1 00:00:08,150 --> 00:00:13,310 OK, this is the second lecture on determinants. 2 00:00:13,310 --> 00:00:15,260 There are only three. 3 00:00:15,260 --> 00:00:19,270 With determinants it's a fascinating, small topic 4 00:00:19,270 --> 00:00:21,480 inside linear algebra. 5 00:00:21,480 --> 00:00:24,000 Used to be determinants were the big thing, 6 00:00:24,000 --> 00:00:27,780 and linear algebra was the little thing, but they -- 7 00:00:27,780 --> 00:00:30,510 those changed, that situation changed. 8 00:00:30,510 --> 00:00:36,170 Now determinants is one specific part, very neat little part. 9 00:00:36,170 --> 00:00:41,740 And my goal today is to find a formula for the determinant. 10 00:00:41,740 --> 00:00:45,480 It'll be a messy formula. 11 00:00:45,480 --> 00:00:49,570 So that's why you didn't see it right away. 12 00:00:49,570 --> 00:00:52,090 But if I'm given this n by n matrix 13 00:00:52,090 --> 00:00:55,890 then I use those entries to create 14 00:00:55,890 --> 00:00:57,450 this number, the determinant. 15 00:00:57,450 --> 00:00:58,620 So there's a formula for it. 16 00:00:58,620 --> 00:01:03,250 In fact, there's another formula, a second formula using 17 00:01:03,250 --> 00:01:05,129 something called cofactors. 18 00:01:05,129 --> 00:01:07,910 So you'll -- you have to know what cofactors are. 19 00:01:07,910 --> 00:01:10,590 And then I'll apply those formulas 20 00:01:10,590 --> 00:01:15,410 for some, some matrices that have a lot of zeros 21 00:01:15,410 --> 00:01:18,430 away from the three diagonals. 22 00:01:18,430 --> 00:01:19,220 OK. 23 00:01:19,220 --> 00:01:24,690 So I'm shooting now for a formula for the determinant. 24 00:01:24,690 --> 00:01:30,470 You remember we started with these three properties, three 25 00:01:30,470 --> 00:01:32,610 simple properties, but out of that we 26 00:01:32,610 --> 00:01:37,440 got all these amazing facts, like the determinant of A B 27 00:01:37,440 --> 00:01:42,210 equals determinant of A times determinant of B. 28 00:01:42,210 --> 00:01:45,280 But the three facts were -- 29 00:01:45,280 --> 00:01:48,950 oh, how about I just take two by twos. 30 00:01:48,950 --> 00:01:52,320 I know, because everybody here knows, the determinant of a two 31 00:01:52,320 --> 00:01:57,770 by two matrix, but let's get it out of these three formulas. 32 00:01:57,770 --> 00:02:00,940 OK, so here's my, my two by two matrix. 33 00:02:00,940 --> 00:02:04,160 I'm looking for a formula for this determinant. 34 00:02:04,160 --> 00:02:07,810 a b c d, OK. 35 00:02:07,810 --> 00:02:14,220 So property one, I know what to do with the identity. 36 00:02:14,220 --> 00:02:14,900 Right? 37 00:02:14,900 --> 00:02:19,190 Property two allows me to exchange rows, 38 00:02:19,190 --> 00:02:20,940 and I know what to do then. 39 00:02:20,940 --> 00:02:23,570 So I know that that determinant is one. 40 00:02:23,570 --> 00:02:27,280 Property two allows me to exchange rows and know 41 00:02:27,280 --> 00:02:32,400 that this determinant is minus one. 42 00:02:32,400 --> 00:02:37,970 And now I want to use property three to get everybody, 43 00:02:37,970 --> 00:02:39,210 to get everybody. 44 00:02:39,210 --> 00:02:40,850 And how will I do that? 45 00:02:40,850 --> 00:02:41,500 OK. 46 00:02:41,500 --> 00:02:46,600 So remember that if I keep the second row the same, 47 00:02:46,600 --> 00:02:52,590 I'm allowed to use linearity in the first row. 48 00:02:52,590 --> 00:02:54,750 And I'll just use it in a simple way. 49 00:02:54,750 --> 00:03:02,020 I'll write this vector a b as a 0 + 0 b. 50 00:03:04,590 --> 00:03:09,750 So that's one step using property three, linearity 51 00:03:09,750 --> 00:03:12,361 in the first row when the second row's the same. 52 00:03:12,361 --> 00:03:12,860 OK. 53 00:03:12,860 --> 00:03:15,650 But now you can guess what I'm going to do next. 54 00:03:15,650 --> 00:03:17,870 I'll -- because I'd like to -- 55 00:03:17,870 --> 00:03:20,040 if I can make the matrices diagonal, 56 00:03:20,040 --> 00:03:22,730 then I'm clearly there. 57 00:03:22,730 --> 00:03:24,810 So I'll take this one. 58 00:03:24,810 --> 00:03:28,470 Now I'll keep the first row fixed and split the second row, 59 00:03:28,470 --> 00:03:32,510 so that'll be an a 0 and I'll split that 60 00:03:32,510 --> 00:03:39,660 into a c 0 and, keeping that first row the same, a 0 d. 61 00:03:39,660 --> 00:03:42,920 I used, for this part, linearity. 62 00:03:42,920 --> 00:03:47,600 And now I'll -- whoops, that's plus because I've got more 63 00:03:47,600 --> 00:03:48,710 coming. 64 00:03:48,710 --> 00:03:50,530 This one I'll do the same. 65 00:03:50,530 --> 00:03:53,710 I'll keep this first row the same 66 00:03:53,710 --> 00:03:59,320 and I'll split c d into c 0 and 0 d. 67 00:03:59,320 --> 00:04:00,170 OK. 68 00:04:00,170 --> 00:04:03,310 Now I've got four easy determinants, 69 00:04:03,310 --> 00:04:05,250 and two of them are -- 70 00:04:05,250 --> 00:04:07,460 well, all four are extremely easy. 71 00:04:07,460 --> 00:04:12,020 Two of them are so easy as to turn into zero, right? 72 00:04:12,020 --> 00:04:17,500 Which two of these determinants are zero right away? 73 00:04:17,500 --> 00:04:20,670 The first guy is zero. 74 00:04:20,670 --> 00:04:21,720 Why is he zero? 75 00:04:21,720 --> 00:04:26,520 Why is that determinant nothing, forget him? 76 00:04:26,520 --> 00:04:30,090 Well, it has a column of zeros. 77 00:04:30,090 --> 00:04:33,630 And by the -- well, so one way to think is, well, 78 00:04:33,630 --> 00:04:35,170 it's a singular matrix. 79 00:04:35,170 --> 00:04:38,150 Oh, for, for like forty-eight different reasons, 80 00:04:38,150 --> 00:04:40,060 that determinant is zero. 81 00:04:40,060 --> 00:04:43,100 It's a singular matrix that has a column of zeros. 82 00:04:43,100 --> 00:04:44,570 It's, it's dead. 83 00:04:44,570 --> 00:04:47,900 And this one is about as dead too. 84 00:04:47,900 --> 00:04:49,191 Column of zeros. 85 00:04:49,191 --> 00:04:49,690 OK. 86 00:04:49,690 --> 00:04:51,650 So that's leaving us with this one. 87 00:04:51,650 --> 00:04:54,370 Now what do I -- how do I know its determinant, 88 00:04:54,370 --> 00:04:56,910 following the rules? 89 00:04:56,910 --> 00:05:00,660 Well, I guess one of the properties that we actually got 90 00:05:00,660 --> 00:05:07,120 to was the determinant of that -- diagonal matrix, then -- 91 00:05:07,120 --> 00:05:12,340 so I, I'm finally getting to that determinant is the a d. 92 00:05:12,340 --> 00:05:16,030 And this determinant is what? 93 00:05:16,030 --> 00:05:18,010 What's this one? 94 00:05:18,010 --> 00:05:23,120 Minus -- because I would use property two to do a flip 95 00:05:23,120 --> 00:05:29,220 to make it c b, then property three to factor out the b, 96 00:05:29,220 --> 00:05:31,370 property c to factor out the c -- 97 00:05:31,370 --> 00:05:35,320 the property again to factor out the c, and that minus, 98 00:05:35,320 --> 00:05:39,860 and of course finally I got the answer that we knew we would 99 00:05:39,860 --> 00:05:42,110 get. 100 00:05:42,110 --> 00:05:44,810 But you see the method. 101 00:05:44,810 --> 00:05:48,250 You see the method, because it's method I'm looking for here, 102 00:05:48,250 --> 00:05:52,190 not just a two by two answer but the method of doing -- 103 00:05:52,190 --> 00:05:59,220 now I can do three by threes and four by fours and any size. 104 00:05:59,220 --> 00:06:05,470 So if you can see the method of taking each row at a time -- 105 00:06:05,470 --> 00:06:07,880 so let's -- what would happen with three by threes? 106 00:06:07,880 --> 00:06:10,190 Can we mentally do it rather than I 107 00:06:10,190 --> 00:06:13,730 write everything on the board for three by threes? 108 00:06:13,730 --> 00:06:16,900 So what would we do if I had three by threes? 109 00:06:16,900 --> 00:06:20,360 I would keep rows two and three the same 110 00:06:20,360 --> 00:06:25,540 and I would split the first row into how many pieces? 111 00:06:25,540 --> 00:06:26,570 Three pieces. 112 00:06:26,570 --> 00:06:30,050 I'd have an A zero zero and a zero B zero 113 00:06:30,050 --> 00:06:35,670 and a zero zero C or something for the first row. 114 00:06:35,670 --> 00:06:40,250 So I would instead of going from one piece to two pieces to four 115 00:06:40,250 --> 00:06:48,560 pieces, I would go from one piece to three pieces to -- 116 00:06:48,560 --> 00:06:49,650 what would it be? 117 00:06:49,650 --> 00:06:54,290 Each of those three, would, would it be nine? 118 00:06:54,290 --> 00:06:56,280 Or twenty-seven? 119 00:06:56,280 --> 00:06:58,940 Oh yeah, I've actually got more steps, 120 00:06:58,940 --> 00:06:59,510 right. 121 00:06:59,510 --> 00:07:02,520 I'd go to nine but then I'd have another row to straighten out, 122 00:07:02,520 --> 00:07:03,200 twenty-seven. 123 00:07:03,200 --> 00:07:04,630 Yes, oh God. 124 00:07:04,630 --> 00:07:06,610 OK, let me say this again then. 125 00:07:06,610 --> 00:07:10,170 If I -- if it was three by three, I would -- 126 00:07:10,170 --> 00:07:12,830 separating out one row into three pieces 127 00:07:12,830 --> 00:07:16,820 would give me three, separating out the second row into three 128 00:07:16,820 --> 00:07:20,230 pieces, then I'd be up to nine, separating out the third row 129 00:07:20,230 --> 00:07:24,560 into its three pieces, I'd be up to twenty-seven, three cubed, 130 00:07:24,560 --> 00:07:25,290 pieces. 131 00:07:25,290 --> 00:07:28,470 But a lot of them would be zero. 132 00:07:28,470 --> 00:07:31,350 So now when would they not be zero? 133 00:07:31,350 --> 00:07:35,480 Tell me the pieces that would not be zero. 134 00:07:35,480 --> 00:07:37,450 Now I will write the non-zero ones. 135 00:07:37,450 --> 00:07:39,790 OK, so I have this matrix. 136 00:07:39,790 --> 00:07:43,520 I think I have to use these, start 137 00:07:43,520 --> 00:07:50,590 using these double symbols here because otherwise I could never 138 00:07:50,590 --> 00:07:52,970 do n by n. 139 00:07:52,970 --> 00:07:54,080 OK. 140 00:07:54,080 --> 00:07:55,220 OK. 141 00:07:55,220 --> 00:07:57,075 So I split this up like crazy. 142 00:08:00,200 --> 00:08:01,730 A bunch of pieces are zero. 143 00:08:01,730 --> 00:08:06,790 Whenever I have a column of zeros, I know I've got zero. 144 00:08:06,790 --> 00:08:08,620 When do I not have zero? 145 00:08:08,620 --> 00:08:12,950 When do I have -- what is it that's like these guys? 146 00:08:12,950 --> 00:08:16,330 These are the survivors, two survivors there. 147 00:08:16,330 --> 00:08:18,080 So my question for three by three 148 00:08:18,080 --> 00:08:20,990 is going to be what are the survivors? 149 00:08:20,990 --> 00:08:22,760 How many survivors are there? 150 00:08:22,760 --> 00:08:24,290 What are they? 151 00:08:24,290 --> 00:08:28,010 And when do I get a survivor. 152 00:08:28,010 --> 00:08:30,780 Well, I would get a survivor -- 153 00:08:30,780 --> 00:08:32,730 for example, one survivor will be 154 00:08:32,730 --> 00:08:35,960 that one times that one times that one, with all zeros 155 00:08:35,960 --> 00:08:37,340 everywhere else. 156 00:08:37,340 --> 00:08:39,320 That would be one survivor. 157 00:08:39,320 --> 00:08:44,330 a one one zero zero zero a two two zero zero 158 00:08:44,330 --> 00:08:47,030 zero a three three. 159 00:08:47,030 --> 00:08:50,990 That's like the a d survivor. 160 00:08:50,990 --> 00:08:53,970 Tell me another survivor. 161 00:08:53,970 --> 00:08:58,560 What other thing -- oh, now here you see the clue. 162 00:08:58,560 --> 00:09:00,750 Now can -- shall I just say the whole clue? 163 00:09:00,750 --> 00:09:02,920 That I'm having -- 164 00:09:02,920 --> 00:09:10,460 the survivors have one entry from each row and each column. 165 00:09:10,460 --> 00:09:14,430 One entry from each row and column. 166 00:09:14,430 --> 00:09:16,760 Because if some column is missing, 167 00:09:16,760 --> 00:09:20,600 then I get a singular matrix. 168 00:09:20,600 --> 00:09:22,960 And that, that's one of these guys. 169 00:09:22,960 --> 00:09:25,470 See, you see what happened with -- 170 00:09:25,470 --> 00:09:27,480 this guy? 171 00:09:27,480 --> 00:09:32,950 Column one never got used in 0 b 0 d. 172 00:09:32,950 --> 00:09:35,620 So its determinant was zero and I forget it. 173 00:09:35,620 --> 00:09:37,510 So I'm going to forget those and just put -- 174 00:09:37,510 --> 00:09:42,750 so tell me one more that would be a survivor? 175 00:09:42,750 --> 00:09:44,960 Well -- well, here's another one. 176 00:09:44,960 --> 00:09:50,690 a one one zero zero -- now OK, that's used up row -- row one 177 00:09:50,690 --> 00:09:51,800 is used. 178 00:09:51,800 --> 00:09:54,420 Column one is already used so it better be zero. 179 00:09:57,510 --> 00:09:58,850 What else could I have? 180 00:09:58,850 --> 00:10:03,050 Where could I pick the guy -- which column shall I use in row 181 00:10:03,050 --> 00:10:04,360 two? 182 00:10:04,360 --> 00:10:08,070 Use column three, because here if I use column -- 183 00:10:08,070 --> 00:10:10,140 here I used column one and row one. 184 00:10:10,140 --> 00:10:12,710 This was like the column -- 185 00:10:12,710 --> 00:10:15,950 numbers were one two three, right in order. 186 00:10:15,950 --> 00:10:23,140 Now the column numbers are going to be one three, column three, 187 00:10:23,140 --> 00:10:26,070 and column two. 188 00:10:26,070 --> 00:10:29,620 So the row numbers are one two three, of course. 189 00:10:29,620 --> 00:10:32,100 The column numbers are some -- 190 00:10:32,100 --> 00:10:35,550 OK, some permutation of one two three, 191 00:10:35,550 --> 00:10:38,880 and here they come in the order one three two. 192 00:10:38,880 --> 00:10:42,060 It's just like having a permutation matrix 193 00:10:42,060 --> 00:10:46,330 with, instead of the ones, with numbers. 194 00:10:46,330 --> 00:10:50,940 And actually, it's very close to having a permutation matrix, 195 00:10:50,940 --> 00:10:55,550 because I, what I do eventually is I factor out these numbers 196 00:10:55,550 --> 00:10:57,630 and then I have got. 197 00:10:57,630 --> 00:10:59,520 So what is that determinant equal? 198 00:10:59,520 --> 00:11:01,120 I factor those numbers out and I've 199 00:11:01,120 --> 00:11:05,400 got a one one times a two two times a three three. 200 00:11:05,400 --> 00:11:08,020 And what does this determinant equal? 201 00:11:08,020 --> 00:11:09,530 Yeah, now tell me the, this -- 202 00:11:09,530 --> 00:11:13,140 I mean, we're really getting to the heart of these formulas 203 00:11:13,140 --> 00:11:13,690 now. 204 00:11:13,690 --> 00:11:16,800 What is that determinant? 205 00:11:16,800 --> 00:11:20,400 By the laws of -- by, by our three properties, 206 00:11:20,400 --> 00:11:24,920 I can factor these out, I can factor out the a one one, 207 00:11:24,920 --> 00:11:27,780 the a two three, and the a three two. 208 00:11:27,780 --> 00:11:28,980 They're in separate rows. 209 00:11:28,980 --> 00:11:31,960 I can do each row separately. 210 00:11:31,960 --> 00:11:35,230 And then I just have to decide is that plus sign 211 00:11:35,230 --> 00:11:37,530 or is that a minus sign? 212 00:11:37,530 --> 00:11:42,640 And the answer is it's a minus. 213 00:11:42,640 --> 00:11:43,490 Why minus? 214 00:11:43,490 --> 00:11:47,350 Because these is one row exchange 215 00:11:47,350 --> 00:11:49,780 to get it back to the identity. 216 00:11:49,780 --> 00:11:52,740 So that's a minus. 217 00:11:52,740 --> 00:11:53,960 Now I through? 218 00:11:53,960 --> 00:11:55,455 No, because there are other ways. 219 00:11:59,020 --> 00:12:02,060 What I'm really through with, what 220 00:12:02,060 --> 00:12:04,510 I've done, what I've, what I've completed 221 00:12:04,510 --> 00:12:08,110 is only the part where the a one one is there. 222 00:12:08,110 --> 00:12:11,810 But now I've got parts where it's a one two. 223 00:12:15,010 --> 00:12:18,820 And now if it's a one two that row is used, that column is 224 00:12:18,820 --> 00:12:19,610 used. 225 00:12:19,610 --> 00:12:21,260 You see that idea? 226 00:12:21,260 --> 00:12:25,380 I could use this row and column. 227 00:12:25,380 --> 00:12:28,070 Now that column is used, that column is used, 228 00:12:28,070 --> 00:12:31,300 and this guy has to be here, a three three. 229 00:12:31,300 --> 00:12:33,190 And what's that determinant? 230 00:12:33,190 --> 00:12:38,510 That's an a one two times an a two one times an a three three, 231 00:12:38,510 --> 00:12:42,510 and does it have a plus or a minus? 232 00:12:42,510 --> 00:12:43,820 A minus is right. 233 00:12:43,820 --> 00:12:45,600 It has a minus. 234 00:12:45,600 --> 00:12:47,690 Because it's one flip away from an id- 235 00:12:47,690 --> 00:12:51,730 from the, regular, the right order, the diagonal order. 236 00:12:51,730 --> 00:12:54,200 And now what's the other guy with a -- with, 237 00:12:54,200 --> 00:12:57,470 a one two up there? 238 00:12:57,470 --> 00:12:59,240 I could have used this row. 239 00:12:59,240 --> 00:13:05,080 I could have put this guy here and this guy here. 240 00:13:05,080 --> 00:13:05,580 Right? 241 00:13:05,580 --> 00:13:07,730 You see the whole deal? 242 00:13:07,730 --> 00:13:14,030 Now that's an a one two, a two three, a three one, 243 00:13:14,030 --> 00:13:17,970 and does that go with a plus or a minus? 244 00:13:17,970 --> 00:13:19,720 Yeah, now that takes a minute of thinking, 245 00:13:19,720 --> 00:13:23,210 doesn't it, because one row exchange doesn't get it 246 00:13:23,210 --> 00:13:24,760 in line. 247 00:13:24,760 --> 00:13:26,300 So what is the answer for this? 248 00:13:26,300 --> 00:13:28,090 Plus or minus? 249 00:13:28,090 --> 00:13:32,120 Plus, because it takes two exchanges. 250 00:13:32,120 --> 00:13:35,920 I could exchange rows one and three and then two and three. 251 00:13:35,920 --> 00:13:40,360 Two exchanges makes this thing a plus. 252 00:13:40,360 --> 00:13:43,300 And then finally we have -- we're going to have two more. 253 00:13:43,300 --> 00:13:43,800 OK. 254 00:13:43,800 --> 00:13:52,770 Zero zero a one three, a two one zero zero, zero a three two 255 00:13:52,770 --> 00:13:54,160 zero. 256 00:13:54,160 --> 00:13:55,720 And one more guy. 257 00:13:55,720 --> 00:14:00,820 Zero zero a one three, zero a two 258 00:14:00,820 --> 00:14:06,750 two zero, A three one zero zero. 259 00:14:06,750 --> 00:14:08,870 And let's put down what we get from those. 260 00:14:08,870 --> 00:14:14,160 An a one three, an a two one, and an a three two, and I 261 00:14:14,160 --> 00:14:16,650 think that one is a plus. 262 00:14:16,650 --> 00:14:21,510 And this guys is a minus because one exchange would put it -- 263 00:14:21,510 --> 00:14:24,864 would order it. 264 00:14:24,864 --> 00:14:25,655 And that's a minus. 265 00:14:29,230 --> 00:14:32,510 All right, that has taken one whole board 266 00:14:32,510 --> 00:14:35,030 just to do the three by three. 267 00:14:35,030 --> 00:14:37,630 But do you agree that we now have 268 00:14:37,630 --> 00:14:42,590 a formula for the determinant which 269 00:14:42,590 --> 00:14:44,080 came from the three properties? 270 00:14:46,840 --> 00:14:50,440 And it must be it. 271 00:14:50,440 --> 00:14:53,090 And I'm going to keep that formula. 272 00:14:53,090 --> 00:14:57,870 That's a famous -- that three by three formula is one that 273 00:14:57,870 --> 00:15:01,570 if, if the cameras will follow me back to the beginning here, 274 00:15:01,570 --> 00:15:07,130 I, I get the ones with the plus sign are the ones that go down 275 00:15:07,130 --> 00:15:08,800 like down this way. 276 00:15:08,800 --> 00:15:10,390 And the ones with the minus signs 277 00:15:10,390 --> 00:15:13,940 are sort of the ones that go this way. 278 00:15:13,940 --> 00:15:17,440 I won't make that precise. 279 00:15:17,440 --> 00:15:20,700 For two reasons, one, it would clutter up 280 00:15:20,700 --> 00:15:24,970 the board, and second reason, it wouldn't be right for four 281 00:15:24,970 --> 00:15:26,300 by fours. 282 00:15:26,300 --> 00:15:29,540 For four by four, let me just say right away, 283 00:15:29,540 --> 00:15:33,360 four by four matrix -- the, the cross diagonal, 284 00:15:33,360 --> 00:15:38,190 the wrong diagonal happens to come out with a plus sign. 285 00:15:38,190 --> 00:15:39,800 Why is that? 286 00:15:39,800 --> 00:15:43,600 If I have a four by four matrix with ones 287 00:15:43,600 --> 00:15:50,080 coming on the counter diagonal, that determinant is plus. 288 00:15:50,080 --> 00:15:50,980 Why? 289 00:15:50,980 --> 00:15:54,980 Why plus for that guy? 290 00:15:54,980 --> 00:15:59,720 Because if I exchange rows one and four and then 291 00:15:59,720 --> 00:16:02,850 I exchange rows two and three, I've got the identity, 292 00:16:02,850 --> 00:16:05,590 and I did two exchanges. 293 00:16:05,590 --> 00:16:10,880 So this down to this, like, you know, down toward Miami 294 00:16:10,880 --> 00:16:16,330 and down toward LA stuff is, like, three by three only. 295 00:16:16,330 --> 00:16:16,830 OK. 296 00:16:19,510 --> 00:16:25,185 But I do want to get now -- 297 00:16:25,185 --> 00:16:27,310 I don't want to go through this for a four by four. 298 00:16:29,990 --> 00:16:34,280 I do want to get now the general formula. 299 00:16:34,280 --> 00:16:39,570 So this is what I refer to in the book as the big formula. 300 00:16:39,570 --> 00:16:43,140 So now this is the big formula for the determinant. 301 00:16:43,140 --> 00:16:47,380 I'm asking you to make a jump from two by two and three 302 00:16:47,380 --> 00:16:50,260 by three to n by n. 303 00:16:50,260 --> 00:16:52,220 OK, so this will be the big formula. 304 00:17:00,250 --> 00:17:07,310 That the determinant of A is the sum of a whole lot of terms. 305 00:17:07,310 --> 00:17:09,550 And what are those terms? 306 00:17:09,550 --> 00:17:12,490 And, and is it a plus or a minus sign, 307 00:17:12,490 --> 00:17:14,869 and I have to tell you which, which it is, 308 00:17:14,869 --> 00:17:18,520 because this came in -- in the three by three case, 309 00:17:18,520 --> 00:17:20,140 I had how many terms? 310 00:17:20,140 --> 00:17:21,859 Six. 311 00:17:21,859 --> 00:17:25,630 And half were plus and half were minus. 312 00:17:25,630 --> 00:17:30,570 How many terms are you figuring for four by four? 313 00:17:30,570 --> 00:17:36,520 If I get two terms in the two by two case, three -- 314 00:17:36,520 --> 00:17:41,760 six terms in the three by three case, what's that pattern? 315 00:17:41,760 --> 00:17:44,210 How many terms in the four by four case? 316 00:17:46,830 --> 00:17:48,320 Twenty-four. 317 00:17:48,320 --> 00:17:49,715 Four factorial. 318 00:17:52,430 --> 00:17:53,750 Why four factorial? 319 00:17:53,750 --> 00:17:56,200 This will be a sum of n factorial terms. 320 00:18:00,000 --> 00:18:01,530 Twenty-four, a hundred and twenty, 321 00:18:01,530 --> 00:18:05,570 seven hundred and twenty, whatever's after that. 322 00:18:05,570 --> 00:18:06,580 OK. 323 00:18:06,580 --> 00:18:08,945 Half plus and half minus. 324 00:18:12,110 --> 00:18:14,720 And where do those n factorial -- terms come from? 325 00:18:14,720 --> 00:18:17,310 This is the moment to listen to this lecture. 326 00:18:17,310 --> 00:18:20,110 Where do those n factorial terms come from? 327 00:18:20,110 --> 00:18:23,690 They come because the first, the guy in the first row 328 00:18:23,690 --> 00:18:26,940 can be chosen n ways. 329 00:18:26,940 --> 00:18:33,390 And after he's chosen, that's used up that, that column. 330 00:18:33,390 --> 00:18:38,440 So the one in the second row can be chosen n minus one ways. 331 00:18:38,440 --> 00:18:42,360 And after she's chosen, that second column has been 332 00:18:42,360 --> 00:18:43,160 used. 333 00:18:43,160 --> 00:18:46,650 And then the one in the third row can be chosen n minus two 334 00:18:46,650 --> 00:18:49,140 ways, and after it's chosen -- 335 00:18:49,140 --> 00:18:52,330 notice how I'm getting these personal pronouns. 336 00:18:52,330 --> 00:18:53,510 But I've run out. 337 00:18:53,510 --> 00:18:59,570 And I'm not willing to stop with three by three, 338 00:18:59,570 --> 00:19:02,370 so I'm just going to write the formula down. 339 00:19:02,370 --> 00:19:07,470 So the one in the first row comes from some column alpha. 340 00:19:07,470 --> 00:19:10,670 I don't know what alpha is. 341 00:19:10,670 --> 00:19:11,600 And the one in the -- 342 00:19:11,600 --> 00:19:14,410 I multiply that by somebody in the second row that comes 343 00:19:14,410 --> 00:19:16,530 from some different column. 344 00:19:16,530 --> 00:19:19,470 And I multiply that by somebody in the third row who comes 345 00:19:19,470 --> 00:19:21,850 from some yet different column. 346 00:19:21,850 --> 00:19:25,269 And then in the n-th row, I don't know what -- 347 00:19:25,269 --> 00:19:26,310 I don't know how to draw. 348 00:19:26,310 --> 00:19:29,570 Maybe omega, for last. 349 00:19:29,570 --> 00:19:32,170 And the whole point is then that -- 350 00:19:32,170 --> 00:19:34,510 that those column numbers are different, 351 00:19:34,510 --> 00:19:40,990 that alpha, beta, gamma, omega, that set of column numbers 352 00:19:40,990 --> 00:19:50,100 is some permutation, permutation of one to n. 353 00:19:50,100 --> 00:19:54,830 It, it, the n column numbers are each used once. 354 00:19:54,830 --> 00:19:57,570 And that gives us n factorial terms. 355 00:19:57,570 --> 00:20:02,130 And when I choose a term, that means 356 00:20:02,130 --> 00:20:04,850 I'm choosing somebody from every row and column. 357 00:20:04,850 --> 00:20:10,300 And then I just -- like the way I had this from row and column 358 00:20:10,300 --> 00:20:14,160 one, row and column two, row and column three, so that -- 359 00:20:14,160 --> 00:20:19,080 what was the alpha beta stuff in that, for that term here? 360 00:20:19,080 --> 00:20:22,360 Alpha was one, beta was two, gamma was three. 361 00:20:22,360 --> 00:20:26,180 The permutation was, was the trivial permutation, one 362 00:20:26,180 --> 00:20:28,140 two three, everybody in the right order. 363 00:20:30,730 --> 00:20:31,750 You see that formula? 364 00:20:34,420 --> 00:20:37,120 It's -- do you see why I didn't want to start with that 365 00:20:37,120 --> 00:20:39,930 the first day, Friday? 366 00:20:39,930 --> 00:20:42,470 I'd rather we understood the properties. 367 00:20:42,470 --> 00:20:44,920 Because out of this formula, presumably I 368 00:20:44,920 --> 00:20:47,720 could figure out all these properties. 369 00:20:47,720 --> 00:20:50,900 How would I know that the determinant of the identity 370 00:20:50,900 --> 00:20:56,630 matrix was one, for example, out of this formula? 371 00:20:56,630 --> 00:20:59,680 Why is -- if A is the identity matrix, 372 00:20:59,680 --> 00:21:04,080 how does this formula give me a plus one? 373 00:21:04,080 --> 00:21:05,160 You see it, right? 374 00:21:05,160 --> 00:21:10,440 Because, because almost all the terms are zeros. 375 00:21:10,440 --> 00:21:15,450 Which term isn't zero, if, if A is the identity matrix? 376 00:21:15,450 --> 00:21:18,501 Almost all the terms are zero because almost all the As are 377 00:21:18,501 --> 00:21:19,000 zero. 378 00:21:19,000 --> 00:21:21,070 It's only, the only time I'll get something 379 00:21:21,070 --> 00:21:25,270 is if it's a one one times a two two times a three three. 380 00:21:25,270 --> 00:21:28,590 Only, only the, only the permutation 381 00:21:28,590 --> 00:21:32,650 that's in the right order will, will give me something. 382 00:21:32,650 --> 00:21:34,400 It'll come with a plus sign. 383 00:21:34,400 --> 00:21:37,490 And the determinant of the identity is one. 384 00:21:37,490 --> 00:21:40,910 So, so we could go back from this formula and prove 385 00:21:40,910 --> 00:21:41,710 everything. 386 00:21:41,710 --> 00:21:45,460 We could even try to prove that the determinant of A B 387 00:21:45,460 --> 00:21:48,860 was the determinant of A times the determinant of B. 388 00:21:48,860 --> 00:21:51,070 But like next week we would still be working on it, 389 00:21:51,070 --> 00:21:54,600 because it's not -- 390 00:21:54,600 --> 00:21:56,500 clear from -- if I took A B, 391 00:21:56,500 --> 00:21:57,030 my God. 392 00:21:57,030 --> 00:21:57,530 You know --. 393 00:21:57,530 --> 00:22:02,210 The entries of A B would be all these pieces. 394 00:22:02,210 --> 00:22:06,630 Well, probably, it's probably -- historically it's been done, 395 00:22:06,630 --> 00:22:09,131 but it won't be repeated in eighteen oh six. 396 00:22:09,131 --> 00:22:09,630 OK. 397 00:22:09,630 --> 00:22:16,480 It would be possible probably to see, why the determinant of A 398 00:22:16,480 --> 00:22:18,190 equals the determinant of A transpose. 399 00:22:18,190 --> 00:22:21,154 That was another, like, miracle property at the end. 400 00:22:21,154 --> 00:22:22,820 That would, that would, that's an easier 401 00:22:22,820 --> 00:22:25,290 one, which we could find. 402 00:22:25,290 --> 00:22:26,200 OK. 403 00:22:26,200 --> 00:22:30,460 Is that all right for the big formula? 404 00:22:30,460 --> 00:22:33,160 I could take you then a, a typical -- 405 00:22:33,160 --> 00:22:36,070 let me do an example. 406 00:22:36,070 --> 00:22:39,280 Which I'll just create. 407 00:22:39,280 --> 00:22:42,410 I'll take a four by four matrix. 408 00:22:42,410 --> 00:22:46,820 I'll put some, I'll put some ones in and some zeros in. 409 00:22:46,820 --> 00:22:47,350 OK. 410 00:22:47,350 --> 00:22:48,530 Let me -- 411 00:22:48,530 --> 00:22:52,690 I don't know how many to put in, to tell the truth. 412 00:22:52,690 --> 00:22:54,100 I've never done this before. 413 00:22:58,840 --> 00:23:02,260 I don't know the determinant of that matrix. 414 00:23:02,260 --> 00:23:05,380 So like mathematics is being done for the first time 415 00:23:05,380 --> 00:23:07,772 in, in front of your eyes. 416 00:23:07,772 --> 00:23:08,730 What's the determinant? 417 00:23:12,090 --> 00:23:15,210 Well, a lot of -- there are twenty-four terms, 418 00:23:15,210 --> 00:23:17,700 because it's four by four. 419 00:23:17,700 --> 00:23:19,460 Many of them will be zero, because I've 420 00:23:19,460 --> 00:23:22,430 got all those zeros there. 421 00:23:22,430 --> 00:23:25,350 Maybe the whole determinant is zero. 422 00:23:25,350 --> 00:23:29,390 I mean, I -- is that a singular matrix? 423 00:23:29,390 --> 00:23:33,470 That possibility definitely exists. 424 00:23:33,470 --> 00:23:37,620 I could, I could, So one way to do it would be elimination. 425 00:23:37,620 --> 00:23:42,090 Actually, that would probably be a fairly reasonable way. 426 00:23:42,090 --> 00:23:44,440 I could use elimination, so I could use -- 427 00:23:44,440 --> 00:23:47,760 go back to those properties, that -- and use elimination, 428 00:23:47,760 --> 00:23:50,990 get down, eliminate it down, do I have a row of zeros 429 00:23:50,990 --> 00:23:53,020 at the end of elimination? 430 00:23:53,020 --> 00:23:54,230 The answer is zero. 431 00:23:54,230 --> 00:23:58,320 I was thinking, shall I try this big formula? 432 00:23:58,320 --> 00:23:59,390 OK. 433 00:23:59,390 --> 00:24:00,810 Let's try the big formula. 434 00:24:00,810 --> 00:24:08,190 How -- tell me one way I can go down the matrix, taking a one, 435 00:24:08,190 --> 00:24:13,250 taking a one from every row and column, and make it to the end? 436 00:24:13,250 --> 00:24:15,640 So it's -- I get something that isn't zero. 437 00:24:15,640 --> 00:24:18,280 Well, one way to do it, I could take that times that times 438 00:24:18,280 --> 00:24:20,360 that times that times that. 439 00:24:20,360 --> 00:24:23,510 That would be one and, and, and I just said, 440 00:24:23,510 --> 00:24:25,990 that comes in with what sign? 441 00:24:25,990 --> 00:24:26,820 Plus. 442 00:24:26,820 --> 00:24:28,690 That comes with a plus sign. 443 00:24:28,690 --> 00:24:32,600 Because, because that permutation -- 444 00:24:32,600 --> 00:24:35,160 I've just written the permutation 445 00:24:35,160 --> 00:24:38,530 about four three two one, and one exchange 446 00:24:38,530 --> 00:24:40,900 and a second exchange, two exchanges 447 00:24:40,900 --> 00:24:42,540 puts it in the correct order. 448 00:24:46,750 --> 00:24:51,620 Keep walking away, don't.... 449 00:24:51,620 --> 00:24:54,550 OK, we're executing a determinant formula here. 450 00:24:54,550 --> 00:25:07,900 Uh as long as it's not periodic, of course. 451 00:25:07,900 --> 00:25:11,290 If he comes back I'm in -- 452 00:25:11,290 --> 00:25:11,790 no. 453 00:25:11,790 --> 00:25:13,620 All right, all right. 454 00:25:13,620 --> 00:25:16,260 OK, so that would give me a plus one. 455 00:25:21,260 --> 00:25:22,700 All right. 456 00:25:22,700 --> 00:25:24,240 Are there any others? 457 00:25:24,240 --> 00:25:26,610 Well, of course we see another one here. 458 00:25:26,610 --> 00:25:29,840 This times this times this times this strikes us right 459 00:25:29,840 --> 00:25:30,340 away. 460 00:25:30,340 --> 00:25:34,470 So that's the order three, the order -- 461 00:25:34,470 --> 00:25:37,450 let me make a little different mark here. 462 00:25:37,450 --> 00:25:41,520 Three two one four. 463 00:25:41,520 --> 00:25:45,330 And is that a plus or a minus, three two one four? 464 00:25:48,030 --> 00:25:52,950 Is that, is that permutation a plus or a minus permutation? 465 00:25:52,950 --> 00:25:53,827 It's a minus. 466 00:25:53,827 --> 00:25:54,660 How do you see that? 467 00:25:54,660 --> 00:25:59,640 What exchange shall I do to get it in the right order? 468 00:25:59,640 --> 00:26:02,480 If I exchange the one and the three I'm in the right orders, 469 00:26:02,480 --> 00:26:05,120 took one exchange to do it, so that would be a plus -- 470 00:26:05,120 --> 00:26:07,040 that would be a minus one. 471 00:26:07,040 --> 00:26:09,700 And now I don't know if there're any more here. 472 00:26:09,700 --> 00:26:10,280 Let's see. 473 00:26:10,280 --> 00:26:15,670 Let me try again starting with this. 474 00:26:15,670 --> 00:26:18,360 Now I've got to pick somebody from -- oh yeah, see, 475 00:26:18,360 --> 00:26:20,400 you see what's happening. 476 00:26:20,400 --> 00:26:24,900 If I I start there, OK, column three is used. 477 00:26:24,900 --> 00:26:27,680 So then when I go to next row, I can't use that, 478 00:26:27,680 --> 00:26:28,850 I must use that. 479 00:26:28,850 --> 00:26:30,810 Now columns two and three are used. 480 00:26:30,810 --> 00:26:33,370 When I come to this row I must use that. 481 00:26:33,370 --> 00:26:34,700 And then I must use that. 482 00:26:34,700 --> 00:26:38,280 So if I start there, this is the only one I get. 483 00:26:38,280 --> 00:26:42,220 And similarly, if I start there, that's the only one I get. 484 00:26:42,220 --> 00:26:45,280 So what's the determinant? 485 00:26:45,280 --> 00:26:47,420 What's the determinant? 486 00:26:47,420 --> 00:26:47,920 Zero. 487 00:26:47,920 --> 00:26:51,940 The determinant is zero for that case. 488 00:26:51,940 --> 00:26:56,080 Because we, we were able to check the twenty-four terms. 489 00:26:56,080 --> 00:26:57,850 Twenty-two of them were zero. 490 00:26:57,850 --> 00:26:59,290 One of them was plus one. 491 00:26:59,290 --> 00:27:01,100 One of them was minus one. 492 00:27:01,100 --> 00:27:04,600 Add up the twenty-four terms, zero is the answer. 493 00:27:04,600 --> 00:27:05,120 OK. 494 00:27:05,120 --> 00:27:07,136 Well, I didn't know it would be zero, I -- 495 00:27:07,136 --> 00:27:08,760 because I wasn't, like, thinking ahead. 496 00:27:08,760 --> 00:27:10,700 I was a little scared, actually. 497 00:27:13,300 --> 00:27:18,360 I said, that, apparition went by. 498 00:27:18,360 --> 00:27:22,420 So and I don't know if the camera caught that. 499 00:27:22,420 --> 00:27:23,970 So whether the rest of the world will 500 00:27:23,970 --> 00:27:27,450 realize that I was in danger or not, we don't know. 501 00:27:27,450 --> 00:27:29,674 But anyway, I guess he just wanted 502 00:27:29,674 --> 00:27:31,590 to be sure that we got the right answer, which 503 00:27:31,590 --> 00:27:33,260 is determinant zero. 504 00:27:33,260 --> 00:27:36,060 And then that makes me think, OK, the matrix 505 00:27:36,060 --> 00:27:41,110 must be, the matrix must be singular. 506 00:27:41,110 --> 00:27:43,020 And then if the matrix is singular, 507 00:27:43,020 --> 00:27:46,000 maybe there's another way to see that it's singular, like find 508 00:27:46,000 --> 00:27:47,225 something in its null space. 509 00:27:50,690 --> 00:27:54,190 Or find a combination of the rows that gives zero. 510 00:27:54,190 --> 00:27:58,320 And like what d- what, what combination of those rows 511 00:27:58,320 --> 00:28:01,070 does give zero. 512 00:28:01,070 --> 00:28:05,980 Suppose I add rows one and rows three. 513 00:28:05,980 --> 00:28:08,660 If I add rows one and rows three, what do I get? 514 00:28:08,660 --> 00:28:10,990 I get a row of all ones. 515 00:28:10,990 --> 00:28:15,340 Then if I add rows two and rows four I get a row of all ones. 516 00:28:15,340 --> 00:28:19,370 So row one minus row two plus row three minus row four 517 00:28:19,370 --> 00:28:21,280 is probably the zero row. 518 00:28:21,280 --> 00:28:24,490 It's a singular matrix. 519 00:28:24,490 --> 00:28:27,330 And I could find something in its null space the same way. 520 00:28:27,330 --> 00:28:29,720 That would be a combination of columns that gives zero. 521 00:28:29,720 --> 00:28:32,040 OK, there's an example. 522 00:28:32,040 --> 00:28:32,950 All right. 523 00:28:32,950 --> 00:28:36,840 So that's, well, that shows two things. 524 00:28:36,840 --> 00:28:39,280 That shows how we get the twenty-four terms 525 00:28:39,280 --> 00:28:41,700 and it shows the great advantage of having 526 00:28:41,700 --> 00:28:43,810 a lot of zeros in there. 527 00:28:43,810 --> 00:28:45,830 OK. 528 00:28:45,830 --> 00:28:48,970 So we'll use this big formula, but I want to pick -- 529 00:28:48,970 --> 00:28:53,710 I want to go onward now to cofactors. 530 00:28:53,710 --> 00:28:56,030 Onward to cofactors. 531 00:28:56,030 --> 00:29:03,330 Cofactors is a way of breaking up this big formula that 532 00:29:03,330 --> 00:29:08,910 connects this n by n -- this is an n by n determinant that 533 00:29:08,910 --> 00:29:13,880 we've just have a formula for, the big formula. 534 00:29:13,880 --> 00:29:18,450 So cofactors is a way to connect this n by n determinant to, 535 00:29:18,450 --> 00:29:22,270 determinants one smaller. 536 00:29:22,270 --> 00:29:24,170 One smaller. 537 00:29:24,170 --> 00:29:29,430 And the way we want to do it is actually going to show up in 538 00:29:29,430 --> 00:29:30,260 this. 539 00:29:30,260 --> 00:29:34,470 Since the three by three is the one that we wrote out in full, 540 00:29:34,470 --> 00:29:38,060 let's, let me do this three by -- 541 00:29:38,060 --> 00:29:43,350 so I'm talking about cofactors, and I'm going to start again 542 00:29:43,350 --> 00:29:44,420 with three by three. 543 00:29:48,180 --> 00:29:51,220 And I'm going to take the, the exact formula, 544 00:29:51,220 --> 00:29:55,690 and I'm just going to write it as a one one -- 545 00:29:55,690 --> 00:29:59,540 this is the determinant I'm writing. 546 00:29:59,540 --> 00:30:03,360 I'm just going to say a one one times what? 547 00:30:03,360 --> 00:30:04,760 A one one times what? 548 00:30:04,760 --> 00:30:08,830 And it's a one one times a two two a three three 549 00:30:08,830 --> 00:30:12,230 minus a two three a three two. 550 00:30:15,410 --> 00:30:22,640 Then I've got the a one two stuff times something. 551 00:30:22,640 --> 00:30:26,910 And I've got the a one three stuff times something. 552 00:30:26,910 --> 00:30:29,500 Do you see what I'm doing? 553 00:30:29,500 --> 00:30:33,990 I'm taking our big formula and I'm saying, OK, 554 00:30:33,990 --> 00:30:37,320 choose column -- 555 00:30:37,320 --> 00:30:40,380 out of the first row, choose column one. 556 00:30:40,380 --> 00:30:43,520 And take all the possibilities. 557 00:30:43,520 --> 00:30:46,240 And those extra factors will be what 558 00:30:46,240 --> 00:30:51,800 we'll call the cofactor, co meaning going with a one one. 559 00:30:51,800 --> 00:30:55,890 So this in parenthesis are, these are in, 560 00:30:55,890 --> 00:30:57,930 the cofactors are in parens. 561 00:31:02,210 --> 00:31:05,660 A one one times something. 562 00:31:05,660 --> 00:31:09,820 And I figured out what that something was by just looking 563 00:31:09,820 --> 00:31:14,290 back -- if I can walk back here to the, to the a one one, 564 00:31:14,290 --> 00:31:17,440 the one that comes down the diagonal minus the one that 565 00:31:17,440 --> 00:31:19,830 comes that way. 566 00:31:19,830 --> 00:31:24,570 That's, those are the two, only two that used a one one. 567 00:31:24,570 --> 00:31:26,970 So there they are, one with a plus and one with a 568 00:31:26,970 --> 00:31:28,240 minus. 569 00:31:28,240 --> 00:31:30,800 And now I can write in the -- 570 00:31:30,800 --> 00:31:33,100 I could look back and see what used a one two 571 00:31:33,100 --> 00:31:34,740 and I can see what used a one three, 572 00:31:34,740 --> 00:31:36,770 and those will give me the cofactors 573 00:31:36,770 --> 00:31:38,920 of a one two and a one 574 00:31:38,920 --> 00:31:41,650 three. 575 00:31:41,650 --> 00:31:45,770 Before I do that, what's this number, what is this cofactor? 576 00:31:48,310 --> 00:31:52,080 What is it there that's multiplying a one one? 577 00:31:52,080 --> 00:31:55,360 Tell me what a two two a three three minus a two three a three 578 00:31:55,360 --> 00:31:59,310 two is, for this -- 579 00:31:59,310 --> 00:32:00,340 do you recognize that? 580 00:32:03,600 --> 00:32:06,500 Do you recognize -- 581 00:32:06,500 --> 00:32:08,780 let's see, I can -- and I'll put it here. 582 00:32:08,780 --> 00:32:11,520 There's the a one one. 583 00:32:11,520 --> 00:32:14,160 That's used column one. 584 00:32:14,160 --> 00:32:17,645 Then there's -- the other factors involved these other 585 00:32:17,645 --> 00:32:18,145 columns. 586 00:32:25,070 --> 00:32:27,360 This row is used. 587 00:32:27,360 --> 00:32:29,140 This column is used. 588 00:32:29,140 --> 00:32:32,870 So this the only things left to use are these. 589 00:32:32,870 --> 00:32:36,250 And this formula uses them, and what's 590 00:32:36,250 --> 00:32:39,860 the, what's the cofactor? 591 00:32:39,860 --> 00:32:42,420 Tell me what it is because you see it, and then -- 592 00:32:42,420 --> 00:32:47,150 I'll be happy you see what the idea of cofactors. 593 00:32:47,150 --> 00:32:50,010 It's the determinant of this smaller guy. 594 00:32:52,920 --> 00:32:55,400 A one one multiplies the determinant 595 00:32:55,400 --> 00:32:56,590 of this smaller guy. 596 00:32:56,590 --> 00:33:02,840 That gives me all the a one one part of the big formula. 597 00:33:02,840 --> 00:33:03,540 You see that? 598 00:33:03,540 --> 00:33:05,500 This, the determinant of this smaller guy 599 00:33:05,500 --> 00:33:10,540 is a two two a three three minus a two three a three two. 600 00:33:10,540 --> 00:33:14,050 In other words, once I've used column one and row 601 00:33:14,050 --> 00:33:20,850 one, what's left is all the ways to use the other n-1 602 00:33:20,850 --> 00:33:25,130 columns and n-1 rows, one of each. 603 00:33:25,130 --> 00:33:28,970 All the other -- and that's the determinant of the smaller guy 604 00:33:28,970 --> 00:33:30,880 of size n-1. 605 00:33:30,880 --> 00:33:33,710 So that's the whole idea of cofactors. 606 00:33:33,710 --> 00:33:37,150 And we just have to remember that with determinants we've 607 00:33:37,150 --> 00:33:39,900 got pluses and minus signs to keep straight. 608 00:33:39,900 --> 00:33:43,260 Can we keep this next one straight? 609 00:33:43,260 --> 00:33:45,090 Let's do the next one. 610 00:33:45,090 --> 00:33:51,640 OK, the next one will be when I use a one two. 611 00:33:51,640 --> 00:33:55,330 I'll have left -- so I can't use that column any more, 612 00:33:55,330 --> 00:34:02,710 but I can use a two one and a two three and I can use a three 613 00:34:02,710 --> 00:34:06,330 one and a three three. 614 00:34:06,330 --> 00:34:08,850 So this one gave me a one times that determinant. 615 00:34:08,850 --> 00:34:13,760 This will give me a one two times this determinant, a two 616 00:34:13,760 --> 00:34:24,690 one a three three minus a two three a three one. 617 00:34:24,690 --> 00:34:27,969 So that's all the stuff involving a one two. 618 00:34:27,969 --> 00:34:32,360 But have I got the sign right? 619 00:34:32,360 --> 00:34:35,280 Is the determinant of that correctly given by that 620 00:34:35,280 --> 00:34:37,510 or is there a minus sign? 621 00:34:37,510 --> 00:34:39,520 There is a minus sign. 622 00:34:39,520 --> 00:34:40,929 I can follow one of these. 623 00:34:40,929 --> 00:34:43,080 If I do that times that times that, 624 00:34:43,080 --> 00:34:45,046 that was one that's showing up here, 625 00:34:45,046 --> 00:34:47,420 but it should have showed -- it should have been a minus. 626 00:34:52,969 --> 00:34:55,675 So I'm going to build that minus sign into the cofactor. 627 00:34:58,580 --> 00:35:01,390 So, so the cofactor -- so I'll put, 628 00:35:01,390 --> 00:35:04,110 put that minus sign in here. 629 00:35:04,110 --> 00:35:07,180 So because the cofactor is going to be strictly 630 00:35:07,180 --> 00:35:09,900 the thing that multiplies the, the factor. 631 00:35:09,900 --> 00:35:12,570 The factor is a one two, the cofactor is this, 632 00:35:12,570 --> 00:35:15,620 is the parens, the stuff in parentheses. 633 00:35:15,620 --> 00:35:18,590 So it's got the minus sign built in. 634 00:35:18,590 --> 00:35:23,960 And if I did -- if I went on to the third guy, there w- 635 00:35:23,960 --> 00:35:26,920 there'll be this and this, this and this. 636 00:35:26,920 --> 00:35:28,410 And it would take its determinant. 637 00:35:28,410 --> 00:35:31,790 It would come out plus the determinant. 638 00:35:31,790 --> 00:35:34,960 So now I'm ready to say what cofactors are. 639 00:35:34,960 --> 00:35:40,620 So this would be a plus and a one three times its cofactor. 640 00:35:40,620 --> 00:35:45,940 And over here we had plus a one one times this determinant. 641 00:35:45,940 --> 00:35:49,400 But and there we had the a one two times its cofactor, 642 00:35:49,400 --> 00:35:53,310 but the -- so the point is the cofactor is either plus 643 00:35:53,310 --> 00:35:56,320 or minus the determinant. 644 00:35:56,320 --> 00:35:57,990 So let me write that underneath them. 645 00:35:57,990 --> 00:36:00,750 What is the, what are cofactors? 646 00:36:00,750 --> 00:36:09,650 The cofactor if any number aij, let's say. 647 00:36:13,500 --> 00:36:18,630 This is, this is all the terms in the, in the big formula that 648 00:36:18,630 --> 00:36:20,260 involve aij. 649 00:36:20,260 --> 00:36:25,490 We're especially interested in a1j, the first row, that's 650 00:36:25,490 --> 00:36:28,790 what I've been talking about, but any row would be all right. 651 00:36:28,790 --> 00:36:30,650 All right, so -- 652 00:36:30,650 --> 00:36:32,900 what terms involve aij? 653 00:36:32,900 --> 00:36:42,563 So -- it's the determinant of the n minus one matrix -- 654 00:36:46,430 --> 00:36:51,470 with row i, column j erased. 655 00:36:56,460 --> 00:37:00,900 So it's the, it's a matrix of size n-1 with -- 656 00:37:00,900 --> 00:37:05,710 of course, because I can't use this row or this column again. 657 00:37:05,710 --> 00:37:08,120 So I have the matrix all there. 658 00:37:08,120 --> 00:37:11,400 But now it's multiplied by a plus or a minus. 659 00:37:11,400 --> 00:37:14,165 This is the cofactor, and I'm going to call that cij. 660 00:37:17,750 --> 00:37:19,990 Capital, I use capital c just to, 661 00:37:19,990 --> 00:37:22,570 just to emphasize that these are important 662 00:37:22,570 --> 00:37:25,850 and emphasize that they're, they're, they're 663 00:37:25,850 --> 00:37:29,591 different from the (a)s. 664 00:37:29,591 --> 00:37:30,090 OK. 665 00:37:30,090 --> 00:37:34,340 So now is it a plus or is it a minus? 666 00:37:34,340 --> 00:37:36,340 Because we see that in this case, 667 00:37:36,340 --> 00:37:41,260 for a one one it was a plus, for a one two I -- this is ij -- 668 00:37:41,260 --> 00:37:43,360 it was a minus. 669 00:37:43,360 --> 00:37:46,530 For this ij it was a plus. 670 00:37:46,530 --> 00:37:50,550 So any any guess on the rule for plus or minus 671 00:37:50,550 --> 00:37:55,650 when we see those examples, ij equal one one or one three 672 00:37:55,650 --> 00:37:58,070 was a plus? 673 00:37:58,070 --> 00:38:04,400 It sounds very like i+j odd or even. 674 00:38:04,400 --> 00:38:06,170 That, that's doesn't surprise us, 675 00:38:06,170 --> 00:38:07,600 and that's the right answer. 676 00:38:07,600 --> 00:38:17,950 So it's a plus if i+j is even and it's a minus if i+j is odd. 677 00:38:24,450 --> 00:38:28,170 So if I go along row one and look at the cofactors, 678 00:38:28,170 --> 00:38:32,430 I just take those determinants, those one smaller determinants, 679 00:38:32,430 --> 00:38:36,830 and they come in order plus minus plus minus plus minus. 680 00:38:36,830 --> 00:38:42,120 But if I go along row two and, and, and take the cofactors 681 00:38:42,120 --> 00:38:46,310 of sub-determinants, they would start with a minus, 682 00:38:46,310 --> 00:38:52,520 because the two one entry, two plus one is odd, so the -- 683 00:38:52,520 --> 00:38:57,560 like there's a pattern plus minus plus minus plus if it was 684 00:38:57,560 --> 00:39:01,180 five by five, but then if I was doing a cofactor then this sign 685 00:39:01,180 --> 00:39:06,087 would be minus plus minus plus minus, plus minus plus -- 686 00:39:06,087 --> 00:39:07,170 it's sort of checkerboard. 687 00:39:12,540 --> 00:39:13,040 OK. 688 00:39:17,551 --> 00:39:18,050 OK. 689 00:39:18,050 --> 00:39:22,220 Those are the signs that, that are given by this rule, 690 00:39:22,220 --> 00:39:24,540 i+j even or odd. 691 00:39:24,540 --> 00:39:27,760 And those are built into the cofactors. 692 00:39:27,760 --> 00:39:31,030 The thing is called a minor without th- 693 00:39:31,030 --> 00:39:34,400 before you've built in the sign, but I don't care about those. 694 00:39:34,400 --> 00:39:39,700 Build in that sign and call it a cofactor. 695 00:39:39,700 --> 00:39:41,950 So what's the cofactor formula? 696 00:39:41,950 --> 00:39:42,450 OK. 697 00:39:42,450 --> 00:39:44,240 What's the cofactor formula then? 698 00:39:44,240 --> 00:39:49,110 Let me come back to this board and say, 699 00:39:49,110 --> 00:39:50,560 what's the cofactor formula? 700 00:39:58,150 --> 00:40:02,840 Determinant of A is -- 701 00:40:02,840 --> 00:40:04,670 let's go along the first row. 702 00:40:04,670 --> 00:40:11,480 It's a one one times its cofactor, 703 00:40:11,480 --> 00:40:15,730 and then the second guy is a one two times its cofactor, 704 00:40:15,730 --> 00:40:19,070 and you just keep going to the end of the row, 705 00:40:19,070 --> 00:40:22,750 a1n times its cofactor. 706 00:40:22,750 --> 00:40:24,840 So that's cofactor for -- 707 00:40:24,840 --> 00:40:30,780 along row one. 708 00:40:30,780 --> 00:40:39,030 And if I went along row I, I would -- those ones would be 709 00:40:39,030 --> 00:40:39,530 Is. 710 00:40:39,530 --> 00:40:43,520 That's worth putting a box over. 711 00:40:43,520 --> 00:40:47,470 That's the cofactor formula. 712 00:40:47,470 --> 00:40:51,180 Do you see that -- 713 00:40:51,180 --> 00:40:53,740 actually, this would give me another way 714 00:40:53,740 --> 00:41:00,150 I could have started the whole topic of determinants. 715 00:41:00,150 --> 00:41:02,000 And some, some people might do it this -- 716 00:41:02,000 --> 00:41:04,440 choose to do it this way. 717 00:41:04,440 --> 00:41:06,540 Because the cofactor formula would 718 00:41:06,540 --> 00:41:09,740 allow me to build up an n by n determinant out 719 00:41:09,740 --> 00:41:14,790 of n-1 sized determinants, build those out of n-2, and so on. 720 00:41:14,790 --> 00:41:17,740 I could boil all the way down to one by ones. 721 00:41:17,740 --> 00:41:21,470 So what's the cofactor formula for two by two matrices? 722 00:41:21,470 --> 00:41:23,300 Yeah, tell me that. 723 00:41:23,300 --> 00:41:24,580 What's the cofactor for us? 724 00:41:24,580 --> 00:41:28,620 Here is the, here is the world's smallest example, practically, 725 00:41:28,620 --> 00:41:32,580 of a cofactor formula. 726 00:41:32,580 --> 00:41:33,170 OK. 727 00:41:33,170 --> 00:41:35,570 Let's go along row one. 728 00:41:35,570 --> 00:41:39,500 I take this first guy times its cofactor. 729 00:41:39,500 --> 00:41:44,350 What's the cofactor of the one one entry? 730 00:41:44,350 --> 00:41:48,460 d, because you strike out the one one row and column 731 00:41:48,460 --> 00:41:50,460 and you're left with d. 732 00:41:50,460 --> 00:41:54,330 Then I take this guy, b, times its cofactor. 733 00:41:54,330 --> 00:41:57,740 What's the cofactor of b? 734 00:41:57,740 --> 00:42:00,070 Is it c or it's -- 735 00:42:00,070 --> 00:42:03,330 minus c, because I strike out this guy, 736 00:42:03,330 --> 00:42:08,360 I take that determinant, and then I follow the i+j rule 737 00:42:08,360 --> 00:42:11,730 and I get a minus, I get an odd. 738 00:42:11,730 --> 00:42:13,450 So it's b times minus c. 739 00:42:17,230 --> 00:42:18,120 OK, it worked. 740 00:42:18,120 --> 00:42:20,430 Of course it, it worked. 741 00:42:20,430 --> 00:42:23,100 And the three by three works. 742 00:42:23,100 --> 00:42:28,200 So that's the cofactor formula, and that is, that's an -- 743 00:42:28,200 --> 00:42:33,230 that's a good formula to know, and now I'm feeling like, wow, 744 00:42:33,230 --> 00:42:38,560 I'm giving you a lot of algebra to swallow here. 745 00:42:38,560 --> 00:42:41,955 Last lecture gave you ten properties. 746 00:42:44,900 --> 00:42:46,750 Now I'm giving you -- 747 00:42:46,750 --> 00:42:50,390 and by the way, those ten properties led us to a formula 748 00:42:50,390 --> 00:42:52,480 for the determinant which was very important, 749 00:42:52,480 --> 00:42:55,850 and I haven't repeated it till now. 750 00:42:55,850 --> 00:42:56,820 What was that? 751 00:42:56,820 --> 00:43:01,140 The, the determinant is the product of the pivots. 752 00:43:01,140 --> 00:43:03,980 So the pivot formula is, is very important. 753 00:43:03,980 --> 00:43:07,440 The pivots have all this complicated mess already 754 00:43:07,440 --> 00:43:08,810 built in. 755 00:43:08,810 --> 00:43:11,460 As you did elimination to get the pivots, 756 00:43:11,460 --> 00:43:17,670 you built in all this horrible stuff, quite efficiently. 757 00:43:17,670 --> 00:43:20,680 Then the big formula with the n factorial terms, 758 00:43:20,680 --> 00:43:24,180 that's got all the horrible stuff spread out. 759 00:43:24,180 --> 00:43:28,870 And the cofactor formula is like in between. 760 00:43:28,870 --> 00:43:35,460 It's got easy stuff times horrible stuff, basically. 761 00:43:35,460 --> 00:43:39,840 But it's, it shows you, how to get determinants 762 00:43:39,840 --> 00:43:42,780 from smaller determinants, and that's the application that I 763 00:43:42,780 --> 00:43:45,290 now want to make. 764 00:43:45,290 --> 00:43:51,030 So may I do one more example? 765 00:43:51,030 --> 00:43:54,670 So I remember the general idea. 766 00:43:54,670 --> 00:43:59,360 But I'm going to use this cofactor formula for a matrix 767 00:43:59,360 --> 00:44:01,230 -- 768 00:44:01,230 --> 00:44:03,640 so here is going to be my example. 769 00:44:03,640 --> 00:44:07,890 It's -- I promised in the, in the lecture, 770 00:44:07,890 --> 00:44:11,090 outline at the very beginning to do an example. 771 00:44:11,090 --> 00:44:12,650 And let me do -- 772 00:44:12,650 --> 00:44:18,280 I'm going to pick tri-diagonal matrix of ones. 773 00:44:21,700 --> 00:44:26,340 I could, I'm drawing here the four by four. 774 00:44:26,340 --> 00:44:27,880 So this will be the matrix. 775 00:44:27,880 --> 00:44:29,260 I could call that A4. 776 00:44:34,830 --> 00:44:40,590 But my real idea is to do n by n. 777 00:44:40,590 --> 00:44:43,100 To do them all. 778 00:44:43,100 --> 00:44:44,930 So A -- I could -- 779 00:44:44,930 --> 00:44:49,010 everybody understands what A1 and A2 are. 780 00:44:49,010 --> 00:44:49,510 Yeah. 781 00:44:49,510 --> 00:44:54,900 Maybe we should just do A1 and A2 and A3 just for -- 782 00:44:54,900 --> 00:44:55,400 so this is 783 00:44:55,400 --> 00:44:57,591 What's the determinant of A1? 784 00:44:57,591 --> 00:44:58,090 A4. 785 00:44:58,090 --> 00:45:01,120 What's the determinant of A1? 786 00:45:01,120 --> 00:45:04,470 So, so what's the matrix A1 in this formula? 787 00:45:04,470 --> 00:45:06,310 It's just got that. 788 00:45:06,310 --> 00:45:08,850 So the determinant is one. 789 00:45:08,850 --> 00:45:10,950 What's the determinant of A2? 790 00:45:10,950 --> 00:45:14,831 So it's just got this two by two, and its determinant is -- 791 00:45:17,780 --> 00:45:19,690 zero. 792 00:45:19,690 --> 00:45:22,350 And then the three by three. 793 00:45:22,350 --> 00:45:23,800 Can we see its determinant? 794 00:45:23,800 --> 00:45:27,740 Can you take the determinant of that three by three? 795 00:45:27,740 --> 00:45:32,910 Well, that's not quite so obvious, at least not to me. 796 00:45:32,910 --> 00:45:35,060 Being three by three, I don't know -- 797 00:45:35,060 --> 00:45:36,710 so here's a, here's a good example. 798 00:45:36,710 --> 00:45:39,590 How would you do that three by three determinant? 799 00:45:39,590 --> 00:45:43,160 We've got, like, n factorial different ways. 800 00:45:43,160 --> 00:45:44,080 Well, three factorial. 801 00:45:44,080 --> 00:45:45,260 So we've got six ways. 802 00:45:45,260 --> 00:45:46,220 OK. 803 00:45:46,220 --> 00:45:49,440 I mean, one way to do it -- 804 00:45:49,440 --> 00:45:51,040 actually the way I would probably 805 00:45:51,040 --> 00:45:52,990 do it, being three by three, I would use 806 00:45:52,990 --> 00:45:55,250 the complete the big formula. 807 00:45:55,250 --> 00:45:57,120 I would say, I've got a one from that, 808 00:45:57,120 --> 00:46:00,710 I've got a zero from that, I've got a zero from that, a zero 809 00:46:00,710 --> 00:46:03,140 from that, and this direction is a minus one, 810 00:46:03,140 --> 00:46:04,410 that direction's a minus one. 811 00:46:04,410 --> 00:46:06,200 I believe the answer is minus one. 812 00:46:06,200 --> 00:46:14,220 Would you do it another way? 813 00:46:14,220 --> 00:46:16,400 Here's another way to do it, look. 814 00:46:16,400 --> 00:46:18,065 Subtract row three from -- 815 00:46:18,065 --> 00:46:19,440 I'm just looking at this three by 816 00:46:19,440 --> 00:46:19,939 three. 817 00:46:19,939 --> 00:46:22,400 Everybody's looking at the three by three. 818 00:46:22,400 --> 00:46:25,630 Subtract row three from row two. 819 00:46:25,630 --> 00:46:26,930 Determinant doesn't change. 820 00:46:26,930 --> 00:46:29,710 So those become zeros. 821 00:46:29,710 --> 00:46:31,710 OK, now use the cofactor formula. 822 00:46:31,710 --> 00:46:33,160 How's that? 823 00:46:33,160 --> 00:46:36,310 How can, how -- if this was now zeros and I'm looking at this 824 00:46:36,310 --> 00:46:39,530 three by three, use the cofactor formula. 825 00:46:39,530 --> 00:46:42,875 Why not use the cofactor formula along that row? 826 00:46:45,890 --> 00:46:49,360 Because then I take that number times its cofactor, 827 00:46:49,360 --> 00:46:52,800 so I take this number -- let me put a box around it -- 828 00:46:52,800 --> 00:46:56,030 times its cofactor, which is the determinant of that and that, 829 00:46:56,030 --> 00:46:56,830 which is what? 830 00:47:02,310 --> 00:47:07,230 That two by two matrix has determinant one. 831 00:47:07,230 --> 00:47:08,625 So what's the cofactor? 832 00:47:11,410 --> 00:47:15,150 What's the cofactor of this guy here? 833 00:47:15,150 --> 00:47:17,220 Looking just at this three by three. 834 00:47:17,220 --> 00:47:21,470 The cofactor of that one is this determinant, 835 00:47:21,470 --> 00:47:26,490 which is one times negative. 836 00:47:26,490 --> 00:47:30,370 So that's why the answer came out minus one. 837 00:47:30,370 --> 00:47:31,280 OK. 838 00:47:31,280 --> 00:47:32,910 So I did the three by three. 839 00:47:32,910 --> 00:47:35,310 I don't know if we want to try the four by four. 840 00:47:35,310 --> 00:47:38,090 Yeah, let's -- I guess that was the point of my example, 841 00:47:38,090 --> 00:47:41,250 of course, so I have to try it. 842 00:47:41,250 --> 00:47:44,120 Sorry, I'm in a good mood today, so you have 843 00:47:44,120 --> 00:47:45,840 to stand for all the bad jokes. 844 00:47:45,840 --> 00:47:46,340 OK. 845 00:47:46,340 --> 00:47:47,200 OK. 846 00:47:47,200 --> 00:47:50,830 So what was the matrix? 847 00:47:50,830 --> 00:47:51,330 Ah. 848 00:47:55,580 --> 00:47:57,660 OK, now I'm ready for four by four. 849 00:47:57,660 --> 00:48:00,720 Who wants to -- who wants to guess the, the -- 850 00:48:00,720 --> 00:48:04,270 I don't know, frankly, this four by four, 851 00:48:04,270 --> 00:48:07,280 what's, what's the determinant. 852 00:48:07,280 --> 00:48:08,575 I plan to use cofactors. 853 00:48:12,640 --> 00:48:14,310 OK, let's use cofactors. 854 00:48:14,310 --> 00:48:17,960 The determinant of A4 is -- 855 00:48:17,960 --> 00:48:20,420 OK, let's use cofactors on the first row. 856 00:48:20,420 --> 00:48:21,630 Those are easy. 857 00:48:21,630 --> 00:48:25,720 So I multiply this number, which is a convenient one, 858 00:48:25,720 --> 00:48:28,240 times this determinant. 859 00:48:28,240 --> 00:48:31,790 So it's, it's one times the, this three by three 860 00:48:31,790 --> 00:48:32,380 determinant. 861 00:48:32,380 --> 00:48:36,600 Now what is -- do you recognize that matrix? 862 00:48:36,600 --> 00:48:37,900 It's A3. 863 00:48:37,900 --> 00:48:41,940 So it's one times the determinant of A3. 864 00:48:41,940 --> 00:48:46,830 Coming along this row is a one times this determinant, 865 00:48:46,830 --> 00:48:49,430 and it goes with a plus, right? 866 00:48:49,430 --> 00:48:50,665 And then we have this one. 867 00:48:53,170 --> 00:48:55,290 And what is its cofactor? 868 00:48:55,290 --> 00:48:58,780 Now I'm looking at, now I'm looking at this three 869 00:48:58,780 --> 00:49:00,990 by three, this three by three, so I'm 870 00:49:00,990 --> 00:49:04,270 looking at the three by three that I haven't X-ed out. 871 00:49:04,270 --> 00:49:08,630 What is that -- oh, now it, we did a plus or a -- 872 00:49:08,630 --> 00:49:10,880 is it plus this determinant, this three by three 873 00:49:10,880 --> 00:49:13,730 determinant, or minus it? 874 00:49:13,730 --> 00:49:16,150 It's minus it, right, because this is -- 875 00:49:16,150 --> 00:49:20,180 I'm starting in a one two position, and that's a minus. 876 00:49:20,180 --> 00:49:22,390 So I want minus this determinant. 877 00:49:22,390 --> 00:49:25,180 But these guys are X-ed out. 878 00:49:25,180 --> 00:49:25,680 OK. 879 00:49:25,680 --> 00:49:27,260 So I've got a three by three. 880 00:49:27,260 --> 00:49:31,170 Well, let's use cofactors again. 881 00:49:31,170 --> 00:49:33,740 Use cofactors of the column, because actually we 882 00:49:33,740 --> 00:49:35,620 could use cofactors of columns just 883 00:49:35,620 --> 00:49:39,930 as well as rows, because, because the transpose rule. 884 00:49:39,930 --> 00:49:43,750 So I'll take this one, which is now sitting in the plus 885 00:49:43,750 --> 00:49:46,440 position, times its determinant -- oh! 886 00:49:46,440 --> 00:49:47,773 Oh, hell. 887 00:49:50,870 --> 00:49:53,990 What -- oh yeah, I shouldn't have said hell, 888 00:49:53,990 --> 00:49:55,260 because it's all right. 889 00:49:55,260 --> 00:49:55,760 OK. 890 00:49:55,760 --> 00:49:58,230 One times the determinant. 891 00:49:58,230 --> 00:50:00,410 What is that matrix now that I'm taking 892 00:50:00,410 --> 00:50:01,920 the, this smaller one of? 893 00:50:01,920 --> 00:50:03,420 Oh, but there's a minus, right? 894 00:50:03,420 --> 00:50:05,630 The minus came from, from the fact 895 00:50:05,630 --> 00:50:10,570 that this was in the one two position and that's odd. 896 00:50:10,570 --> 00:50:14,750 So this is a minus one times -- and what's -- 897 00:50:14,750 --> 00:50:17,130 and then this one is the upper left, 898 00:50:17,130 --> 00:50:21,880 that's the one one position in its matrix, so plus. 899 00:50:21,880 --> 00:50:23,910 And what's this matrix here? 900 00:50:23,910 --> 00:50:26,460 Do you recognize that? 901 00:50:26,460 --> 00:50:28,840 That matrix is -- 902 00:50:28,840 --> 00:50:30,275 yes, please say it -- 903 00:50:30,275 --> 00:50:30,775 A2. 904 00:50:36,320 --> 00:50:39,110 And we -- that's our formula for any case. 905 00:50:39,110 --> 00:50:46,790 A of any size n is equal to the determinant of A n minus one, 906 00:50:46,790 --> 00:50:49,310 that's what came from taking the one in the upper corner, 907 00:50:49,310 --> 00:50:55,520 the first cofactor, minus the determinant of A n minus two. 908 00:50:55,520 --> 00:51:02,020 What we discovered there is true for all n. 909 00:51:02,020 --> 00:51:04,680 I didn't even mention it, but I stopped taking 910 00:51:04,680 --> 00:51:06,610 cofactors when I got this one. 911 00:51:06,610 --> 00:51:08,920 Why did I stop? 912 00:51:08,920 --> 00:51:12,660 Why didn't I take the cofactor of this guy? 913 00:51:12,660 --> 00:51:16,490 Because he's going to get multiplied by zero, and no, 914 00:51:16,490 --> 00:51:18,130 no use wasting time. 915 00:51:18,130 --> 00:51:20,260 Or this one too. 916 00:51:20,260 --> 00:51:22,570 The cofactor, her cofactor will be 917 00:51:22,570 --> 00:51:24,400 whatever that determinant is, but it'll 918 00:51:24,400 --> 00:51:26,980 be multiplied by zero, so I won't bother. 919 00:51:26,980 --> 00:51:30,010 OK, there is the formula. 920 00:51:30,010 --> 00:51:32,160 And that now tells us -- 921 00:51:32,160 --> 00:51:34,620 I could figure out what A4 is now. 922 00:51:34,620 --> 00:51:37,570 Oh yeah, finally I can get A4. 923 00:51:37,570 --> 00:51:43,770 Because it's A3, which is minus one, minus A2, which is zero, 924 00:51:43,770 --> 00:51:44,880 so it's minus one. 925 00:51:47,750 --> 00:51:50,090 You see how we're getting kind of numbers 926 00:51:50,090 --> 00:51:51,710 that you might not have guessed. 927 00:51:51,710 --> 00:51:55,800 So our sequence right now is one zero minus one minus one. 928 00:51:55,800 --> 00:52:00,760 What's the next one in the sequence, A5? 929 00:52:00,760 --> 00:52:06,300 A5 is this minus this, so it is zero. 930 00:52:06,300 --> 00:52:09,260 What's A6? 931 00:52:09,260 --> 00:52:15,140 A6 is this minus this, which is one. 932 00:52:15,140 --> 00:52:18,150 What's A7? 933 00:52:18,150 --> 00:52:21,640 I'm, I'm going to be stopped by either the time runs out 934 00:52:21,640 --> 00:52:23,230 or the board runs out. 935 00:52:23,230 --> 00:52:27,740 OK, A7 is this minus this, which is one. 936 00:52:27,740 --> 00:52:30,790 I'll stop here, because time is out, but let me tell you 937 00:52:30,790 --> 00:52:32,180 what we've got. 938 00:52:32,180 --> 00:52:35,850 What -- these determinants have this series, 939 00:52:35,850 --> 00:52:39,730 one zero minus one minus one zero one, 940 00:52:39,730 --> 00:52:42,760 and then it starts repeating. 941 00:52:42,760 --> 00:52:44,600 It's pretty fantastic. 942 00:52:44,600 --> 00:52:48,480 These determinants have period six. 943 00:52:48,480 --> 00:52:51,880 So the determinant of A sixty-one 944 00:52:51,880 --> 00:52:55,771 would be the determinant of A1, which would be one. 945 00:52:55,771 --> 00:52:56,270 OK. 946 00:52:56,270 --> 00:52:58,190 I hope you liked that example. 947 00:52:58,190 --> 00:53:04,640 A non-trivial example of a tri-diagonal determinant. 948 00:53:04,640 --> 00:53:05,520 Thanks. 949 00:53:05,520 --> 00:53:08,260 See you on Wednesday.