> 00:22:34,100 OK, so they had to wait until the last minute. 346 00:22:34,100 --> 00:22:37,280 Suppose, suppose I -- all right, rule seven. 347 00:22:37,280 --> 00:22:42,430 So suppose my matrix is now triangular. 348 00:22:42,430 --> 00:22:44,660 So it's got -- 349 00:22:44,660 --> 00:22:48,870 so I even give these the names of the pivots, d1, d2, to dn, 350 00:22:48,870 --> 00:22:54,710 and stuff is up here, I don't know what that is, 351 00:22:54,710 --> 00:22:57,970 but what I do know is this is all zeroes. 352 00:22:57,970 --> 00:23:04,750 That's all zeroes, and I would like to know the determinant, 353 00:23:04,750 --> 00:23:08,370 because elimination will get me to this. 354 00:23:08,370 --> 00:23:11,610 So once I'm here, what's the determinant then? 355 00:23:11,610 --> 00:23:16,560 Let me use an eraser to get those, get that vertical bar 356 00:23:16,560 --> 00:23:22,350 again, so that I'm taking the determinant of U so that, so, 357 00:23:22,350 --> 00:23:28,510 what is the determinant of an upper triangular matrix? 358 00:23:28,510 --> 00:23:33,215 Do you know the answer? 359 00:23:36,090 --> 00:23:40,220 It's just the product of the d's. for it. 360 00:23:40,220 --> 00:23:44,740 The -- these things that I don't even put in letters 361 00:23:44,740 --> 00:23:53,020 for, because they don't matter, the determinant is d1 times d2 362 00:23:53,020 --> 00:23:54,197 times dn. 363 00:23:57,120 --> 00:24:02,100 If I have a triangular matrix, then the diagonal 364 00:24:02,100 --> 00:24:04,770 is all I have to work with. 365 00:24:04,770 --> 00:24:06,550 And that's, that's telling us then. 366 00:24:06,550 --> 00:24:15,120 We now have the way that MATLAB, any reasonable software, 367 00:24:15,120 --> 00:24:17,250 would compute a determinant. 368 00:24:17,250 --> 00:24:20,990 If I have a matrix of size a hundred, 369 00:24:20,990 --> 00:24:24,817 the way I would actually compute its determinant would be 370 00:24:24,817 --> 00:24:27,400 elimination, make it triangular, multiply the pivots together, 371 00:24:27,400 --> 00:24:29,940 but it -- would it be possible t- 372 00:24:29,940 --> 00:24:33,110 to produce the same matrix the product of the pivots, 373 00:24:33,110 --> 00:24:34,080 the product of pivots. 374 00:24:34,080 --> 00:24:39,080 Now, there's always in determinants a plus or minus 375 00:24:39,080 --> 00:24:44,630 and cross every T in that proof, but that's really 376 00:24:44,630 --> 00:24:46,850 the sign to remember. 377 00:24:46,850 --> 00:24:51,840 Where -- where does that come into this rule? 378 00:24:51,840 --> 00:24:54,500 Could it be, could the determinant 379 00:24:54,500 --> 00:24:58,210 be minus the product of the pivots? 380 00:24:58,210 --> 00:25:00,740 The determinant is d1, d2, to dn. 381 00:25:00,740 --> 00:25:02,380 No doubt about that. 382 00:25:02,380 --> 00:25:05,370 But to get to this triangular form, 383 00:25:05,370 --> 00:25:12,660 we may have had to do a row exchange, so, so this -- 384 00:25:12,660 --> 00:25:15,890 it's the product of the pivots if there were no row exchanges. 385 00:25:15,890 --> 00:25:19,370 If, if SLU code, the simple LU code, 386 00:25:19,370 --> 00:25:21,770 the square LU went right through. 387 00:25:21,770 --> 00:25:24,070 If we had to do some row exchanges, 388 00:25:24,070 --> 00:25:26,840 then we've got to watch plus or minus. 389 00:25:26,840 --> 00:25:31,680 OK, but though -- this law is simply that. 390 00:25:31,680 --> 00:25:33,180 OK, how do I prove that? 391 00:25:37,420 --> 00:25:42,730 Let's see, let me suppose that d's are not zeroes. 392 00:25:42,730 --> 00:25:44,620 The pivots are not zeroes. 393 00:25:44,620 --> 00:25:50,550 And tell me, how do I show that none of this upper stuff 394 00:25:50,550 --> 00:25:53,840 makes any difference? 395 00:25:53,840 --> 00:25:56,550 How do I get zeroes there? 396 00:25:56,550 --> 00:25:58,630 By elimination, right? 397 00:25:58,630 --> 00:26:01,940 I just multiply this row by the right number, 398 00:26:01,940 --> 00:26:05,950 subtract from that row, kills that. 399 00:26:05,950 --> 00:26:09,030 I multiply this row by the right number, kills that, 400 00:26:09,030 --> 00:26:11,010 by the right number, kills that. 401 00:26:11,010 --> 00:26:15,850 Now, you kill every one of these off-diagonal terms, no problem 402 00:26:15,850 --> 00:26:17,450 and I'm just left with the diagonal. 403 00:26:20,690 --> 00:26:22,940 Well, strictly speaking, I still have 404 00:26:22,940 --> 00:26:26,680 to figure out why is, for a diagonal matrix 405 00:26:26,680 --> 00:26:28,745 now, why is that the right determinant? 406 00:26:31,540 --> 00:26:37,260 I mean, we sure hope it is, but why? 407 00:26:37,260 --> 00:26:41,270 I have to go back to properties one, two, three. 408 00:26:41,270 --> 00:26:46,710 Why is -- now that the matrix is suddenly diagonal, 409 00:26:46,710 --> 00:26:48,950 how do I know that the determinant is just 410 00:26:48,950 --> 00:26:51,070 a product of That's my proof, really, 411 00:26:51,070 --> 00:26:53,200 that once I've got those diagonal entries? 412 00:26:53,200 --> 00:26:55,030 Well, what I going to use? 413 00:26:55,030 --> 00:26:57,760 I'm going to use property 3A, is that right? 414 00:26:57,760 --> 00:27:01,250 I'll factor this, I'll factor this, 415 00:27:01,250 --> 00:27:05,380 I'll factor that d1 out and have one and have 416 00:27:05,380 --> 00:27:07,280 the first row will be that. 417 00:27:07,280 --> 00:27:09,710 And then I'll factor out the d2, shall I shall 418 00:27:09,710 --> 00:27:13,160 I put the d2 here, and the second row 419 00:27:13,160 --> 00:27:16,110 will look like that, and so on. 420 00:27:16,110 --> 00:27:20,150 So I've factored out all the d's and what I left with? 421 00:27:20,150 --> 00:27:21,370 The identity. 422 00:27:21,370 --> 00:27:24,970 And what rule do I finally get to use? 423 00:27:24,970 --> 00:27:26,030 Rule one. 424 00:27:26,030 --> 00:27:29,900 Finally, this is the point where rule one finally chips 425 00:27:29,900 --> 00:27:33,480 in and says that this determinant is one, 426 00:27:33,480 --> 00:27:35,520 so it's the product of the d's. 427 00:27:35,520 --> 00:27:40,950 So this was rules five, to do elimination, 428 00:27:40,950 --> 00:27:48,680 3A to factor out the D's, and, and our best friend, rule one. 429 00:27:48,680 --> 00:27:52,240 And possibly rule two, the exchanges 430 00:27:52,240 --> 00:27:53,710 may have been needed also. 431 00:27:53,710 --> 00:27:54,210 OK. 432 00:27:56,940 --> 00:28:01,350 Now I guess I have to consider also the case if some d is 433 00:28:01,350 --> 00:28:06,680 zero, because I was assuming I could use the d's to clean out 434 00:28:06,680 --> 00:28:08,490 and make a diagonal, but what if -- 435 00:28:08,490 --> 00:28:13,390 what if one of those diagonal entries is zero? 436 00:28:13,390 --> 00:28:16,380 Well, then with elimination we know 437 00:28:16,380 --> 00:28:21,440 that we can get a row of zeroes, and for a row 438 00:28:21,440 --> 00:28:25,200 of zeroes I'm using rule six, the determinant is zero, 439 00:28:25,200 --> 00:28:26,020 and that's right. 440 00:28:26,020 --> 00:28:28,690 So I can check the singular case. 441 00:28:28,690 --> 00:28:36,250 In fact, I can now get to the key point that determinant of A 442 00:28:36,250 --> 00:28:44,920 is zero, exactly when, exactly when A is singular. 443 00:28:48,790 --> 00:28:52,880 And otherwise is not singular, so that the determinant 444 00:28:52,880 --> 00:28:58,800 is a fair test for invertibility or non-invertibility. 445 00:28:58,800 --> 00:29:03,610 So, I must be close to that because I can take any matrix 446 00:29:03,610 --> 00:29:05,290 and get there. 447 00:29:05,290 --> 00:29:06,870 Do I -- did I have anything to say? 448 00:29:09,570 --> 00:29:12,970 The, the proofs, it starts by saying by elimination 449 00:29:12,970 --> 00:29:14,680 go from A to U. 450 00:29:14,680 --> 00:29:15,290 Oh, yeah. 451 00:29:15,290 --> 00:29:17,940 Actually looks to me like I don't -- 452 00:29:17,940 --> 00:29:22,450 haven't said anything brand-new here, that, that really, 453 00:29:22,450 --> 00:29:28,950 I've got this, because let's just remember the 454 00:29:28,950 --> 00:29:37,980 By elimination, I can go from the original A to reason. 455 00:29:37,980 --> 00:29:43,440 Well, OK, now suppose the matrix is U. singular. 456 00:29:43,440 --> 00:29:46,630 If the matrix is singular, what happens? 457 00:29:46,630 --> 00:29:50,480 Then by elimination I get a row of zeroes 458 00:29:50,480 --> 00:29:55,240 and therefore the determinant is zero. 459 00:29:55,240 --> 00:29:59,170 And if the matrix is not singular, I don't get zero, 460 00:29:59,170 --> 00:30:02,220 so maybe -- do you want me to put this, like, in two parts? 461 00:30:02,220 --> 00:30:10,100 The determinant of A is not zero when A is invertible. 462 00:30:15,480 --> 00:30:18,750 Because I've already -- 463 00:30:18,750 --> 00:30:23,240 in chapter two we figured out when is the matrix invertible. 464 00:30:23,240 --> 00:30:27,770 It's invertible when elimination produces a full set of pivots 465 00:30:27,770 --> 00:30:31,420 and now, and we now, we know the determinant is the product 466 00:30:31,420 --> 00:30:34,160 of those non-zero numbers. 467 00:30:34,160 --> 00:30:36,410 So those are the two cases. 468 00:30:36,410 --> 00:30:39,180 If it's singular, I go to a row of zeroes. 469 00:30:43,370 --> 00:30:49,680 If it's invertible, I go to U and then to the diagonal D, 470 00:30:49,680 --> 00:30:57,150 and then which -- and then to d1, d2, up to dn. 471 00:30:57,150 --> 00:31:00,050 As the formula -- so we have a formula now. 472 00:31:02,700 --> 00:31:05,110 We have a formula for the determinant 473 00:31:05,110 --> 00:31:08,520 and it's actually a very much more practical 474 00:31:08,520 --> 00:31:12,100 formula than the but they didn't matter anyway. ad-bc formula. 475 00:31:12,100 --> 00:31:18,050 Is it correct, maybe you should just -- let's just check that. 476 00:31:18,050 --> 00:31:18,970 Two-by-two. 477 00:31:18,970 --> 00:31:23,850 What are the pivots of a two-by-two matrix? 478 00:31:23,850 --> 00:31:28,270 What does elimination do with a two-by-two matrix? 479 00:31:28,270 --> 00:31:30,370 It -- there's the first pivot, fine. 480 00:31:30,370 --> 00:31:33,880 What's the second pivot? 481 00:31:33,880 --> 00:31:38,970 We subtract, so I'm putting it in this upper triangular form. 482 00:31:38,970 --> 00:31:44,270 What do I -- my multiplier is c over a, right? 483 00:31:44,270 --> 00:31:46,790 I multiply that row by c over a and I 484 00:31:46,790 --> 00:31:50,590 subtract to get that zero, and here I 485 00:31:50,590 --> 00:31:54,470 have d minus c over a times b. 486 00:31:58,500 --> 00:32:01,870 That's the elimination on a two-by-two. 487 00:32:01,870 --> 00:32:07,440 So I've finally discovered that the determinant of this matrix 488 00:32:07,440 --> 00:32:07,940 -- 489 00:32:07,940 --> 00:32:12,240 I've got it from the properties, not by knowing the answer 490 00:32:12,240 --> 00:32:18,830 from last year, that the determinant of this two-by-two 491 00:32:18,830 --> 00:32:22,730 is the product of A times that, and of course 492 00:32:22,730 --> 00:32:26,180 when I multiply A by that, the product of that and that 493 00:32:26,180 --> 00:32:30,485 is ad minus, the a is canceled, 494 00:32:30,485 --> 00:32:30,985 bc. 495 00:32:34,270 --> 00:32:36,320 So that's great, provided a isn't zero. 496 00:32:36,320 --> 00:32:38,960 because all math professors watching this will be waiting 497 00:32:38,960 --> 00:32:42,190 If a was zero, that step wasn't allowed, with seven row 498 00:32:42,190 --> 00:32:45,130 exchanges and with ten row exchanges? zero wasn't a pivot. 499 00:32:45,130 --> 00:32:46,600 So that's what I -- 500 00:32:46,600 --> 00:32:48,850 I've covered all the bases. 501 00:32:48,850 --> 00:32:53,090 I have to -- if a is zero, then I have to do the exchange, 502 00:32:53,090 --> 00:32:56,570 and if the exchange doesn't work, it's because a is proof. 503 00:32:56,570 --> 00:32:57,730 singular. 504 00:32:57,730 --> 00:33:03,130 OK, those are -- 505 00:33:03,130 --> 00:33:06,520 those are the direct properties of the determinant. 506 00:33:06,520 --> 00:33:11,120 And now, finally, I've got two more, nine and ten. 507 00:33:11,120 --> 00:33:13,920 And that's -- 508 00:33:13,920 --> 00:33:15,030 I think you can... 509 00:33:15,030 --> 00:33:25,210 Like, the ones we've got here are 510 00:33:25,210 --> 00:33:28,870 totally connected with our elimination process 511 00:33:28,870 --> 00:33:35,340 and whether pivots are available and whether we 512 00:33:35,340 --> 00:33:36,820 get a row of zeroes. 513 00:33:36,820 --> 00:33:40,360 I think all that you can swallow in one shot. 514 00:33:40,360 --> 00:33:43,730 Let me tell you properties nine and ten. 515 00:33:46,990 --> 00:33:50,000 They're quick to write down. 516 00:33:50,000 --> 00:33:53,940 They're very, very useful. 517 00:33:53,940 --> 00:33:56,260 So I'll just write them down and use them. 518 00:33:56,260 --> 00:34:01,410 Property nine says that the determinant of a product -- 519 00:34:01,410 --> 00:34:05,610 if I That's the, like, concrete proof that, 520 00:34:05,610 --> 00:34:07,180 multiply two matrices. 521 00:34:07,180 --> 00:34:11,909 So if I multiply two matrices, A and B, 522 00:34:11,909 --> 00:34:14,050 that the determinant of the product 523 00:34:14,050 --> 00:34:26,730 is determinant of A times determinant of B, and for me 524 00:34:26,730 --> 00:34:31,750 that one is like, that's a very valuable property, 525 00:34:31,750 --> 00:34:34,449 and it's sort of like partly coming out of the blue, 526 00:34:34,449 --> 00:34:37,050 because we haven't been multiplying matrices 527 00:34:37,050 --> 00:34:41,159 and here suddenly this determinant 528 00:34:41,159 --> 00:34:44,590 has this multiplying property. 529 00:34:44,590 --> 00:34:46,750 Remember, it didn't have the linear property, 530 00:34:46,750 --> 00:34:48,810 it didn't have the adding property. 531 00:34:48,810 --> 00:34:52,389 The determinant of A plus B is not 532 00:34:52,389 --> 00:34:57,210 the sum of the determinants, but the determinant of A times B 533 00:34:57,210 --> 00:35:01,220 is the product, is the product of the determinants. 534 00:35:01,220 --> 00:35:06,955 OK, so for example, what's the determinant of A inverse? 535 00:35:12,560 --> 00:35:14,240 Using that property nine. 536 00:35:19,360 --> 00:35:21,230 Let me, let me put that under here 537 00:35:21,230 --> 00:35:27,800 because the camera is happier if it can focus on both at once. 538 00:35:27,800 --> 00:35:29,140 So let me put it underneath. 539 00:35:29,140 --> 00:35:34,510 The determinant of A inverse, because property ten 540 00:35:34,510 --> 00:35:40,420 will come in that space. 541 00:35:40,420 --> 00:35:44,530 What does this tell me about A inverse, its determinant? 542 00:35:44,530 --> 00:35:47,730 OK, well, what do I know about A inverse? 543 00:35:47,730 --> 00:35:54,980 I know that A inverse times A is odd. 544 00:35:54,980 --> 00:35:55,960 So what I going to do? 545 00:35:59,200 --> 00:36:02,450 I'm going to take determinants of both sides. 546 00:36:02,450 --> 00:36:06,290 The determinant of I is one, and what's 547 00:36:06,290 --> 00:36:10,390 the determinant of A inverse A? 548 00:36:10,390 --> 00:36:13,670 That's a product of two matrices, A and B. 549 00:36:13,670 --> 00:36:15,510 So it's the product of the determinant, 550 00:36:15,510 --> 00:36:16,830 so what I learning? 551 00:36:16,830 --> 00:36:18,840 I'm learning that the determinant 552 00:36:18,840 --> 00:36:24,070 of A inverse times the determinant of A 553 00:36:24,070 --> 00:36:27,800 is the determinant of I, that's this one. 554 00:36:27,800 --> 00:36:31,870 Again, I happily use property one. 555 00:36:31,870 --> 00:36:35,950 OK, so that tells me that the determinant of A inverse 556 00:36:35,950 --> 00:36:37,270 is one over. 557 00:36:37,270 --> 00:36:40,000 Here's my -- here's my conclusion -- 558 00:36:40,000 --> 00:36:53,910 is one over the determinant of A. 559 00:36:53,910 --> 00:36:55,810 I guess that that -- 560 00:36:55,810 --> 00:36:59,620 I, I always try to think, well, do we know some cases of that? 561 00:36:59,620 --> 00:37:04,450 Of course, we know it's right already if A is diagonal. 562 00:37:04,450 --> 00:37:09,000 If A is a diagonal matrix, then its determinant 563 00:37:09,000 --> 00:37:10,480 is just a product of those numbers. 564 00:37:10,480 --> 00:37:14,890 So if A is, for example, two-three, 565 00:37:14,890 --> 00:37:20,740 then we know that A-inverse is one-half one-third, 566 00:37:20,740 --> 00:37:26,440 and sure enough, that has determinant six, 567 00:37:26,440 --> 00:37:29,430 and that has determinant one-sixth. 568 00:37:29,430 --> 00:37:32,360 And our rule checks. 569 00:37:32,360 --> 00:37:39,490 So somehow this proof, this property has to -- 570 00:37:39,490 --> 00:37:41,960 somehow the proof of that property -- 571 00:37:41,960 --> 00:37:45,950 if we can boil it down to diagonal matrices then we can 572 00:37:45,950 --> 00:37:49,140 read it off, whether it's A and A-inverse, 573 00:37:49,140 --> 00:37:52,660 or two different diagonal matrices A and B. 574 00:37:52,660 --> 00:37:54,430 For diagonal -- so what I saying? 575 00:37:54,430 --> 00:37:59,050 I'm saying for a diagonal matrices, check. 576 00:37:59,050 --> 00:38:02,380 But we'd have to do elimination steps, 577 00:38:02,380 --> 00:38:08,510 we'd have to patiently do the, the, argument 578 00:38:08,510 --> 00:38:11,810 if we want to use these previous properties to get it 579 00:38:11,810 --> 00:38:12,980 for other matrices. 580 00:38:12,980 --> 00:38:18,094 And it also tells me -- what, just let's, see what else 581 00:38:18,094 --> 00:38:18,760 it's telling me. 582 00:38:18,760 --> 00:38:21,900 What's the determinant of, of A-squared? 583 00:38:21,900 --> 00:38:26,750 If I take a matrix and square it? 584 00:38:26,750 --> 00:38:30,460 Then the determinant just got squared. 585 00:38:30,460 --> 00:38:31,140 Right? 586 00:38:31,140 --> 00:38:34,180 The determinant of A-squared is the determinant 587 00:38:34,180 --> 00:38:35,864 of A times the determinant of A. 588 00:38:35,864 --> 00:38:38,030 So if I square the matrix, I square the determinant. 589 00:38:38,030 --> 00:38:43,180 If I double the matrix, what do I do to the non-zeroes flipped 590 00:38:43,180 --> 00:38:46,350 to the other side of the diagonal, determinant? 591 00:38:46,350 --> 00:38:47,930 Think about that one. 592 00:38:47,930 --> 00:38:52,690 If I double the matrix, what -- so the determinant of A, 593 00:38:52,690 --> 00:38:56,150 since I'm writing down, like, facts that follow, 594 00:38:56,150 --> 00:39:02,500 the determinant of A-squared is the determinant of A, 595 00:39:02,500 --> 00:39:04,740 all squared. 596 00:39:04,740 --> 00:39:09,580 The determinant of 2A is what? 597 00:39:12,250 --> 00:39:16,380 That's A plus A. 598 00:39:16,380 --> 00:39:19,910 But wait, er, I don't want the answer 599 00:39:19,910 --> 00:39:22,410 to determinant of A here. 600 00:39:22,410 --> 00:39:23,150 That's wrong. 601 00:39:23,150 --> 00:39:25,860 It's not two determinant of A, What is it? 602 00:39:25,860 --> 00:39:28,880 OK, one more coming, which I I have to make, 603 00:39:28,880 --> 00:39:32,190 what's the number that I have to multiply determinant of A 604 00:39:32,190 --> 00:39:34,900 by if I double the whole matrix, if I 605 00:39:34,900 --> 00:39:36,710 double every entry in the matrix? 606 00:39:36,710 --> 00:39:38,220 What happens to the determinant? 607 00:39:38,220 --> 00:39:41,230 If that were possible, that would be a bad thing, 608 00:39:41,230 --> 00:39:43,640 Supposed it's an n-by-n matrix. that gets -- 609 00:39:43,640 --> 00:39:44,850 get down to triangular 610 00:39:44,850 --> 00:39:46,690 Two to the n, right. 611 00:39:46,690 --> 00:39:48,120 Two to the nth. 612 00:39:48,120 --> 00:39:50,650 Now, why is it two to the nth, and not just two? 613 00:39:54,700 --> 00:39:58,070 So why is it two to the nth? 614 00:39:58,070 --> 00:40:02,230 Because I'm factoring out two from every row. 615 00:40:02,230 --> 00:40:05,610 There's a factor -- this has a factor two in every row, 616 00:40:05,610 --> 00:40:08,640 so I can factor two out of the first row. 617 00:40:08,640 --> 00:40:12,120 I factor a different two out of the second row, a different two 618 00:40:12,120 --> 00:40:15,640 out of the nth row, so I've got all those twos coming out. 619 00:40:15,640 --> 00:40:20,290 So it's like volume, really, and that's 620 00:40:20,290 --> 00:40:23,570 one of the great applications of determinants. 621 00:40:23,570 --> 00:40:30,860 If I -- if I have a box and I double all the sides, 622 00:40:30,860 --> 00:40:35,800 I multiply the volume by two to the nth. 623 00:40:35,800 --> 00:40:38,440 If it's a box in three dimensions, 624 00:40:38,440 --> 00:40:40,970 I multiply the volume by 8. 625 00:40:43,710 --> 00:40:47,190 So this is rule 3A here. 626 00:40:47,190 --> 00:40:49,480 This is rule nine. 627 00:40:49,480 --> 00:40:55,100 And notice the way this rule sort of checks out with 628 00:40:55,100 --> 00:41:02,550 the singular/non-singular stuff, that if A is invertible, 629 00:41:02,550 --> 00:41:05,650 what does that mean about its determinant? 630 00:41:05,650 --> 00:41:08,020 It's not zero, and therefore this makes sense. 631 00:41:10,770 --> 00:41:12,940 The case when determinant of A is 632 00:41:12,940 --> 00:41:19,870 zero, that's the case where my formula doesn't work anymore. 633 00:41:19,870 --> 00:41:24,030 If determinant of A is zero, this is ridiculous, 634 00:41:24,030 --> 00:41:25,800 and that's ridiculous. 635 00:41:25,800 --> 00:41:31,300 A-inverse doesn't exist, and one over zero doesn't make sense. 636 00:41:31,300 --> 00:41:36,090 So don't miss this property. 637 00:41:36,090 --> 00:41:38,560 It's sort of, like, amazing that it can... 638 00:41:38,560 --> 00:41:44,660 And the tenth property is equally simple to state, 639 00:41:44,660 --> 00:41:47,720 that the determinant of A transposed 640 00:41:47,720 --> 00:41:57,010 equals the determinant of A. 641 00:41:57,010 --> 00:42:03,030 And of course, let's just check it on the ab cd guy. 642 00:42:03,030 --> 00:42:07,075 We could check that sure enough, that's ab cd, it works. 643 00:42:09,710 --> 00:42:14,350 It's ad - bc, it's ad - bc, sure enough. 644 00:42:14,350 --> 00:42:19,140 So that transposing did not change the determinant. 645 00:42:19,140 --> 00:42:24,790 But what it does change is -- 646 00:42:24,790 --> 00:42:28,400 well, what it does is it lists, so all -- 647 00:42:28,400 --> 00:42:31,340 I've been working with rows. 648 00:42:31,340 --> 00:42:35,690 If a row is all zeroes, the determinant is zero. 649 00:42:35,690 --> 00:42:40,360 But now, with rule ten, I know what to do 650 00:42:40,360 --> 00:42:42,750 is a column is all zero. 651 00:42:42,750 --> 00:42:46,550 If a column is all zero, what's the determinant? 652 00:42:46,550 --> 00:42:48,260 Zero, again. 653 00:42:48,260 --> 00:42:53,250 So, like all those properties about rows, exchanging two rows 654 00:42:53,250 --> 00:42:55,080 reverses the sign. 655 00:42:55,080 --> 00:42:58,210 Now, exchanging two columns reverses 656 00:42:58,210 --> 00:43:00,990 the sign, because I can always, if I 657 00:43:00,990 --> 00:43:03,690 want to see why, I can transpose, 658 00:43:03,690 --> 00:43:08,680 those columns become rows, I do the exchange, I transpose back. 659 00:43:08,680 --> 00:43:11,760 And I've done a column operation. 660 00:43:11,760 --> 00:43:17,530 So, in, in conclusion, there was nothing special about row one, 661 00:43:17,530 --> 00:43:20,610 'cause I could exchange rows, and now there's 662 00:43:20,610 --> 00:43:25,060 nothing special about rows that isn't equally true for columns 663 00:43:25,060 --> 00:43:26,980 because this is the same. 664 00:43:26,980 --> 00:43:27,580 OK. 665 00:43:27,580 --> 00:43:32,080 And again, maybe I won't -- 666 00:43:32,080 --> 00:43:33,320 oh, let's see. 667 00:43:33,320 --> 00:43:33,820 Do we...? 668 00:43:33,820 --> 00:43:37,930 Maybe it's worth seeing a quick proof of this number ten, 669 00:43:37,930 --> 00:43:44,620 quick, quick, er, proof of number ten. 670 00:43:44,620 --> 00:43:48,970 Er, let me take the -- this is number ten. 671 00:43:48,970 --> 00:43:51,380 A transposed equals A. 672 00:43:51,380 --> 00:43:56,480 Determinate of A transposed equals determinate of A. 673 00:43:56,480 --> 00:43:58,110 That's what I should have said. 674 00:43:58,110 --> 00:43:59,280 OK. 675 00:43:59,280 --> 00:44:07,820 So, let's just, er. 676 00:44:07,820 --> 00:44:11,450 A typical matrix A, if I use elimination, 677 00:44:11,450 --> 00:44:16,100 this factors into LU. 678 00:44:16,100 --> 00:44:21,710 And the transpose is U transpose, l transpose. 679 00:44:25,430 --> 00:44:26,370 Er... let me. 680 00:44:29,150 --> 00:44:36,870 So this is proof, this is proof number ten, using -- 681 00:44:36,870 --> 00:44:39,820 well, I don't know which ones I'll use, so I'll put 682 00:44:39,820 --> 00:44:42,840 'em all in, one to nine. 683 00:44:42,840 --> 00:44:43,650 OK. 684 00:44:43,650 --> 00:44:47,400 I'm going to prove number ten by using one to nine. 685 00:44:47,400 --> 00:44:50,910 I won't cover every case, but I'll cover almost every 686 00:44:50,910 --> 00:44:51,550 case. 687 00:44:51,550 --> 00:44:55,400 So in almost every case, A can factor into LU, 688 00:44:55,400 --> 00:44:57,710 and A transposed can factor into that. 689 00:44:57,710 --> 00:45:00,070 Now, what do I do next? 690 00:45:00,070 --> 00:45:03,910 So I want to prove that these are the same. 691 00:45:03,910 --> 00:45:06,610 I see a product here. 692 00:45:06,610 --> 00:45:09,560 So I use rule nine. 693 00:45:09,560 --> 00:45:14,860 So, now what I want to prove is, so now I know that this is LU, 694 00:45:14,860 --> 00:45:19,810 this is U transposed and l transposed. 695 00:45:19,810 --> 00:45:24,010 Now, just for a practice, what are all those determinants? 696 00:45:24,010 --> 00:45:28,710 So this is, this is, this is prove this, prove this, prove 697 00:45:28,710 --> 00:45:32,460 this, and now I'm ready to do it. 698 00:45:32,460 --> 00:45:34,460 What's the determinant of l? 699 00:45:34,460 --> 00:45:40,240 You remember what l is, it's this lower triangular matrix 700 00:45:40,240 --> 00:45:43,085 with ones on the diagonals. 701 00:45:43,085 --> 00:45:44,710 So what is the determinant of that guy? 702 00:45:44,710 --> 00:45:45,210 I- It's one. 703 00:45:48,640 --> 00:45:53,000 Any time I have this triangular matrix, 704 00:45:53,000 --> 00:46:00,390 I can get rid of all the non-zeroes, 705 00:46:00,390 --> 00:46:07,920 down to the diagonal case. 706 00:46:07,920 --> 00:46:12,750 The determinate of l is one. 707 00:46:12,750 --> 00:46:21,810 How about the determinant of l transposed? 708 00:46:21,810 --> 00:46:25,410 That's triangular also, right? 709 00:46:25,410 --> 00:46:28,050 Still got those ones on the diagonal, 710 00:46:28,050 --> 00:46:59,050 it's just the matrices and then get down to diagonal matrices. 711 00:46:59,050 --> 00:47:00,600 right? 712 00:47:00,600 --> 00:47:09,290 If If I could -- why would it be bad? 713 00:47:09,290 --> 00:47:11,800 My whole lecture would die, right? 714 00:47:11,800 --> 00:47:37,360 Because rule two said that if you do seven row exchanges, 715 00:47:37,360 --> 00:47:47,700 then the sign of the determinant reverses. 716 00:47:47,700 --> 00:47:56,820 But if you do ten row exchanges, the sign of the determinant 717 00:47:56,820 --> 00:48:02,520 stays the same, because minus one ten times is plus one. 718 00:48:02,520 --> 00:48:16,630 OK, so there's a hidden fact here, that every -- 719 00:48:16,630 --> 00:48:19,150 like, every permutation, the permutations 720 00:48:19,150 --> 00:48:23,310 are either odd or even. 721 00:48:23,310 --> 00:48:26,360 I could get the permutation with seven row exchanges, 722 00:48:26,360 --> 00:48:27,770 then I could probably get it with 723 00:48:27,770 --> 00:48:31,430 twenty-one, or twenty-three, or a hundred and one, 724 00:48:31,430 --> 00:48:33,430 if it's an odd one. 725 00:48:33,430 --> 00:48:35,980 Or an even number of permutations, so, 726 00:48:35,980 --> 00:48:39,070 but that's the key fact that just takes 727 00:48:39,070 --> 00:48:43,760 another little algebraic trick to see, 728 00:48:43,760 --> 00:48:45,947 and that means that the determinant is well-defined 729 00:48:45,947 --> 00:48:47,530 by properties one, two, three and it's 730 00:48:47,530 --> 00:48:47,580 got properties four to ten. 731 00:48:47,580 --> 00:48:47,660 OK, that's today and I'll try to get 732 00:48:47,660 --> 00:48:47,890 the homework for next Wednesday onto the web this afternoon. 733 00:48:47,890 --> 00:48:49,440 Thanks.