1 00:00:00 --> 00:00:05 OK, this is the second lecture on determinants. 2 00:00:05 --> 00:00:08 There are only three. 3 00:00:08 --> 00:00:15 With determinants it's a fascinating, small topic inside 4 00:00:15 --> 00:00:17 linear algebra. 5 00:00:17 --> 00:00:23 Used to be determinants were the big thing, 6 00:00:23 --> 00:00:28 and linear algebra was the little thing, 7 00:00:28 --> 00:00:32 but they -- those changed, 8 00:00:32 --> 00:00:34 that situation changed. 9 00:00:34 --> 00:00:37 Now determinants is one specific part, 10 00:00:37 --> 00:00:39 very neat little part. 11 00:00:39 --> 00:00:45 And my goal today is to find a formula for the determinant. 12 00:00:45 --> 00:00:47 It'll be a messy formula. 13 00:00:47 --> 00:00:51 So that's why you didn't see it right away. 14 00:00:51 --> 00:00:55 But if I'm given this n by n matrix 15 00:00:55 --> 00:00:58 then I use those entries to create this number, 16 00:00:58 --> 00:00:59 the determinant. 17 00:00:59 --> 00:01:01 So there's a formula for it. 18 00:01:01 --> 00:01:05 In fact, there's another formula, a second formula using 19 00:01:05 --> 00:01:07 something called cofactors. 20 00:01:07 --> 00:01:10 So you'll -- you have to know what cofactors are. 21 00:01:10 --> 00:01:13 And then I'll apply those formulas for some, 22 00:01:13 --> 00:01:18 some matrices that have a lot of zeros away from the 23 00:01:18 --> 00:01:20 three diagonals. 24 00:01:20 --> 00:01:20 OK. 25 00:01:20 --> 00:01:25 So I'm shooting now for a formula for the determinant. 26 00:01:25 --> 00:01:30 You remember we started with these three properties, 27 00:01:30 --> 00:01:35 three simple properties, but out of that we got all 28 00:01:35 --> 00:01:40 these amazing facts, like the determinant of A B 29 00:01:40 --> 00:01:44 equals determinant of A times determinant of B. 30 00:01:44 --> 00:01:49 But the three facts were -- oh, how about I just take two by 31 00:01:49 --> 00:01:50 twos. 32 00:01:50 --> 00:01:55.16 I know, because everybody here knows, the determinant of a two 33 00:01:55.16 --> 00:01:59.11 by two matrix, but let's get it out of 34 00:01:59.11 --> 00:02:01 these three formulas. 35 00:02:01 --> 00:02:05 OK, so here's my, my two by two matrix. 36 00:02:05 --> 00:02:09.8 I'm looking for a formula for this determinant. 37 00:02:09.8 --> 00:02:11 a b c d, OK. 38 00:02:11 --> 00:02:15 So property one, I know what to do with the 39 00:02:15 --> 00:02:16 identity. 40 00:02:16 --> 00:02:16 Right? 41 00:02:16 --> 00:02:21 Property two allows me to exchange 42 00:02:21 --> 00:02:24 rows, and I know what to do then. 43 00:02:24 --> 00:02:27 So I know that that determinant is one. 44 00:02:27 --> 00:02:32 Property two allows me to exchange rows and know that this 45 00:02:32 --> 00:02:35 determinant is minus one. 46 00:02:35 --> 00:02:39 And now I want to use property three to get everybody, 47 00:02:39 --> 00:02:41 to get everybody. 48 00:02:41 --> 00:02:43 And how will I do that? 49 00:02:43 --> 00:02:44 OK. 50 00:02:44 --> 00:02:50 So remember that if I keep the second row the same, 51 00:02:50 --> 00:02:55 I'm allowed to use linearity in the first row. 52 00:02:55 --> 00:02:59 And I'll just use it in a simple way. 53 00:02:59 --> 00:03:04 I'll write this vector a b as a 0 + 0 b. 54 00:03:04 --> 00:03:08 So that's one step using property three, 55 00:03:08 --> 00:03:15 linearity in the first row when the second row's the same. 56 00:03:15 --> 00:03:15.59 OK. 57 00:03:15.59 --> 00:03:19.22 But now you can guess what I'm going to do next. 58 00:03:19.22 --> 00:03:23 I'll -- because I'd like to -- if I can make the matrices 59 00:03:23 --> 00:03:26 diagonal, then I'm clearly there. 60 00:03:26 --> 00:03:27 So I'll take this one. 61 00:03:27 --> 00:03:33 Now I'll keep the first row fixed and split the second row, 62 00:03:33 --> 00:03:38 so that'll be an a 0 and I'll split that into a c 0 and, 63 00:03:38 --> 00:03:41 keeping that first row the same, a 0 d. 64 00:03:41 --> 00:03:44 I used, for this part, linearity. 65 00:03:44 --> 00:03:49 And now I'll -- whoops, that's plus because I've got 66 00:03:49 --> 00:03:50 more coming. 67 00:03:50 --> 00:03:52 This one I'll do the same. 68 00:03:52 --> 00:03:56 I'll keep this first row the same 69 00:03:56 --> 00:04:00 and I'll split c d into c 0 and 0 d. 70 00:04:00 --> 00:04:00 OK. 71 00:04:00 --> 00:04:03 Now I've got four easy determinants, 72 00:04:03 --> 00:04:09 and two of them are -- well, all four are extremely easy. 73 00:04:09 --> 00:04:13 Two of them are so easy as to turn into zero, 74 00:04:13 --> 00:04:14 right? 75 00:04:14 --> 00:04:20 Which two of these determinants are zero right away? 76 00:04:20 --> 00:04:22 The first guy is zero. 77 00:04:22 --> 00:04:23.54 Why is he zero? 78 00:04:23.54 --> 00:04:27 Why is that determinant nothing, forget him? 79 00:04:27 --> 00:04:30 Well, it has a column of zeros. 80 00:04:30 --> 00:04:33 And by the -- well, so one way to think is, 81 00:04:33 --> 00:04:36 well, it's a singular matrix. 82 00:04:36 --> 00:04:40 Oh, for, for like forty-eight different reasons, 83 00:04:40 --> 00:04:43 that determinant is zero. 84 00:04:43 --> 00:04:46 It's a singular matrix that has a column of zeros. 85 00:04:46 --> 00:04:48 It's, it's dead. 86 00:04:48 --> 00:04:50.51 And this one is about as dead too. 87 00:04:50.51 --> 00:04:51 Column of zeros. 88 00:04:51 --> 00:04:51 OK. 89 00:04:51 --> 00:04:54 So that's leaving us with this one. 90 00:04:54 --> 00:04:58 Now what do I -- how do I know its determinant, 91 00:04:58 --> 00:05:00.44 following the rules? 92 00:05:00.44 --> 00:05:05 Well, I guess one of the properties that we actually got 93 00:05:05 --> 00:05:09 to was the determinant of that -- diagonal matrix, 94 00:05:09 --> 00:05:13 then -- so I, I'm finally getting to that 95 00:05:13 --> 00:05:15 determinant is the a d. 96 00:05:15 --> 00:05:18 And this determinant is what? 97 00:05:18 --> 00:05:19 What's this one? 98 00:05:19 --> 00:05:26 Minus -- because I would use property two to do a flip to 99 00:05:26 --> 00:05:30 make it c b, then property three to factor out the b, 100 00:05:30 --> 00:05:36 property c to factor out the c -- the property again to factor 101 00:05:36 --> 00:05:41 out the c, and that minus, and of course finally I got the 102 00:05:41 --> 00:05:44 answer that we knew we would get. 103 00:05:44 --> 00:05:47 But you see the method. 104 00:05:47 --> 00:05:51 You see the method, because it's method I'm looking 105 00:05:51 --> 00:05:56 for here, not just a two by two answer but the method of doing 106 00:05:56 --> 00:06:01 -- now I can do three by threes and four by fours and any size. 107 00:06:01 --> 00:06:06.85 So if you can see the method of taking each row at a time -- so 108 00:06:06.85 --> 00:06:11 let's -- what would happen with three by threes? 109 00:06:11 --> 00:06:15 Can we mentally do it rather than I write everything on the 110 00:06:15 --> 00:06:17 board for three by threes? 111 00:06:17 --> 00:06:21 So what would we do if I had three by threes? 112 00:06:21 --> 00:06:25 I would keep rows two and three the same and I would split the 113 00:06:25 --> 00:06:28 first row into how many pieces? 114 00:06:28 --> 00:06:29 Three pieces. 115 00:06:29 --> 00:06:32 I'd have an A zero zero and a zero B 116 00:06:32 --> 00:06:38 zero and a zero zero C or something for the first row. 117 00:06:38 --> 00:06:44 So I would instead of going from one piece to two pieces to 118 00:06:44 --> 00:06:50 four pieces, I would go from one piece to three pieces to -- what 119 00:06:50 --> 00:06:51 would it be? 120 00:06:51 --> 00:06:56 Each of those three, would, would it be nine? 121 00:06:56 --> 00:06:58.09 Or twenty-seven? 122 00:06:58.09 --> 00:07:02 Oh yeah, I've actually got more steps, 123 00:07:02 --> 00:07:02 right. 124 00:07:02 --> 00:07:06 I'd go to nine but then I'd have another row to straighten 125 00:07:06 --> 00:07:07 out, twenty-seven. 126 00:07:07 --> 00:07:07 Yes, oh God. 127 00:07:07 --> 00:07:09 OK, let me say this again then. 128 00:07:09 --> 00:07:13 If I -- if it was three by three, I would -- separating out 129 00:07:13 --> 00:07:15 one row into three pieces would give me three, 130 00:07:15 --> 00:07:19 separating out the second row into three pieces, 131 00:07:19 --> 00:07:23 then I'd be up to nine, separating out the third row 132 00:07:23 --> 00:07:27.57 into its three pieces, I'd be up to twenty-seven, 133 00:07:27.57 --> 00:07:29 three cubed, pieces. 134 00:07:29 --> 00:07:31 But a lot of them would be zero. 135 00:07:31 --> 00:07:34 So now when would they not be zero? 136 00:07:34 --> 00:07:38.6 Tell me the pieces that would not be zero. 137 00:07:38.6 --> 00:07:42.39 Now I will write the non-zero ones. 138 00:07:42.39 --> 00:07:45 OK, so I have this matrix. 139 00:07:45 --> 00:07:51 I think I have to use these, start using these double 140 00:07:51 --> 00:07:57 symbols here because otherwise I could never do n by n. 141 00:07:57 --> 00:07:57 OK. 142 00:07:57 --> 00:07:57 OK. 143 00:07:57 --> 00:08:01 So I split this up like crazy. 144 00:08:01 --> 00:08:05 A bunch of pieces are zero. 145 00:08:05 --> 00:08:09 Whenever I have a column of zeros, I know I've got zero. 146 00:08:09 --> 00:08:11 When do I not have zero? 147 00:08:11 --> 00:08:14.97 When do I have -- what is it that's like these guys? 148 00:08:14.97 --> 00:08:18 These are the survivors, two survivors there. 149 00:08:18 --> 00:08:22 So my question for three by three is going to be what are 150 00:08:22 --> 00:08:23 the survivors? 151 00:08:23 --> 00:08:26 How many survivors are there? 152 00:08:26 --> 00:08:27 What are they? 153 00:08:27 --> 00:08:30 And when do I get a survivor. 154 00:08:30 --> 00:08:35 Well, I would get a survivor -- for example, one survivor will 155 00:08:35 --> 00:08:41 be that one times that one times that one, with all zeros 156 00:08:41 --> 00:08:42 everywhere else. 157 00:08:42 --> 00:08:47.69 That would be one survivor. a one one zero zero zero a two 158 00:08:47.69 --> 00:08:50.62 two zero zero zero a three three. 159 00:08:50.62 --> 00:08:54 That's like the a d survivor. 160 00:08:54 --> 00:08:56 Tell me another survivor. 161 00:08:56 --> 00:09:01 What other thing -- oh, now here you see the clue. 162 00:09:01 --> 00:09:05 Now can -- shall I just say the whole clue? 163 00:09:05 --> 00:09:10 That I'm having -- the survivors have one entry from 164 00:09:10 --> 00:09:12.91 each row and each column. 165 00:09:12.91 --> 00:09:16 One entry from each row and column. 166 00:09:16 --> 00:09:20 Because if some column is missing, 167 00:09:20 --> 00:09:23 then I get a singular matrix. 168 00:09:23 --> 00:09:26 And that, that's one of these guys. 169 00:09:26 --> 00:09:29 See, you see what happened with -- this guy? 170 00:09:29 --> 00:09:32 Column one never got used in 0 b 0 d. 171 00:09:32 --> 00:09:36 So its determinant was zero and I forget it. 172 00:09:36 --> 00:09:40.39 So I'm going to forget those and 173 00:09:40.39 --> 00:09:45 just put -- so tell me one more that would be a survivor? 174 00:09:45 --> 00:09:48.58 Well -- well, here's another one. 175 00:09:48.58 --> 00:09:54 a one one zero zero -- now OK, that's used up row -- row one 176 00:09:54 --> 00:09:54 is used. 177 00:09:54 --> 00:10:00 Column one is already used so it better be zero. 178 00:10:00 --> 00:10:01 What else could I have? 179 00:10:01 --> 00:10:06 Where could I pick the guy -- which column shall I use in row 180 00:10:06 --> 00:10:06 two? 181 00:10:06 --> 00:10:10 Use column three, because here if I use column -- 182 00:10:10 --> 00:10:13 here I used column one and row one. 183 00:10:13 --> 00:10:17 This was like the column -- numbers were one two three, 184 00:10:17 --> 00:10:19 right in order. 185 00:10:19 --> 00:10:22 Now the column numbers are going to 186 00:10:22 --> 00:10:25 be one three, column three, 187 00:10:25 --> 00:10:26 and column two. 188 00:10:26 --> 00:10:31 So the row numbers are one two three, of course. 189 00:10:31 --> 00:10:36 The column numbers are some -- OK, some permutation of one two 190 00:10:36 --> 00:10:41 three, and here they come in the order one three two. 191 00:10:41 --> 00:10:47.34 It's just like having a permutation matrix with, 192 00:10:47.34 --> 00:10:49 instead of the ones, with numbers. 193 00:10:49 --> 00:10:52 And actually, it's very close to having a 194 00:10:52 --> 00:10:55 permutation matrix, because I, what I do eventually 195 00:10:55 --> 00:10:59 is I factor out these numbers and then I have got. 196 00:10:59 --> 00:11:01 So what is that determinant equal? 197 00:11:01 --> 00:11:04 I factor those numbers out and I've 198 00:11:04 --> 00:11:08 got a one one times a two two times a three three. 199 00:11:08 --> 00:11:11 And what does this determinant equal? 200 00:11:11 --> 00:11:14 Yeah, now tell me the, this -- I mean, 201 00:11:14 --> 00:11:18 we're really getting to the heart of these formulas now. 202 00:11:18 --> 00:11:20 What is that determinant? 203 00:11:20 --> 00:11:24 By the laws of -- by, by our three properties, 204 00:11:24 --> 00:11:28 I can factor these out, I can factor out the a one one, 205 00:11:28 --> 00:11:31 the a two three, and the a three two. 206 00:11:31 --> 00:11:34 They're in separate rows. 207 00:11:34 --> 00:11:36.54 I can do each row separately. 208 00:11:36.54 --> 00:11:41 And then I just have to decide is that plus sign or is that a 209 00:11:41 --> 00:11:42.59 minus sign? 210 00:11:42.59 --> 00:11:45 And the answer is it's a minus. 211 00:11:45 --> 00:11:47.02 Why minus? 212 00:11:47.02 --> 00:11:52 Because these is one row exchange to get it back to the 213 00:11:52 --> 00:11:53 identity. 214 00:11:53 --> 00:11:54 So that's a minus. 215 00:11:54 --> 00:11:56 Now I through? 216 00:11:56 --> 00:11:59.53 No, because there are other ways. 217 00:11:59.53 --> 00:12:03 What I'm really through with, what I've done, 218 00:12:03 --> 00:12:09 what I've, what I've completed is only the part where the a one 219 00:12:09 --> 00:12:11 one is there. 220 00:12:11 --> 00:12:16 But now I've got parts where it's a one two. 221 00:12:16 --> 00:12:21 And now if it's a one two that row is used, that column is 222 00:12:21 --> 00:12:21 used. 223 00:12:21 --> 00:12:23 You see that idea? 224 00:12:23 --> 00:12:26 I could use this row and column. 225 00:12:26 --> 00:12:29 Now that column is used, that column is used, 226 00:12:29 --> 00:12:33 and this guy has to be here, a three three. 227 00:12:33 --> 00:12:36 And what's that determinant? 228 00:12:36 --> 00:12:41 That's an a one two times an a two one times an a three three, 229 00:12:41 --> 00:12:44 and does it have a plus or a minus? 230 00:12:44 --> 00:12:45 A minus is right. 231 00:12:45 --> 00:12:46.68 It has a minus. 232 00:12:46.68 --> 00:12:50 Because it's one flip away from an id- from the, 233 00:12:50 --> 00:12:53 regular, the right order, the diagonal order. 234 00:12:53 --> 00:12:57 And now what's the other guy with a -- 235 00:12:57 --> 00:13:00 with, a one two up there? 236 00:13:00 --> 00:13:02 I could have used this row. 237 00:13:02 --> 00:13:07 I could have put this guy here and this guy here. 238 00:13:07 --> 00:13:08 Right? 239 00:13:08 --> 00:13:10 You see the whole deal? 240 00:13:10 --> 00:13:14 Now that's an a one two, a two three, 241 00:13:14 --> 00:13:18 a three one, and does that go with a plus or 242 00:13:18 --> 00:13:20 a minus? 243 00:13:20 --> 00:13:24 Yeah, now that takes a minute of thinking, doesn't it, 244 00:13:24 --> 00:13:28 because one row exchange doesn't get it in line. 245 00:13:28 --> 00:13:30 So what is the answer for this? 246 00:13:30 --> 00:13:31 Plus or minus? 247 00:13:31 --> 00:13:34 Plus, because it takes two exchanges. 248 00:13:34 --> 00:13:39 I could exchange rows one and three and then two and three. 249 00:13:39 --> 00:13:43.63 Two exchanges makes this thing a plus. 250 00:13:43.63 --> 00:13:43 OK. 251 00:13:43 --> 00:13:50 And then finally we have -- we're going to have two more. 252 00:13:50 --> 00:13:54 Zero zero a one three, a two one zero zero, 253 00:13:54 --> 00:13:57 zero a three two zero. 254 00:13:57 --> 00:13:59.32 And one more guy. 255 00:13:59.32 --> 00:14:04 Zero zero a one three, zero a two two zero, 256 00:14:04 --> 00:14:07 A three one zero zero. 257 00:14:07 --> 00:14:11 And let's put down what we get from those. 258 00:14:11 --> 00:14:13 An a one three, an a two one, 259 00:14:13 --> 00:14:18 and an a three two, and I think that one is a plus. 260 00:14:18 --> 00:14:24 And this guys is a minus because one exchange would put 261 00:14:24 --> 00:14:27 it -- would order it. 262 00:14:27 --> 00:14:29 And that's a minus. 263 00:14:29 --> 00:14:36 All right, that has taken one whole board just to do the three 264 00:14:36 --> 00:14:37 by three. 265 00:14:37 --> 00:14:43 But do you agree that we now have a formula for the 266 00:14:43 --> 00:14:49 determinant which came from the three properties? 267 00:14:49 --> 00:14:51 And it must be it. 268 00:14:51 --> 00:14:55 And I'm going to keep that formula. 269 00:14:55 --> 00:15:01 That's a famous -- that three by three formula is 270 00:15:01 --> 00:15:05.86 one that if, if the cameras will follow me back to the beginning 271 00:15:05.86 --> 00:15:10 here, I, I get the ones with the plus sign are the ones that go 272 00:15:10 --> 00:15:12 down like down this way. 273 00:15:12 --> 00:15:16 And the ones with the minus signs are sort of the ones that 274 00:15:16 --> 00:15:17 go this way. 275 00:15:17 --> 00:15:20 I won't make that precise. 276 00:15:20 --> 00:15:23 For two reasons, one, it would clutter up the 277 00:15:23 --> 00:15:28 board, and second reason, it wouldn't be right for four 278 00:15:28 --> 00:15:28 by fours. 279 00:15:28 --> 00:15:32 For four by four, let me just say right away, 280 00:15:32 --> 00:15:35 four by four matrix -- the, the cross diagonal, 281 00:15:35 --> 00:15:41 the wrong diagonal happens to come out with a plus sign. 282 00:15:41 --> 00:15:42 Why is that? 283 00:15:42 --> 00:15:48 If I have a four by four matrix with ones coming on the counter 284 00:15:48 --> 00:15:52 diagonal, that determinant is plus. 285 00:15:52 --> 00:15:52 Why? 286 00:15:52 --> 00:15:55 Why plus for that guy? 287 00:15:55 --> 00:16:01.08 Because if I exchange rows one and four and then I exchange 288 00:16:01.08 --> 00:16:06 rows two and three, I've got the identity, 289 00:16:06 --> 00:16:09 and I did two exchanges. 290 00:16:09 --> 00:16:13 So this down to this, like, you know, 291 00:16:13 --> 00:16:18 down toward Miami and down toward LA stuff is, 292 00:16:18 --> 00:16:22 like, three by three only. 293 00:16:22 --> 00:16:22 OK. 294 00:16:22 --> 00:16:29 But I do want to get now -- I don't want to go through this 295 00:16:29 --> 00:16:31 for a four by four. 296 00:16:31 --> 00:16:37 I do want to get now the general formula. 297 00:16:37 --> 00:16:43.79 So this is what I refer to in the book as the big formula. 298 00:16:43.79 --> 00:16:49 So now this is the big formula for the determinant. 299 00:16:49 --> 00:16:55.74 I'm asking you to make a jump from two by two and three by 300 00:16:55.74 --> 00:16:57 three to n by n. 301 00:16:57 --> 00:17:01 OK, so this will be the big formula. 302 00:17:01 --> 00:17:06 That the determinant of A is the sum 303 00:17:06 --> 00:17:08.75 of a whole lot of terms. 304 00:17:08.75 --> 00:17:10.9 And what are those terms? 305 00:17:10.9 --> 00:17:15 And, and is it a plus or a minus sign, and I have to tell 306 00:17:15 --> 00:17:20 you which, which it is, because this came in -- in the 307 00:17:20 --> 00:17:23 three by three case, I had how many terms? 308 00:17:23 --> 00:17:24 Six. 309 00:17:24 --> 00:17:28 And half were plus and half were minus. 310 00:17:28 --> 00:17:33 How many terms are you figuring for four by four? 311 00:17:33 --> 00:17:40 If I get two terms in the two by two case, three -- six terms 312 00:17:40 --> 00:17:46 in the three by three case, what's that pattern? 313 00:17:46 --> 00:17:50 How many terms in the four by four case? 314 00:17:50 --> 00:17:51 Twenty-four. 315 00:17:51 --> 00:17:54 Four factorial. 316 00:17:54 --> 00:17:56 Why four factorial? 317 00:17:56 --> 00:18:00.79 This will be a sum of n factorial terms. 318 00:18:00.79 --> 00:18:04 Twenty-four, a hundred and twenty, 319 00:18:04 --> 00:18:09 seven hundred and twenty, whatever's after that. 320 00:18:09 --> 00:18:09 OK. 321 00:18:09 --> 00:18:12 Half plus and half minus. 322 00:18:12 --> 00:18:16 And where do those n factorial -- 323 00:18:16 --> 00:18:17 terms come from? 324 00:18:17 --> 00:18:21 This is the moment to listen to this lecture. 325 00:18:21 --> 00:18:24 Where do those n factorial terms come from? 326 00:18:24 --> 00:18:29 They come because the first, the guy in the first row can be 327 00:18:29 --> 00:18:31 chosen n ways. 328 00:18:31 --> 00:18:34 And after he's chosen, that's used up that, 329 00:18:34 --> 00:18:36 that column. 330 00:18:36 --> 00:18:40 So the one in the second row can be chosen n minus one ways. 331 00:18:40 --> 00:18:44 And after she's chosen, that second column has been 332 00:18:44 --> 00:18:44 used. 333 00:18:44 --> 00:18:49.35 And then the one in the third row can be chosen n minus two 334 00:18:49.35 --> 00:18:53 ways, and after it's chosen -- notice how I'm getting these 335 00:18:53 --> 00:18:55 personal pronouns. 336 00:18:55 --> 00:18:57 But I've run out. 337 00:18:57 --> 00:19:00 And I'm not willing to stop with three by three, 338 00:19:00 --> 00:19:04 so I'm just going to write the formula down. 339 00:19:04 --> 00:19:08.57 So the one in the first row comes from some column alpha. 340 00:19:08.57 --> 00:19:10 I don't know what alpha is. 341 00:19:10 --> 00:19:14 And the one in the -- I multiply that by somebody in the 342 00:19:14 --> 00:19:19.5 second row that comes from some different column. 343 00:19:19.5 --> 00:19:24 And I multiply that by somebody in the third row who comes from 344 00:19:24 --> 00:19:26 some yet different column. 345 00:19:26 --> 00:19:30 And then in the n-th row, I don't know what -- I don't 346 00:19:30 --> 00:19:31 know how to draw. 347 00:19:31 --> 00:19:33 Maybe omega, for last. 348 00:19:33 --> 00:19:38 And the whole point is then that -- that those column 349 00:19:38 --> 00:19:42.14 numbers are different, that alpha, beta, 350 00:19:42.14 --> 00:19:46 gamma, omega, that set of column numbers is 351 00:19:46 --> 00:19:50 some permutation, permutation of one to n. 352 00:19:50 --> 00:19:55 It, it, the n column numbers are each used once. 353 00:19:55 --> 00:19:58 And that gives us n factorial terms. 354 00:19:58 --> 00:20:03 And when I choose a term, that means I'm choosing 355 00:20:03 --> 00:20:07 somebody from every row and column. 356 00:20:07 --> 00:20:12 And then I just -- like the way I had this from row and column 357 00:20:12 --> 00:20:16 one, row and column two, row and column three, 358 00:20:16 --> 00:20:20 so that -- what was the alpha beta stuff in that, 359 00:20:20 --> 00:20:21 for that term here? 360 00:20:21 --> 00:20:23 Alpha was one, beta was two, 361 00:20:23 --> 00:20:25 gamma was three. 362 00:20:25 --> 00:20:30 The permutation was, was the trivial permutation, 363 00:20:30 --> 00:20:33 one two three, everybody in the right order. 364 00:20:33 --> 00:20:35.26 You see that formula? 365 00:20:35.26 --> 00:20:39 It's -- do you see why I didn't want to start with that the 366 00:20:39 --> 00:20:41 first day, Friday? 367 00:20:41 --> 00:20:44 I'd rather we understood the properties. 368 00:20:44 --> 00:20:50 Because out of this formula, presumably I could figure out 369 00:20:50 --> 00:20:51.76 all these properties. 370 00:20:51.76 --> 00:20:56.03 How would I know that the determinant of the identity 371 00:20:56.03 --> 00:20:58 matrix was one, for example, 372 00:20:58 --> 00:20:59 out of this formula? 373 00:20:59 --> 00:21:04 Why is -- if A is the identity matrix, how does this formula 374 00:21:04 --> 00:21:06 give me a plus one? 375 00:21:06 --> 00:21:08 You see it, right? 376 00:21:08 --> 00:21:12 Because, because almost all the terms are zeros. 377 00:21:12 --> 00:21:15 Which term isn't zero, if, if A is the identity 378 00:21:15 --> 00:21:16 matrix? 379 00:21:16 --> 00:21:20 Almost all the terms are zero because almost all the As are 380 00:21:20 --> 00:21:20 zero. 381 00:21:20 --> 00:21:25 It's only, the only time I'll get something is if it's a one 382 00:21:25 --> 00:21:28 one times a two two times a three three. 383 00:21:28 --> 00:21:31 Only, only the, only the permutation that's in 384 00:21:31 --> 00:21:34 the right order will, will give me something. 385 00:21:34 --> 00:21:36 It'll come with a plus sign. 386 00:21:36 --> 00:21:39.7 And the determinant of the identity is one. 387 00:21:39.7 --> 00:21:43 So, so we could go back from this formula and prove 388 00:21:43 --> 00:21:43 everything. 389 00:21:43 --> 00:21:46 We could even try to prove that the 390 00:21:46 --> 00:21:50 determinant of A B was the determinant of A times the 391 00:21:50 --> 00:21:52 determinant of B. 392 00:21:52 --> 00:21:55 But like next week we would still be working on it, 393 00:21:55 --> 00:21:59 because it's not -- clear from -- if I took A B, 394 00:21:59 --> 00:21:59 my God. 395 00:21:59 --> 00:22:00 You know --. 396 00:22:00 --> 00:22:04 The entries of A B would be all these pieces. 397 00:22:04 --> 00:22:08.25 Well, probably, it's probably -- historically 398 00:22:08.25 --> 00:22:11.64 it's been done, but it won't be repeated in 399 00:22:11.64 --> 00:22:12 eighteen oh six. 400 00:22:12 --> 00:22:13 OK. 401 00:22:13 --> 00:22:18 It would be possible probably to see, why the determinant of A 402 00:22:18 --> 00:22:21 equals the determinant of A transpose. 403 00:22:21 --> 00:22:24 That was another, like, miracle property at the 404 00:22:24 --> 00:22:25 end. 405 00:22:25 --> 00:22:28 That would, that would, that's an easier one, 406 00:22:28 --> 00:22:29 which we could find. 407 00:22:29 --> 00:22:29 OK. 408 00:22:29 --> 00:22:32 Is that all right for the big formula? 409 00:22:32 --> 00:22:37.36 I could take you then a, a typical -- let me do an 410 00:22:37.36 --> 00:22:38 example. 411 00:22:38 --> 00:22:40 Which I'll just create. 412 00:22:40 --> 00:22:43 I'll take a four by four matrix. 413 00:22:43 --> 00:22:47 I'll put some, I'll put some ones in and some 414 00:22:47 --> 00:22:48 zeros in. 415 00:22:48 --> 00:22:48 OK. 416 00:22:48 --> 00:22:53.13 Let me -- I don't know how many to 417 00:22:53.13 --> 00:22:55 put in, to tell the truth. 418 00:22:55 --> 00:22:58 I've never done this before. 419 00:22:58 --> 00:23:02.93 I don't know the determinant of that matrix. 420 00:23:02.93 --> 00:23:08 So like mathematics is being done for the first time in, 421 00:23:08 --> 00:23:10 in front of your eyes. 422 00:23:10 --> 00:23:13 What's the determinant? 423 00:23:13 --> 00:23:18 Well, a lot of -- there are twenty-four terms, 424 00:23:18 --> 00:23:20 because it's four by four. 425 00:23:20 --> 00:23:25 Many of them will be zero, because I've got all those 426 00:23:25 --> 00:23:26 zeros there. 427 00:23:26 --> 00:23:29 Maybe the whole determinant is zero. 428 00:23:29 --> 00:23:32 I mean, I -- is that a singular matrix? 429 00:23:32 --> 00:23:35 That possibility definitely exists. 430 00:23:35 --> 00:23:39 I could, I could, So one way to do it would be 431 00:23:39 --> 00:23:40 elimination. 432 00:23:40 --> 00:23:44 Actually, that would probably be a fairly reasonable way. 433 00:23:44 --> 00:23:47 I could use elimination, so I could use -- go back to 434 00:23:47 --> 00:23:50 those properties, that -- and use elimination, 435 00:23:50 --> 00:23:54 get down, eliminate it down, do I have a row of zeros at the 436 00:23:54 --> 00:23:56.55 end of elimination? 437 00:23:56.55 --> 00:23:58 The answer is zero. 438 00:23:58 --> 00:24:01 I was thinking, shall I try this big formula? 439 00:24:01 --> 00:24:02 OK. 440 00:24:02 --> 00:24:04 Let's try the big formula. 441 00:24:04 --> 00:24:08 How -- tell me one way I can go down the matrix, 442 00:24:08 --> 00:24:11 taking a one, taking a one from 443 00:24:11 --> 00:24:14 every row and column, and make it to the end? 444 00:24:14 --> 00:24:17 So it's -- I get something that isn't zero. 445 00:24:17 --> 00:24:21 Well, one way to do it, I could take that times that 446 00:24:21 --> 00:24:23 times that times that times that. 447 00:24:23 --> 00:24:26 That would be one and, and, and I just said, 448 00:24:26 --> 00:24:29 that comes in with what sign? 449 00:24:29 --> 00:24:29 Plus. 450 00:24:29 --> 00:24:32 That comes with a plus sign. 451 00:24:32 --> 00:24:38 Because, because that permutation -- I've just written 452 00:24:38 --> 00:24:44 the permutation about four three two one, and one exchange and a 453 00:24:44 --> 00:24:49 second exchange, two exchanges puts it in the 454 00:24:49 --> 00:24:50 correct order. 455 00:24:50 --> 00:24:53 Keep walking away, don't.... 456 00:24:53 --> 00:24:58 OK, we're executing a determinant 457 00:24:58 --> 00:25:00 formula here. 458 00:25:00 --> 00:25:06 Uh as long as it's not periodic, of course. 459 00:25:06 --> 00:25:11 If he comes back I'm in -- no. 460 00:25:11 --> 00:25:14 All right, all right. 461 00:25:14 --> 00:25:19 OK, so that would give me a plus one. 462 00:25:19 --> 00:25:21 All right. 463 00:25:21 --> 00:25:24 Are there any others? 464 00:25:24 --> 00:25:30 Well, of course we see another one here. 465 00:25:30 --> 00:25:35 This times this times this times this strikes us right 466 00:25:35 --> 00:25:35 away. 467 00:25:35 --> 00:25:40.5 So that's the order three, the order -- let me make a 468 00:25:40.5 --> 00:25:42 little different mark here. 469 00:25:42 --> 00:25:44 Three two one four. 470 00:25:44 --> 00:25:49 And is that a plus or a minus, three two one four? 471 00:25:49 --> 00:25:53 Is that, is that permutation a plus 472 00:25:53 --> 00:25:54.78 or a minus permutation? 473 00:25:54.78 --> 00:25:55 It's a minus. 474 00:25:55 --> 00:25:56 How do you see that? 475 00:25:56 --> 00:26:00 What exchange shall I do to get it in the right order? 476 00:26:00 --> 00:26:04 If I exchange the one and the three I'm in the right orders, 477 00:26:04 --> 00:26:08 took one exchange to do it, so that would be a plus -- that 478 00:26:08 --> 00:26:10 would be a minus one. 479 00:26:10 --> 00:26:14 And now I don't know if there're any more here. 480 00:26:14 --> 00:26:14 Let's see. 481 00:26:14 --> 00:26:17 Let me try again starting with this. 482 00:26:17 --> 00:26:21 Now I've got to pick somebody from -- oh yeah, 483 00:26:21 --> 00:26:24 see, you see what's happening. 484 00:26:24 --> 00:26:28 If I I start there, OK, column three is used. 485 00:26:28 --> 00:26:31 So then when I go to next row, I can't use that, 486 00:26:31 --> 00:26:31 I must use that. 487 00:26:31 --> 00:26:34 Now columns two and three are used. 488 00:26:34 --> 00:26:36 When I come to this row I must use that. 489 00:26:36 --> 00:26:37 And then I must use that. 490 00:26:37 --> 00:26:41 So if I start there, this is the only one I get. 491 00:26:41 --> 00:26:44 And similarly, if I start there, 492 00:26:44 --> 00:26:46.28 that's the only one I get. 493 00:26:46.28 --> 00:26:48 So what's the determinant? 494 00:26:48 --> 00:26:50 What's the determinant? 495 00:26:50 --> 00:26:50 Zero. 496 00:26:50 --> 00:26:53 The determinant is zero for that case. 497 00:26:53 --> 00:26:57.76 Because we, we were able to check the 498 00:26:57.76 --> 00:26:58 twenty-four terms. 499 00:26:58 --> 00:27:00.63 Twenty-two of them were zero. 500 00:27:00.63 --> 00:27:02 One of them was plus one. 501 00:27:02 --> 00:27:03 One of them was minus one. 502 00:27:03 --> 00:27:06 Add up the twenty-four terms, zero is the answer. 503 00:27:06 --> 00:27:06 OK. 504 00:27:06 --> 00:27:10 Well, I didn't know it would be zero, I -- because I wasn't, 505 00:27:10 --> 00:27:12 like, thinking ahead. 506 00:27:12 --> 00:27:15 I was a little scared, actually. 507 00:27:15 --> 00:27:17 I said, that, apparition went by. 508 00:27:17 --> 00:27:21 So and I don't know if the camera caught that. 509 00:27:21 --> 00:27:26 So whether the rest of the world will realize that I was in 510 00:27:26 --> 00:27:29 danger or not, we don't know. 511 00:27:29 --> 00:27:33.82 But anyway, I guess he just wanted to be sure 512 00:27:33.82 --> 00:27:37 that we got the right answer, which is determinant zero. 513 00:27:37 --> 00:27:41 And then that makes me think, OK, the matrix must be, 514 00:27:41 --> 00:27:43 the matrix must be singular. 515 00:27:43 --> 00:27:47 And then if the matrix is singular, maybe there's another 516 00:27:47 --> 00:27:52 way to see that it's singular, like find something in its null 517 00:27:52 --> 00:27:53 space. 518 00:27:53 --> 00:27:57 Or find a combination of the rows that gives zero. 519 00:27:57 --> 00:28:02 And like what d- what, what combination of those rows 520 00:28:02 --> 00:28:03 does give zero. 521 00:28:03 --> 00:28:06 Suppose I add rows one and rows three. 522 00:28:06 --> 00:28:11 If I add rows one and rows three, what do I get? 523 00:28:11 --> 00:28:14 I get a row of all ones. 524 00:28:14 --> 00:28:18 Then if I add rows two and rows four I get a row of all ones. 525 00:28:18 --> 00:28:22 So row one minus row two plus row three minus row four is 526 00:28:22 --> 00:28:23 probably the zero row. 527 00:28:23 --> 00:28:25 It's a singular matrix. 528 00:28:25 --> 00:28:28 And I could find something in its null space the same way. 529 00:28:28 --> 00:28:33 That would be a combination of columns that gives zero. 530 00:28:33 --> 00:28:35 OK, there's an example. 531 00:28:35 --> 00:28:35.93 All right. 532 00:28:35.93 --> 00:28:39 So that's, well, that shows two things. 533 00:28:39 --> 00:28:43 That shows how we get the twenty-four terms and it shows 534 00:28:43 --> 00:28:47 the great advantage of having a lot of zeros in there. 535 00:28:47 --> 00:28:47.98 OK. 536 00:28:47.98 --> 00:28:54 So we'll use this big formula, but I want to pick -- I want to 537 00:28:54 --> 00:28:57 go onward now to cofactors. 538 00:28:57 --> 00:28:59 Onward to cofactors. 539 00:28:59 --> 00:29:04.62 Cofactors is a way of breaking up this big formula that 540 00:29:04.62 --> 00:29:11 connects this n by n -- this is an n by n determinant that we've 541 00:29:11 --> 00:29:15.08 just have a formula for, the big formula. 542 00:29:15.08 --> 00:29:20 So cofactors is a way to connect this n by n determinant 543 00:29:20 --> 00:29:24 to, determinants one smaller. 544 00:29:24 --> 00:29:25 One smaller. 545 00:29:25 --> 00:29:32 And the way we want to do it is actually going to show up in 546 00:29:32 --> 00:29:32 this. 547 00:29:32 --> 00:29:38 Since the three by three is the one that we wrote out in full, 548 00:29:38 --> 00:29:44 let's, let me do this three by -- so I'm talking about 549 00:29:44 --> 00:29:51 cofactors, and I'm going to start again with three by three. 550 00:29:51 --> 00:29:55 And I'm going to take the, the exact formula, 551 00:29:55 --> 00:30:00 and I'm just going to write it as a one one -- this is the 552 00:30:00 --> 00:30:02 determinant I'm writing. 553 00:30:02 --> 00:30:06 I'm just going to say a one one times what? 554 00:30:06 --> 00:30:08 A one one times what? 555 00:30:08 --> 00:30:14 And it's a one one times a two two a three three minus a two 556 00:30:14 --> 00:30:16 three a three two. 557 00:30:16 --> 00:30:22 Then I've got the a one two stuff times something. 558 00:30:22 --> 00:30:28 And I've got the a one three stuff times something. 559 00:30:28 --> 00:30:30 Do you see what I'm doing? 560 00:30:30 --> 00:30:36 I'm taking our big formula and I'm saying, OK, 561 00:30:36 --> 00:30:41 choose column -- out of the first row, 562 00:30:41 --> 00:30:43 choose column one. 563 00:30:43 --> 00:30:46 And take all the possibilities. 564 00:30:46 --> 00:30:53 And those extra factors will be what we'll call the cofactor, 565 00:30:53 --> 00:30:56 co meaning going with a one one. 566 00:30:56 --> 00:31:00 So this in parenthesis are, these are in, 567 00:31:00 --> 00:31:03.81 the cofactors are in parens. 568 00:31:03.81 --> 00:31:06 A one one times something. 569 00:31:06 --> 00:31:12 And I figured out what that something was by just looking 570 00:31:12 --> 00:31:17 back -- if I can walk back here to the, to the a one one, 571 00:31:17 --> 00:31:21 the one that comes down the diagonal minus the one that 572 00:31:21 --> 00:31:22 comes that way. 573 00:31:22 --> 00:31:26.76 That's, those are the two, only two that used a one one. 574 00:31:26.76 --> 00:31:30 So there they are, one with a plus and one with a 575 00:31:30 --> 00:31:30 minus. 576 00:31:30 --> 00:31:36 And now I can write in the -- I could look back and see what 577 00:31:36 --> 00:31:40 used a one two and I can see what used a one three, 578 00:31:40 --> 00:31:45 and those will give me the cofactors of a one two and a one 579 00:31:45 --> 00:31:45 three. 580 00:31:45 --> 00:31:48 Before I do that, what's this number, 581 00:31:48 --> 00:31:50 what is this cofactor? 582 00:31:50 --> 00:31:55.13 What is it there that's multiplying a one one? 583 00:31:55.13 --> 00:32:01 Tell me what a two two a three three minus a two three a three 584 00:32:01 --> 00:32:05.57 two is, for this -- do you recognize that? 585 00:32:05.57 --> 00:32:11 Do you recognize -- let's see, I can -- and I'll put it here. 586 00:32:11 --> 00:32:13 There's the a one one. 587 00:32:13 --> 00:32:16 That's used column one. 588 00:32:16 --> 00:32:21.64 Then there's -- the other factors involved 589 00:32:21.64 --> 00:32:23 these other columns. 590 00:32:23 --> 00:32:25 This row is used. 591 00:32:25 --> 00:32:27 This column is used. 592 00:32:27 --> 00:32:32 So this the only things left to use are these. 593 00:32:32 --> 00:32:37 And this formula uses them, and what's the, 594 00:32:37 --> 00:32:39 what's the cofactor? 595 00:32:39 --> 00:32:46.33 Tell me what it is because you see it, and then -- 596 00:32:46.33 --> 00:32:50 I'll be happy you see what the idea of cofactors. 597 00:32:50 --> 00:32:54.39 It's the determinant of this smaller guy. 598 00:32:54.39 --> 00:32:59 A one one multiplies the determinant of this smaller guy. 599 00:32:59 --> 00:33:04 That gives me all the a one one part of the big formula. 600 00:33:04 --> 00:33:05 You see that? 601 00:33:05 --> 00:33:11 This, the determinant of this smaller guy is a two two a 602 00:33:11 --> 00:33:15 three three minus a two three a three two. 603 00:33:15 --> 00:33:19 In other words, once I've used column one and 604 00:33:19 --> 00:33:24 row one, what's left is all the ways to use the other n-1 605 00:33:24 --> 00:33:26 columns and n-1 rows, one of each. 606 00:33:26 --> 00:33:32 All the other -- and that's the determinant of the smaller guy 607 00:33:32 --> 00:33:34 of size n-1. 608 00:33:34 --> 00:33:37 So that's the whole idea of cofactors. 609 00:33:37 --> 00:33:42 And we just have to remember that with determinants we've got 610 00:33:42 --> 00:33:46 pluses and minus signs to keep straight. 611 00:33:46 --> 00:33:49 Can we keep this next one straight? 612 00:33:49 --> 00:33:51 Let's do the next one. 613 00:33:51 --> 00:33:54 OK, the next one will be when I use a one two. 614 00:33:54 --> 00:34:00 I'll have left -- so I can't use that column any 615 00:34:00 --> 00:34:07 more, but I can use a two one and a two three and I can use a 616 00:34:07 --> 00:34:10 three one and a three three. 617 00:34:10 --> 00:34:15 So this one gave me a one times that determinant. 618 00:34:15 --> 00:34:20 This will give me a one two times this determinant, 619 00:34:20 --> 00:34:27 a two one a three three minus a two three a three one. 620 00:34:27 --> 00:34:30 So that's all the stuff involving a one two. 621 00:34:30 --> 00:34:33.46 But have I got the sign right? 622 00:34:33.46 --> 00:34:38 Is the determinant of that correctly given by that or is 623 00:34:38 --> 00:34:39 there a minus sign? 624 00:34:39 --> 00:34:41.54 There is a minus sign. 625 00:34:41.54 --> 00:34:43 I can follow one of these. 626 00:34:43 --> 00:34:48 If I do that times that times that, that was one that's 627 00:34:48 --> 00:34:52 showing up here, but it should have showed 628 00:34:52 --> 00:34:55 -- it should have been a minus. 629 00:34:55 --> 00:34:59.91 So I'm going to build that minus sign into the cofactor. 630 00:34:59.91 --> 00:35:04 So, so the cofactor -- so I'll put, put that minus sign in 631 00:35:04 --> 00:35:05.14 here. 632 00:35:05.14 --> 00:35:09 So because the cofactor is going to be strictly the thing 633 00:35:09 --> 00:35:13 that multiplies the, the factor. 634 00:35:13 --> 00:35:16 The factor is a one two, the cofactor is this, 635 00:35:16 --> 00:35:19.43 is the parens, the stuff in parentheses. 636 00:35:19.43 --> 00:35:21 So it's got the minus sign built in. 637 00:35:21 --> 00:35:25 And if I did -- if I went on to the third guy, 638 00:35:25 --> 00:35:28 there w- there'll be this and this, this and this. 639 00:35:28 --> 00:35:31 And it would take its determinant. 640 00:35:31 --> 00:35:34 It would come out plus the determinant. 641 00:35:34 --> 00:35:38 So now I'm ready to say what cofactors are. 642 00:35:38 --> 00:35:42 So this would be a plus and a one three times its cofactor. 643 00:35:42 --> 00:35:47 And over here we had plus a one one times this determinant. 644 00:35:47 --> 00:35:51 But and there we had the a one two times its cofactor, 645 00:35:51 --> 00:35:58 but the -- so the point is the cofactor is either plus or 646 00:35:58 --> 00:36:00 minus the determinant. 647 00:36:00 --> 00:36:05 So let me write that underneath them. 648 00:36:05 --> 00:36:09 What is the, what are cofactors? 649 00:36:09 --> 00:36:14 The cofactor if any number aij, let's say. 650 00:36:14 --> 00:36:19 This is, this is all the terms in the, 651 00:36:19 --> 00:36:23 in the big formula that involve aij. 652 00:36:23 --> 00:36:29 We're especially interested in a1j, the first row, 653 00:36:29 --> 00:36:35 that's what I've been talking about, but any row would be all 654 00:36:35 --> 00:36:36 right. 655 00:36:36 --> 00:36:40 All right, so -- what terms involve aij? 656 00:36:40 --> 00:36:45 So -- it's the determinant of the n 657 00:36:45 --> 00:36:51 minus one matrix -- with row i, column j erased. 658 00:36:51 --> 00:36:57.23 So it's the, it's a matrix of size n-1 with 659 00:36:57.23 --> 00:37:02.86 -- of course, because I can't use this row or 660 00:37:02.86 --> 00:37:05 this column again. 661 00:37:05 --> 00:37:09 So I have the matrix all there. 662 00:37:09 --> 00:37:15 But now it's multiplied by a plus or a minus. 663 00:37:15 --> 00:37:19 This is the cofactor, and I'm going to call that cij. 664 00:37:19 --> 00:37:25 Capital, I use capital c just to, just to emphasize that these 665 00:37:25 --> 00:37:29 are important and emphasize that they're, they're, 666 00:37:29 --> 00:37:32 they're different from the (a)s. 667 00:37:32 --> 00:37:32 OK. 668 00:37:32 --> 00:37:36 So now is it a plus or is it a minus? 669 00:37:36 --> 00:37:42 Because we see that in this case, for a one one it was a 670 00:37:42 --> 00:37:46 plus, for a one two I -- this is ij -- it was a minus. 671 00:37:46 --> 00:37:49 For this ij it was a plus. 672 00:37:49 --> 00:37:54 So any any guess on the rule for plus or minus when we see 673 00:37:54 --> 00:37:58 those examples, ij equal one one or one three 674 00:37:58 --> 00:38:00 was a plus? 675 00:38:00 --> 00:38:05 It sounds very like i+j odd or even. 676 00:38:05 --> 00:38:12 That, that's doesn't surprise us, and that's the right answer. 677 00:38:12 --> 00:38:20 So it's a plus if i+j is even and it's a minus if i+j is odd. 678 00:38:20 --> 00:38:26 So if I go along row one and look at the cofactors, 679 00:38:26 --> 00:38:34 I just take those determinants, those one smaller determinants, 680 00:38:34 --> 00:38:38 and they come in order plus minus 681 00:38:38 --> 00:38:40 plus minus plus minus. 682 00:38:40 --> 00:38:45 But if I go along row two and, and, and take the cofactors of 683 00:38:45 --> 00:38:49 sub-determinants, they would start with a minus, 684 00:38:49 --> 00:38:53 because the two one entry, two plus one is odd, 685 00:38:53 --> 00:38:58 so the -- like there's a pattern plus minus plus minus 686 00:38:58 --> 00:39:05 plus if it was five by five, but then if I was doing a 687 00:39:05 --> 00:39:12 cofactor then this sign would be minus plus minus plus minus, 688 00:39:12 --> 00:39:17 plus minus plus -- it's sort of checkerboard. 689 00:39:17 --> 00:39:17 OK. 690 00:39:17 --> 00:39:17 OK. 691 00:39:17 --> 00:39:24.13 Those are the signs that, that are given by this rule, 692 00:39:24.13 --> 00:39:25 i+j even or odd. 693 00:39:25 --> 00:39:31 And those are built into the cofactors. 694 00:39:31 --> 00:39:36 The thing is called a minor without th- before you've built 695 00:39:36 --> 00:39:40 in the sign, but I don't care about those. 696 00:39:40 --> 00:39:44.02 Build in that sign and call it a cofactor. 697 00:39:44.02 --> 00:39:44 OK. 698 00:39:44 --> 00:39:47 So what's the cofactor formula? 699 00:39:47 --> 00:39:50 What's the cofactor formula then? 700 00:39:50 --> 00:39:54 Let me come back to this board and 701 00:39:54 --> 00:39:58 say, what's the cofactor formula? 702 00:39:58 --> 00:40:04 Determinant of A is -- let's go along the first row. 703 00:40:04 --> 00:40:11 It's a one one times its cofactor, and then the second 704 00:40:11 --> 00:40:18 guy is a one two times its cofactor, and you just keep 705 00:40:18 --> 00:40:25 going to the end of the row, a1n times its cofactor. 706 00:40:25 --> 00:40:31 So that's cofactor for -- along row one. 707 00:40:31 --> 00:40:38 And if I went along row I, I would -- those ones would be 708 00:40:38 --> 00:40:38 Is. 709 00:40:38 --> 00:40:42 That's worth putting a box over. 710 00:40:42 --> 00:40:46 That's the cofactor formula. 711 00:40:46 --> 00:40:53 Do you see that -- actually, this would give me another way 712 00:40:53 --> 00:40:59 I could have started the whole topic of determinants. 713 00:40:59 --> 00:41:05 And some, some people might do it 714 00:41:05 --> 00:41:07 this -- choose to do it this way. 715 00:41:07 --> 00:41:11 Because the cofactor formula would allow me to build up an n 716 00:41:11 --> 00:41:14 by n determinant out of n-1 sized determinants, 717 00:41:14 --> 00:41:17 build those out of n-2, and so on. 718 00:41:17 --> 00:41:19.99 I could boil all the way down to one by ones. 719 00:41:19.99 --> 00:41:23 So what's the cofactor formula for 720 00:41:23 --> 00:41:24 two by two matrices? 721 00:41:24 --> 00:41:26 Yeah, tell me that. 722 00:41:26 --> 00:41:28 What's the cofactor for us? 723 00:41:28 --> 00:41:32 Here is the, here is the world's smallest 724 00:41:32 --> 00:41:35 example, practically, of a cofactor formula. 725 00:41:35 --> 00:41:36 OK. 726 00:41:36 --> 00:41:38 Let's go along row one. 727 00:41:38 --> 00:41:42 I take this first guy times its cofactor. 728 00:41:42 --> 00:41:46 What's the cofactor of the one one entry? 729 00:41:46 --> 00:41:51 d, because you strike out the one one row and column and 730 00:41:51 --> 00:41:53 you're left with d. 731 00:41:53 --> 00:41:57 Then I take this guy, b, times its cofactor. 732 00:41:57 --> 00:42:00 What's the cofactor of b? 733 00:42:00 --> 00:42:05 Is it c or it's -- minus c, because I strike out 734 00:42:05 --> 00:42:10 this guy, I take that determinant, and then I follow 735 00:42:10 --> 00:42:15 the i+j rule and I get a minus, I get an odd. 736 00:42:15 --> 00:42:17 So it's b times minus c. 737 00:42:17 --> 00:42:19 OK, it worked. 738 00:42:19 --> 00:42:21 Of course it, it worked. 739 00:42:21 --> 00:42:24 And the three by three works. 740 00:42:24 --> 00:42:29 So that's the cofactor formula, and that is, 741 00:42:29 --> 00:42:33 that's an -- that's a good formula to know, 742 00:42:33 --> 00:42:39 and now I'm feeling like, wow, I'm giving you a lot of 743 00:42:39 --> 00:42:41 algebra to swallow here. 744 00:42:41 --> 00:42:45 Last lecture gave you ten properties. 745 00:42:45 --> 00:42:51 Now I'm giving you -- and by the way, those ten 746 00:42:51 --> 00:42:55 properties led us to a formula for the determinant which was 747 00:42:55 --> 00:42:58 very important, and I haven't repeated it till 748 00:42:58 --> 00:42:59 now. 749 00:42:59 --> 00:43:00 What was that? 750 00:43:00 --> 00:43:03 The, the determinant is the product of the pivots. 751 00:43:03 --> 00:43:06 So the pivot formula is, is very important. 752 00:43:06 --> 00:43:09 The pivots have all this complicated 753 00:43:09 --> 00:43:11 mess already built in. 754 00:43:11 --> 00:43:16 As you did elimination to get the pivots, you built in all 755 00:43:16 --> 00:43:19 this horrible stuff, quite efficiently. 756 00:43:19 --> 00:43:23 Then the big formula with the n factorial terms, 757 00:43:23 --> 00:43:27 that's got all the horrible stuff spread out. 758 00:43:27 --> 00:43:32 And the cofactor formula is like in between. 759 00:43:32 --> 00:43:36 It's got easy stuff times horrible stuff, 760 00:43:36 --> 00:43:37 basically. 761 00:43:37 --> 00:43:42 But it's, it shows you, how to get determinants from 762 00:43:42 --> 00:43:47 smaller determinants, and that's the application that 763 00:43:47 --> 00:43:49 I now want to make. 764 00:43:49 --> 00:43:52.7 So may I do one more example? 765 00:43:52.7 --> 00:43:55 So I remember the general idea. 766 00:43:55 --> 00:44:00 But I'm going to use this cofactor 767 00:44:00 --> 00:44:06 formula for a matrix -- so here is going to be my example. 768 00:44:06 --> 00:44:11.4 It's -- I promised in the, in the lecture, 769 00:44:11.4 --> 00:44:16 outline at the very beginning to do an example. 770 00:44:16 --> 00:44:22 And let me do -- I'm going to pick tri-diagonal matrix of 771 00:44:22 --> 00:44:24 ones. 772 00:44:24 --> 00:44:30.01 I could, I'm drawing here the four by four. 773 00:44:30.01 --> 00:44:33 So this will be the matrix. 774 00:44:33 --> 00:44:36 I could call that A4. 775 00:44:36 --> 00:44:40.93 But my real idea is to do n by n. 776 00:44:40.93 --> 00:44:42 To do them all. 777 00:44:42 --> 00:44:51 So A -- I could -- everybody understands what A1 and A2 are. 778 00:44:51 --> 00:44:53 Yeah. 779 00:44:53 --> 00:44:58.63 Maybe we should just do A1 and A2 and A3 just for -- so this is 780 00:44:58.63 --> 00:44:58 A4. 781 00:44:58 --> 00:45:01 What's the determinant of A1? 782 00:45:01 --> 00:45:04.09 What's the determinant of A1? 783 00:45:04.09 --> 00:45:07 So, so what's the matrix A1 in this formula? 784 00:45:07 --> 00:45:09.64 It's just got that. 785 00:45:09.64 --> 00:45:11 So the determinant is one. 786 00:45:11 --> 00:45:14 What's the determinant of A2? 787 00:45:14 --> 00:45:18 So it's just got this two by two, 788 00:45:18 --> 00:45:20 and its determinant is -- zero. 789 00:45:20 --> 00:45:23 And then the three by three. 790 00:45:23 --> 00:45:25 Can we see its determinant? 791 00:45:25 --> 00:45:29 Can you take the determinant of that three by three? 792 00:45:29 --> 00:45:33 Well, that's not quite so obvious, at least not to me. 793 00:45:33 --> 00:45:37 Being three by three, I don't know -- so here's a, 794 00:45:37 --> 00:45:40 here's a good example. 795 00:45:40 --> 00:45:43 How would you do that three by three determinant? 796 00:45:43 --> 00:45:46 We've got, like, n factorial different ways. 797 00:45:46 --> 00:45:47 Well, three factorial. 798 00:45:47 --> 00:45:49 So we've got six ways. 799 00:45:49 --> 00:45:49 OK. 800 00:45:49 --> 00:45:53 I mean, one way to do it -- actually the way I would 801 00:45:53 --> 00:45:55 probably do it, being three by three, 802 00:45:55 --> 00:45:58 I would use the complete the big formula. 803 00:45:58 --> 00:46:00 I would say, I've got a one from that, 804 00:46:00 --> 00:46:03 I've got a zero from that, I've got a zero from that, 805 00:46:03 --> 00:46:05 a zero from that, and this direction is a minus 806 00:46:05 --> 00:46:09.01 one, that direction's a minus one. 807 00:46:09.01 --> 00:46:11 I believe the answer is minus one. 808 00:46:11 --> 00:46:13 Would you do it another way? 809 00:46:13 --> 00:46:16 Here's another way to do it, look. 810 00:46:16 --> 00:46:21 Subtract row three from -- I'm just looking at this three by 811 00:46:21 --> 00:46:21 three. 812 00:46:21 --> 00:46:25 Everybody's looking at the three by three. 813 00:46:25 --> 00:46:28 Subtract row three from row two. 814 00:46:28 --> 00:46:30 Determinant doesn't change. 815 00:46:30 --> 00:46:32.44 So those become zeros. 816 00:46:32.44 --> 00:46:34 OK, now use the cofactor formula. 817 00:46:34 --> 00:46:35 How's that? 818 00:46:35 --> 00:46:40 How can, how -- if this was now zeros and I'm looking at this 819 00:46:40 --> 00:46:43 three by three, use the cofactor formula. 820 00:46:43 --> 00:46:48 Why not use the cofactor formula along that row? 821 00:46:48 --> 00:46:53 Because then I take that number times its cofactor, 822 00:46:53 --> 00:46:59 so I take this number -- let me put a box around it -- times its 823 00:46:59 --> 00:47:04.97 cofactor, which is the determinant of that and that, 824 00:47:04.97 --> 00:47:06 which is what? 825 00:47:06 --> 00:47:10 That two by two matrix has determinant one. 826 00:47:10 --> 00:47:13 So what's the cofactor? 827 00:47:13 --> 00:47:17 What's the cofactor of this guy here? 828 00:47:17 --> 00:47:21 Looking just at this three by three. 829 00:47:21 --> 00:47:25 The cofactor of that one is this determinant, 830 00:47:25 --> 00:47:28 which is one times negative. 831 00:47:28 --> 00:47:33 So that's why the answer came out minus one. 832 00:47:33 --> 00:47:34.59 OK. 833 00:47:34.59 --> 00:47:36 So I did the three by three. 834 00:47:36 --> 00:47:39 I don't know if we want to try the four by four. 835 00:47:39 --> 00:47:42 Yeah, let's -- I guess that was the point of my example, 836 00:47:42 --> 00:47:44 of course, so I have to try it. 837 00:47:44 --> 00:47:47 Sorry, I'm in a good mood today, so you have to stand for 838 00:47:47 --> 00:47:49.59 all the bad jokes. 839 00:47:49.59 --> 00:47:49 OK. 840 00:47:49 --> 00:47:50 OK. 841 00:47:50 --> 00:47:52.95 So what was the matrix? 842 00:47:52.95 --> 00:47:53 Ah. 843 00:47:53 --> 00:47:57 OK, now I'm ready for four by four. 844 00:47:57 --> 00:48:04 Who wants to -- who wants to guess the, the -- I don't know, 845 00:48:04 --> 00:48:10 frankly, this four by four, what's, what's the determinant. 846 00:48:10 --> 00:48:14 I plan to use cofactors. 847 00:48:14 --> 00:48:16 OK, let's use cofactors. 848 00:48:16 --> 00:48:21 The determinant of A4 is -- OK, let's use cofactors on the 849 00:48:21 --> 00:48:22 first row. 850 00:48:22 --> 00:48:23.77 Those are easy. 851 00:48:23.77 --> 00:48:28 So I multiply this number, which is a convenient one, 852 00:48:28 --> 00:48:30 times this determinant. 853 00:48:30 --> 00:48:34 So it's, it's one times the, this three by three 854 00:48:34 --> 00:48:35 determinant. 855 00:48:35 --> 00:48:40 Now what is -- do you recognize that matrix? 856 00:48:40 --> 00:48:40 It's A3. 857 00:48:40 --> 00:48:44 So it's one times the determinant of A3. 858 00:48:44 --> 00:48:49 Coming along this row is a one times this determinant, 859 00:48:49 --> 00:48:52 and it goes with a plus, right? 860 00:48:52 --> 00:48:55 And then we have this one. 861 00:48:55 --> 00:48:58 And what is its cofactor? 862 00:48:58 --> 00:49:01 Now I'm looking at, now I'm looking at this three 863 00:49:01 --> 00:49:05 by three, this three by three, so I'm looking at the three by 864 00:49:05 --> 00:49:07 three that I haven't X-ed out. 865 00:49:07 --> 00:49:11.12 What is that -- oh, now it, we did a plus or a -- 866 00:49:11.12 --> 00:49:14 is it plus this determinant, this three by three 867 00:49:14 --> 00:49:16 determinant, or minus it? 868 00:49:16 --> 00:49:19 It's minus it, right, because this is -- I'm 869 00:49:19 --> 00:49:23 starting in a one two position, and that's a minus. 870 00:49:23 --> 00:49:26 So I want minus this determinant. 871 00:49:26 --> 00:49:28 But these guys are X-ed out. 872 00:49:28 --> 00:49:28 OK. 873 00:49:28 --> 00:49:30 So I've got a three by three. 874 00:49:30 --> 00:49:32 Well, let's use cofactors again. 875 00:49:32 --> 00:49:38 Use cofactors of the column, because actually we could use 876 00:49:38 --> 00:49:41 cofactors of columns just as well as rows, 877 00:49:41 --> 00:49:44.64 because, because the transpose rule. 878 00:49:44.64 --> 00:49:48 So I'll take this one, which is now sitting in the 879 00:49:48 --> 00:49:52 plus position, times its determinant -- oh! 880 00:49:52 --> 00:49:52 Oh, hell. 881 00:49:52 --> 00:49:57 What -- oh yeah, I shouldn't have said hell, 882 00:49:57 --> 00:49:58 because it's all right. 883 00:49:58 --> 00:49:58 OK. 884 00:49:58 --> 00:50:00 One times the determinant. 885 00:50:00 --> 00:50:03 What is that matrix now that I'm taking the, 886 00:50:03 --> 00:50:04 this smaller one of? 887 00:50:04 --> 00:50:06 Oh, but there's a minus, right? 888 00:50:06 --> 00:50:10.4 The minus came from, from the fact that this was in 889 00:50:10.4 --> 00:50:13 the one two position and that's odd. 890 00:50:13 --> 00:50:20 So this is a minus one times -- and what's -- and then this one 891 00:50:20 --> 00:50:25 is the upper left, that's the one one position in 892 00:50:25 --> 00:50:27 its matrix, so plus. 893 00:50:27 --> 00:50:30 And what's this matrix here? 894 00:50:30 --> 00:50:32 Do you recognize that? 895 00:50:32 --> 00:50:36 That matrix is -- yes, please say it -- A2. 896 00:50:36 --> 00:50:41 And we -- that's our formula for any case. 897 00:50:41 --> 00:50:47.81 A of any size n is equal to the determinant of A n minus one, 898 00:50:47.81 --> 00:50:53 that's what came from taking the one in the upper corner, 899 00:50:53 --> 00:50:57 the first cofactor, minus the determinant of A n 900 00:50:57 --> 00:50:58 minus two. 901 00:50:58 --> 00:51:02 What we discovered there is true for all n. 902 00:51:02 --> 00:51:08 I didn't even mention it, but I stopped taking cofactors 903 00:51:08 --> 00:51:10 when I got this one. 904 00:51:10 --> 00:51:11 Why did I stop? 905 00:51:11 --> 00:51:15 Why didn't I take the cofactor of this guy? 906 00:51:15 --> 00:51:19 Because he's going to get multiplied by zero, 907 00:51:19 --> 00:51:21 and no, no use wasting time. 908 00:51:21 --> 00:51:22 Or this one too. 909 00:51:22 --> 00:51:27 The cofactor, her cofactor will be whatever 910 00:51:27 --> 00:51:30 that determinant is, but it'll be multiplied by 911 00:51:30 --> 00:51:32 zero, so I won't bother. 912 00:51:32 --> 00:51:34 OK, there is the formula. 913 00:51:34 --> 00:51:39 And that now tells us -- I could figure out what A4 is now. 914 00:51:39 --> 00:51:42 Oh yeah, finally I can get A4. 915 00:51:42 --> 00:51:45 Because it's A3, which is minus one, 916 00:51:45 --> 00:51:49 minus A2, which is zero, so it's minus one. 917 00:51:49 --> 00:51:54 You see how we're getting kind of numbers that you might not 918 00:51:54 --> 00:51:55 have guessed. 919 00:51:55 --> 00:52:00 So our sequence right now is one zero minus one minus one. 920 00:52:00 --> 00:52:03.71 What's the next one in the sequence, A5? 921 00:52:03.71 --> 00:52:08 A5 is this minus this, so it is zero. 922 00:52:08 --> 00:52:09 What's A6? 923 00:52:09 --> 00:52:13 A6 is this minus this, which is one. 924 00:52:13 --> 00:52:14 What's A7? 925 00:52:14 --> 00:52:22 I'm, I'm going to be stopped by either the time runs out or the 926 00:52:22 --> 00:52:24 board runs out. 927 00:52:24 --> 00:52:28 OK, A7 is this minus this, which is one. 928 00:52:28 --> 00:52:33 I'll stop here, because time is out, 929 00:52:33 --> 00:52:36.61 but let me tell you what we've got. 930 00:52:36.61 --> 00:52:41 What -- these determinants have this series, one zero minus one 931 00:52:41 --> 00:52:45 minus one zero one, and then it starts repeating. 932 00:52:45 --> 00:52:47 It's pretty fantastic. 933 00:52:47 --> 00:52:50.41 These determinants have period six. 934 00:52:50.41 --> 00:52:54 So the determinant of A sixty-one 935 00:52:54 --> 00:52:58 would be the determinant of A1, which would be one. 936 00:52:58 --> 00:52:58 OK. 937 00:52:58 --> 00:53:01 I hope you liked that example. 938 00:53:01 --> 00:53:05 A non-trivial example of a tri-diagonal determinant. 939 00:53:05 --> 00:53:05 Thanks. 940 00:53:05 --> 00:53:08 See you on Wednesday.