WEBVTT 00:00:08.150 --> 00:00:13.310 OK, this is the second lecture on determinants. 00:00:13.310 --> 00:00:15.260 There are only three. 00:00:15.260 --> 00:00:19.270 With determinants it's a fascinating, small topic 00:00:19.270 --> 00:00:21.480 inside linear algebra. 00:00:21.480 --> 00:00:24.000 Used to be determinants were the big thing, 00:00:24.000 --> 00:00:27.780 and linear algebra was the little thing, but they -- 00:00:27.780 --> 00:00:30.510 those changed, that situation changed. 00:00:30.510 --> 00:00:36.170 Now determinants is one specific part, very neat little part. 00:00:36.170 --> 00:00:41.740 And my goal today is to find a formula for the determinant. 00:00:41.740 --> 00:00:45.480 It'll be a messy formula. 00:00:45.480 --> 00:00:49.570 So that's why you didn't see it right away. 00:00:49.570 --> 00:00:52.090 But if I'm given this n by n matrix 00:00:52.090 --> 00:00:55.890 then I use those entries to create 00:00:55.890 --> 00:00:57.450 this number, the determinant. 00:00:57.450 --> 00:00:58.620 So there's a formula for it. 00:00:58.620 --> 00:01:03.250 In fact, there's another formula, a second formula using 00:01:03.250 --> 00:01:05.129 something called cofactors. 00:01:05.129 --> 00:01:07.910 So you'll -- you have to know what cofactors are. 00:01:07.910 --> 00:01:10.590 And then I'll apply those formulas 00:01:10.590 --> 00:01:15.410 for some, some matrices that have a lot of zeros 00:01:15.410 --> 00:01:18.430 away from the three diagonals. 00:01:18.430 --> 00:01:19.220 OK. 00:01:19.220 --> 00:01:24.690 So I'm shooting now for a formula for the determinant. 00:01:24.690 --> 00:01:30.470 You remember we started with these three properties, three 00:01:30.470 --> 00:01:32.610 simple properties, but out of that we 00:01:32.610 --> 00:01:37.440 got all these amazing facts, like the determinant of A B 00:01:37.440 --> 00:01:42.210 equals determinant of A times determinant of B. 00:01:42.210 --> 00:01:45.280 But the three facts were -- 00:01:45.280 --> 00:01:48.950 oh, how about I just take two by twos. 00:01:48.950 --> 00:01:52.320 I know, because everybody here knows, the determinant of a two 00:01:52.320 --> 00:01:57.770 by two matrix, but let's get it out of these three formulas. 00:01:57.770 --> 00:02:00.940 OK, so here's my, my two by two matrix. 00:02:00.940 --> 00:02:04.160 I'm looking for a formula for this determinant. 00:02:04.160 --> 00:02:07.810 a b c d, OK. 00:02:07.810 --> 00:02:14.220 So property one, I know what to do with the identity. 00:02:14.220 --> 00:02:14.900 Right? 00:02:14.900 --> 00:02:19.190 Property two allows me to exchange rows, 00:02:19.190 --> 00:02:20.940 and I know what to do then. 00:02:20.940 --> 00:02:23.570 So I know that that determinant is one. 00:02:23.570 --> 00:02:27.280 Property two allows me to exchange rows and know 00:02:27.280 --> 00:02:32.400 that this determinant is minus one. 00:02:32.400 --> 00:02:37.970 And now I want to use property three to get everybody, 00:02:37.970 --> 00:02:39.210 to get everybody. 00:02:39.210 --> 00:02:40.850 And how will I do that? 00:02:40.850 --> 00:02:41.500 OK. 00:02:41.500 --> 00:02:46.600 So remember that if I keep the second row the same, 00:02:46.600 --> 00:02:52.590 I'm allowed to use linearity in the first row. 00:02:52.590 --> 00:02:54.750 And I'll just use it in a simple way. 00:02:54.750 --> 00:03:02.020 I'll write this vector a b as a 0 + 0 b. 00:03:04.590 --> 00:03:09.750 So that's one step using property three, linearity 00:03:09.750 --> 00:03:12.361 in the first row when the second row's the same. 00:03:12.361 --> 00:03:12.860 OK. 00:03:12.860 --> 00:03:15.650 But now you can guess what I'm going to do next. 00:03:15.650 --> 00:03:17.870 I'll -- because I'd like to -- 00:03:17.870 --> 00:03:20.040 if I can make the matrices diagonal, 00:03:20.040 --> 00:03:22.730 then I'm clearly there. 00:03:22.730 --> 00:03:24.810 So I'll take this one. 00:03:24.810 --> 00:03:28.470 Now I'll keep the first row fixed and split the second row, 00:03:28.470 --> 00:03:32.510 so that'll be an a 0 and I'll split that 00:03:32.510 --> 00:03:39.660 into a c 0 and, keeping that first row the same, a 0 d. 00:03:39.660 --> 00:03:42.920 I used, for this part, linearity. 00:03:42.920 --> 00:03:47.600 And now I'll -- whoops, that's plus because I've got more 00:03:47.600 --> 00:03:48.710 coming. 00:03:48.710 --> 00:03:50.530 This one I'll do the same. 00:03:50.530 --> 00:03:53.710 I'll keep this first row the same 00:03:53.710 --> 00:03:59.320 and I'll split c d into c 0 and 0 d. 00:03:59.320 --> 00:04:00.170 OK. 00:04:00.170 --> 00:04:03.310 Now I've got four easy determinants, 00:04:03.310 --> 00:04:05.250 and two of them are -- 00:04:05.250 --> 00:04:07.460 well, all four are extremely easy. 00:04:07.460 --> 00:04:12.020 Two of them are so easy as to turn into zero, right? 00:04:12.020 --> 00:04:17.500 Which two of these determinants are zero right away? 00:04:17.500 --> 00:04:20.670 The first guy is zero. 00:04:20.670 --> 00:04:21.720 Why is he zero? 00:04:21.720 --> 00:04:26.520 Why is that determinant nothing, forget him? 00:04:26.520 --> 00:04:30.090 Well, it has a column of zeros. 00:04:30.090 --> 00:04:33.630 And by the -- well, so one way to think is, well, 00:04:33.630 --> 00:04:35.170 it's a singular matrix. 00:04:35.170 --> 00:04:38.150 Oh, for, for like forty-eight different reasons, 00:04:38.150 --> 00:04:40.060 that determinant is zero. 00:04:40.060 --> 00:04:43.100 It's a singular matrix that has a column of zeros. 00:04:43.100 --> 00:04:44.570 It's, it's dead. 00:04:44.570 --> 00:04:47.900 And this one is about as dead too. 00:04:47.900 --> 00:04:49.191 Column of zeros. 00:04:49.191 --> 00:04:49.690 OK. 00:04:49.690 --> 00:04:51.650 So that's leaving us with this one. 00:04:51.650 --> 00:04:54.370 Now what do I -- how do I know its determinant, 00:04:54.370 --> 00:04:56.910 following the rules? 00:04:56.910 --> 00:05:00.660 Well, I guess one of the properties that we actually got 00:05:00.660 --> 00:05:07.120 to was the determinant of that -- diagonal matrix, then -- 00:05:07.120 --> 00:05:12.340 so I, I'm finally getting to that determinant is the a d. 00:05:12.340 --> 00:05:16.030 And this determinant is what? 00:05:16.030 --> 00:05:18.010 What's this one? 00:05:18.010 --> 00:05:23.120 Minus -- because I would use property two to do a flip 00:05:23.120 --> 00:05:29.220 to make it c b, then property three to factor out the b, 00:05:29.220 --> 00:05:31.370 property c to factor out the c -- 00:05:31.370 --> 00:05:35.320 the property again to factor out the c, and that minus, 00:05:35.320 --> 00:05:39.860 and of course finally I got the answer that we knew we would 00:05:39.860 --> 00:05:42.110 get. 00:05:42.110 --> 00:05:44.810 But you see the method. 00:05:44.810 --> 00:05:48.250 You see the method, because it's method I'm looking for here, 00:05:48.250 --> 00:05:52.190 not just a two by two answer but the method of doing -- 00:05:52.190 --> 00:05:59.220 now I can do three by threes and four by fours and any size. 00:05:59.220 --> 00:06:05.470 So if you can see the method of taking each row at a time -- 00:06:05.470 --> 00:06:07.880 so let's -- what would happen with three by threes? 00:06:07.880 --> 00:06:10.190 Can we mentally do it rather than I 00:06:10.190 --> 00:06:13.730 write everything on the board for three by threes? 00:06:13.730 --> 00:06:16.900 So what would we do if I had three by threes? 00:06:16.900 --> 00:06:20.360 I would keep rows two and three the same 00:06:20.360 --> 00:06:25.540 and I would split the first row into how many pieces? 00:06:25.540 --> 00:06:26.570 Three pieces. 00:06:26.570 --> 00:06:30.050 I'd have an A zero zero and a zero B zero 00:06:30.050 --> 00:06:35.670 and a zero zero C or something for the first row. 00:06:35.670 --> 00:06:40.250 So I would instead of going from one piece to two pieces to four 00:06:40.250 --> 00:06:48.560 pieces, I would go from one piece to three pieces to -- 00:06:48.560 --> 00:06:49.650 what would it be? 00:06:49.650 --> 00:06:54.290 Each of those three, would, would it be nine? 00:06:54.290 --> 00:06:56.280 Or twenty-seven? 00:06:56.280 --> 00:06:58.940 Oh yeah, I've actually got more steps, 00:06:58.940 --> 00:06:59.510 right. 00:06:59.510 --> 00:07:02.520 I'd go to nine but then I'd have another row to straighten out, 00:07:02.520 --> 00:07:03.200 twenty-seven. 00:07:03.200 --> 00:07:04.630 Yes, oh God. 00:07:04.630 --> 00:07:06.610 OK, let me say this again then. 00:07:06.610 --> 00:07:10.170 If I -- if it was three by three, I would -- 00:07:10.170 --> 00:07:12.830 separating out one row into three pieces 00:07:12.830 --> 00:07:16.820 would give me three, separating out the second row into three 00:07:16.820 --> 00:07:20.230 pieces, then I'd be up to nine, separating out the third row 00:07:20.230 --> 00:07:24.560 into its three pieces, I'd be up to twenty-seven, three cubed, 00:07:24.560 --> 00:07:25.290 pieces. 00:07:25.290 --> 00:07:28.470 But a lot of them would be zero. 00:07:28.470 --> 00:07:31.350 So now when would they not be zero? 00:07:31.350 --> 00:07:35.480 Tell me the pieces that would not be zero. 00:07:35.480 --> 00:07:37.450 Now I will write the non-zero ones. 00:07:37.450 --> 00:07:39.790 OK, so I have this matrix. 00:07:39.790 --> 00:07:43.520 I think I have to use these, start 00:07:43.520 --> 00:07:50.590 using these double symbols here because otherwise I could never 00:07:50.590 --> 00:07:52.970 do n by n. 00:07:52.970 --> 00:07:54.080 OK. 00:07:54.080 --> 00:07:55.220 OK. 00:07:55.220 --> 00:07:57.075 So I split this up like crazy. 00:08:00.200 --> 00:08:01.730 A bunch of pieces are zero. 00:08:01.730 --> 00:08:06.790 Whenever I have a column of zeros, I know I've got zero. 00:08:06.790 --> 00:08:08.620 When do I not have zero? 00:08:08.620 --> 00:08:12.950 When do I have -- what is it that's like these guys? 00:08:12.950 --> 00:08:16.330 These are the survivors, two survivors there. 00:08:16.330 --> 00:08:18.080 So my question for three by three 00:08:18.080 --> 00:08:20.990 is going to be what are the survivors? 00:08:20.990 --> 00:08:22.760 How many survivors are there? 00:08:22.760 --> 00:08:24.290 What are they? 00:08:24.290 --> 00:08:28.010 And when do I get a survivor. 00:08:28.010 --> 00:08:30.780 Well, I would get a survivor -- 00:08:30.780 --> 00:08:32.730 for example, one survivor will be 00:08:32.730 --> 00:08:35.960 that one times that one times that one, with all zeros 00:08:35.960 --> 00:08:37.340 everywhere else. 00:08:37.340 --> 00:08:39.320 That would be one survivor. 00:08:39.320 --> 00:08:44.330 a one one zero zero zero a two two zero zero 00:08:44.330 --> 00:08:47.030 zero a three three. 00:08:47.030 --> 00:08:50.990 That's like the a d survivor. 00:08:50.990 --> 00:08:53.970 Tell me another survivor. 00:08:53.970 --> 00:08:58.560 What other thing -- oh, now here you see the clue. 00:08:58.560 --> 00:09:00.750 Now can -- shall I just say the whole clue? 00:09:00.750 --> 00:09:02.920 That I'm having -- 00:09:02.920 --> 00:09:10.460 the survivors have one entry from each row and each column. 00:09:10.460 --> 00:09:14.430 One entry from each row and column. 00:09:14.430 --> 00:09:16.760 Because if some column is missing, 00:09:16.760 --> 00:09:20.600 then I get a singular matrix. 00:09:20.600 --> 00:09:22.960 And that, that's one of these guys. 00:09:22.960 --> 00:09:25.470 See, you see what happened with -- 00:09:25.470 --> 00:09:27.480 this guy? 00:09:27.480 --> 00:09:32.950 Column one never got used in 0 b 0 d. 00:09:32.950 --> 00:09:35.620 So its determinant was zero and I forget it. 00:09:35.620 --> 00:09:37.510 So I'm going to forget those and just put -- 00:09:37.510 --> 00:09:42.750 so tell me one more that would be a survivor? 00:09:42.750 --> 00:09:44.960 Well -- well, here's another one. 00:09:44.960 --> 00:09:50.690 a one one zero zero -- now OK, that's used up row -- row one 00:09:50.690 --> 00:09:51.800 is used. 00:09:51.800 --> 00:09:54.420 Column one is already used so it better be zero. 00:09:57.510 --> 00:09:58.850 What else could I have? 00:09:58.850 --> 00:10:03.050 Where could I pick the guy -- which column shall I use in row 00:10:03.050 --> 00:10:04.360 two? 00:10:04.360 --> 00:10:08.070 Use column three, because here if I use column -- 00:10:08.070 --> 00:10:10.140 here I used column one and row one. 00:10:10.140 --> 00:10:12.710 This was like the column -- 00:10:12.710 --> 00:10:15.950 numbers were one two three, right in order. 00:10:15.950 --> 00:10:23.140 Now the column numbers are going to be one three, column three, 00:10:23.140 --> 00:10:26.070 and column two. 00:10:26.070 --> 00:10:29.620 So the row numbers are one two three, of course. 00:10:29.620 --> 00:10:32.100 The column numbers are some -- 00:10:32.100 --> 00:10:35.550 OK, some permutation of one two three, 00:10:35.550 --> 00:10:38.880 and here they come in the order one three two. 00:10:38.880 --> 00:10:42.060 It's just like having a permutation matrix 00:10:42.060 --> 00:10:46.330 with, instead of the ones, with numbers. 00:10:46.330 --> 00:10:50.940 And actually, it's very close to having a permutation matrix, 00:10:50.940 --> 00:10:55.550 because I, what I do eventually is I factor out these numbers 00:10:55.550 --> 00:10:57.630 and then I have got. 00:10:57.630 --> 00:10:59.520 So what is that determinant equal? 00:10:59.520 --> 00:11:01.120 I factor those numbers out and I've 00:11:01.120 --> 00:11:05.400 got a one one times a two two times a three three. 00:11:05.400 --> 00:11:08.020 And what does this determinant equal? 00:11:08.020 --> 00:11:09.530 Yeah, now tell me the, this -- 00:11:09.530 --> 00:11:13.140 I mean, we're really getting to the heart of these formulas 00:11:13.140 --> 00:11:13.690 now. 00:11:13.690 --> 00:11:16.800 What is that determinant? 00:11:16.800 --> 00:11:20.400 By the laws of -- by, by our three properties, 00:11:20.400 --> 00:11:24.920 I can factor these out, I can factor out the a one one, 00:11:24.920 --> 00:11:27.780 the a two three, and the a three two. 00:11:27.780 --> 00:11:28.980 They're in separate rows. 00:11:28.980 --> 00:11:31.960 I can do each row separately. 00:11:31.960 --> 00:11:35.230 And then I just have to decide is that plus sign 00:11:35.230 --> 00:11:37.530 or is that a minus sign? 00:11:37.530 --> 00:11:42.640 And the answer is it's a minus. 00:11:42.640 --> 00:11:43.490 Why minus? 00:11:43.490 --> 00:11:47.350 Because these is one row exchange 00:11:47.350 --> 00:11:49.780 to get it back to the identity. 00:11:49.780 --> 00:11:52.740 So that's a minus. 00:11:52.740 --> 00:11:53.960 Now I through? 00:11:53.960 --> 00:11:55.455 No, because there are other ways. 00:11:59.020 --> 00:12:02.060 What I'm really through with, what 00:12:02.060 --> 00:12:04.510 I've done, what I've, what I've completed 00:12:04.510 --> 00:12:08.110 is only the part where the a one one is there. 00:12:08.110 --> 00:12:11.810 But now I've got parts where it's a one two. 00:12:15.010 --> 00:12:18.820 And now if it's a one two that row is used, that column is 00:12:18.820 --> 00:12:19.610 used. 00:12:19.610 --> 00:12:21.260 You see that idea? 00:12:21.260 --> 00:12:25.380 I could use this row and column. 00:12:25.380 --> 00:12:28.070 Now that column is used, that column is used, 00:12:28.070 --> 00:12:31.300 and this guy has to be here, a three three. 00:12:31.300 --> 00:12:33.190 And what's that determinant? 00:12:33.190 --> 00:12:38.510 That's an a one two times an a two one times an a three three, 00:12:38.510 --> 00:12:42.510 and does it have a plus or a minus? 00:12:42.510 --> 00:12:43.820 A minus is right. 00:12:43.820 --> 00:12:45.600 It has a minus. 00:12:45.600 --> 00:12:47.690 Because it's one flip away from an id- 00:12:47.690 --> 00:12:51.730 from the, regular, the right order, the diagonal order. 00:12:51.730 --> 00:12:54.200 And now what's the other guy with a -- with, 00:12:54.200 --> 00:12:57.470 a one two up there? 00:12:57.470 --> 00:12:59.240 I could have used this row. 00:12:59.240 --> 00:13:05.080 I could have put this guy here and this guy here. 00:13:05.080 --> 00:13:05.580 Right? 00:13:05.580 --> 00:13:07.730 You see the whole deal? 00:13:07.730 --> 00:13:14.030 Now that's an a one two, a two three, a three one, 00:13:14.030 --> 00:13:17.970 and does that go with a plus or a minus? 00:13:17.970 --> 00:13:19.720 Yeah, now that takes a minute of thinking, 00:13:19.720 --> 00:13:23.210 doesn't it, because one row exchange doesn't get it 00:13:23.210 --> 00:13:24.760 in line. 00:13:24.760 --> 00:13:26.300 So what is the answer for this? 00:13:26.300 --> 00:13:28.090 Plus or minus? 00:13:28.090 --> 00:13:32.120 Plus, because it takes two exchanges. 00:13:32.120 --> 00:13:35.920 I could exchange rows one and three and then two and three. 00:13:35.920 --> 00:13:40.360 Two exchanges makes this thing a plus. 00:13:40.360 --> 00:13:43.300 And then finally we have -- we're going to have two more. 00:13:43.300 --> 00:13:43.800 OK. 00:13:43.800 --> 00:13:52.770 Zero zero a one three, a two one zero zero, zero a three two 00:13:52.770 --> 00:13:54.160 zero. 00:13:54.160 --> 00:13:55.720 And one more guy. 00:13:55.720 --> 00:14:00.820 Zero zero a one three, zero a two 00:14:00.820 --> 00:14:06.750 two zero, A three one zero zero. 00:14:06.750 --> 00:14:08.870 And let's put down what we get from those. 00:14:08.870 --> 00:14:14.160 An a one three, an a two one, and an a three two, and I 00:14:14.160 --> 00:14:16.650 think that one is a plus. 00:14:16.650 --> 00:14:21.510 And this guys is a minus because one exchange would put it -- 00:14:21.510 --> 00:14:24.864 would order it. 00:14:24.864 --> 00:14:25.655 And that's a minus. 00:14:29.230 --> 00:14:32.510 All right, that has taken one whole board 00:14:32.510 --> 00:14:35.030 just to do the three by three. 00:14:35.030 --> 00:14:37.630 But do you agree that we now have 00:14:37.630 --> 00:14:42.590 a formula for the determinant which 00:14:42.590 --> 00:14:44.080 came from the three properties? 00:14:46.840 --> 00:14:50.440 And it must be it. 00:14:50.440 --> 00:14:53.090 And I'm going to keep that formula. 00:14:53.090 --> 00:14:57.870 That's a famous -- that three by three formula is one that 00:14:57.870 --> 00:15:01.570 if, if the cameras will follow me back to the beginning here, 00:15:01.570 --> 00:15:07.130 I, I get the ones with the plus sign are the ones that go down 00:15:07.130 --> 00:15:08.800 like down this way. 00:15:08.800 --> 00:15:10.390 And the ones with the minus signs 00:15:10.390 --> 00:15:13.940 are sort of the ones that go this way. 00:15:13.940 --> 00:15:17.440 I won't make that precise. 00:15:17.440 --> 00:15:20.700 For two reasons, one, it would clutter up 00:15:20.700 --> 00:15:24.970 the board, and second reason, it wouldn't be right for four 00:15:24.970 --> 00:15:26.300 by fours. 00:15:26.300 --> 00:15:29.540 For four by four, let me just say right away, 00:15:29.540 --> 00:15:33.360 four by four matrix -- the, the cross diagonal, 00:15:33.360 --> 00:15:38.190 the wrong diagonal happens to come out with a plus sign. 00:15:38.190 --> 00:15:39.800 Why is that? 00:15:39.800 --> 00:15:43.600 If I have a four by four matrix with ones 00:15:43.600 --> 00:15:50.080 coming on the counter diagonal, that determinant is plus. 00:15:50.080 --> 00:15:50.980 Why? 00:15:50.980 --> 00:15:54.980 Why plus for that guy? 00:15:54.980 --> 00:15:59.720 Because if I exchange rows one and four and then 00:15:59.720 --> 00:16:02.850 I exchange rows two and three, I've got the identity, 00:16:02.850 --> 00:16:05.590 and I did two exchanges. 00:16:05.590 --> 00:16:10.880 So this down to this, like, you know, down toward Miami 00:16:10.880 --> 00:16:16.330 and down toward LA stuff is, like, three by three only. 00:16:16.330 --> 00:16:16.830 OK. 00:16:19.510 --> 00:16:25.185 But I do want to get now -- 00:16:25.185 --> 00:16:27.310 I don't want to go through this for a four by four. 00:16:29.990 --> 00:16:34.280 I do want to get now the general formula. 00:16:34.280 --> 00:16:39.570 So this is what I refer to in the book as the big formula. 00:16:39.570 --> 00:16:43.140 So now this is the big formula for the determinant. 00:16:43.140 --> 00:16:47.380 I'm asking you to make a jump from two by two and three 00:16:47.380 --> 00:16:50.260 by three to n by n. 00:16:50.260 --> 00:16:52.220 OK, so this will be the big formula. 00:17:00.250 --> 00:17:07.310 That the determinant of A is the sum of a whole lot of terms. 00:17:07.310 --> 00:17:09.550 And what are those terms? 00:17:09.550 --> 00:17:12.490 And, and is it a plus or a minus sign, 00:17:12.490 --> 00:17:14.869 and I have to tell you which, which it is, 00:17:14.869 --> 00:17:18.520 because this came in -- in the three by three case, 00:17:18.520 --> 00:17:20.140 I had how many terms? 00:17:20.140 --> 00:17:21.859 Six. 00:17:21.859 --> 00:17:25.630 And half were plus and half were minus. 00:17:25.630 --> 00:17:30.570 How many terms are you figuring for four by four? 00:17:30.570 --> 00:17:36.520 If I get two terms in the two by two case, three -- 00:17:36.520 --> 00:17:41.760 six terms in the three by three case, what's that pattern? 00:17:41.760 --> 00:17:44.210 How many terms in the four by four case? 00:17:46.830 --> 00:17:48.320 Twenty-four. 00:17:48.320 --> 00:17:49.715 Four factorial. 00:17:52.430 --> 00:17:53.750 Why four factorial? 00:17:53.750 --> 00:17:56.200 This will be a sum of n factorial terms. 00:18:00.000 --> 00:18:01.530 Twenty-four, a hundred and twenty, 00:18:01.530 --> 00:18:05.570 seven hundred and twenty, whatever's after that. 00:18:05.570 --> 00:18:06.580 OK. 00:18:06.580 --> 00:18:08.945 Half plus and half minus. 00:18:12.110 --> 00:18:14.720 And where do those n factorial -- terms come from? 00:18:14.720 --> 00:18:17.310 This is the moment to listen to this lecture. 00:18:17.310 --> 00:18:20.110 Where do those n factorial terms come from? 00:18:20.110 --> 00:18:23.690 They come because the first, the guy in the first row 00:18:23.690 --> 00:18:26.940 can be chosen n ways. 00:18:26.940 --> 00:18:33.390 And after he's chosen, that's used up that, that column. 00:18:33.390 --> 00:18:38.440 So the one in the second row can be chosen n minus one ways. 00:18:38.440 --> 00:18:42.360 And after she's chosen, that second column has been 00:18:42.360 --> 00:18:43.160 used. 00:18:43.160 --> 00:18:46.650 And then the one in the third row can be chosen n minus two 00:18:46.650 --> 00:18:49.140 ways, and after it's chosen -- 00:18:49.140 --> 00:18:52.330 notice how I'm getting these personal pronouns. 00:18:52.330 --> 00:18:53.510 But I've run out. 00:18:53.510 --> 00:18:59.570 And I'm not willing to stop with three by three, 00:18:59.570 --> 00:19:02.370 so I'm just going to write the formula down. 00:19:02.370 --> 00:19:07.470 So the one in the first row comes from some column alpha. 00:19:07.470 --> 00:19:10.670 I don't know what alpha is. 00:19:10.670 --> 00:19:11.600 And the one in the -- 00:19:11.600 --> 00:19:14.410 I multiply that by somebody in the second row that comes 00:19:14.410 --> 00:19:16.530 from some different column. 00:19:16.530 --> 00:19:19.470 And I multiply that by somebody in the third row who comes 00:19:19.470 --> 00:19:21.850 from some yet different column. 00:19:21.850 --> 00:19:25.269 And then in the n-th row, I don't know what -- 00:19:25.269 --> 00:19:26.310 I don't know how to draw. 00:19:26.310 --> 00:19:29.570 Maybe omega, for last. 00:19:29.570 --> 00:19:32.170 And the whole point is then that -- 00:19:32.170 --> 00:19:34.510 that those column numbers are different, 00:19:34.510 --> 00:19:40.990 that alpha, beta, gamma, omega, that set of column numbers 00:19:40.990 --> 00:19:50.100 is some permutation, permutation of one to n. 00:19:50.100 --> 00:19:54.830 It, it, the n column numbers are each used once. 00:19:54.830 --> 00:19:57.570 And that gives us n factorial terms. 00:19:57.570 --> 00:20:02.130 And when I choose a term, that means 00:20:02.130 --> 00:20:04.850 I'm choosing somebody from every row and column. 00:20:04.850 --> 00:20:10.300 And then I just -- like the way I had this from row and column 00:20:10.300 --> 00:20:14.160 one, row and column two, row and column three, so that -- 00:20:14.160 --> 00:20:19.080 what was the alpha beta stuff in that, for that term here? 00:20:19.080 --> 00:20:22.360 Alpha was one, beta was two, gamma was three. 00:20:22.360 --> 00:20:26.180 The permutation was, was the trivial permutation, one 00:20:26.180 --> 00:20:28.140 two three, everybody in the right order. 00:20:30.730 --> 00:20:31.750 You see that formula? 00:20:34.420 --> 00:20:37.120 It's -- do you see why I didn't want to start with that 00:20:37.120 --> 00:20:39.930 the first day, Friday? 00:20:39.930 --> 00:20:42.470 I'd rather we understood the properties. 00:20:42.470 --> 00:20:44.920 Because out of this formula, presumably I 00:20:44.920 --> 00:20:47.720 could figure out all these properties. 00:20:47.720 --> 00:20:50.900 How would I know that the determinant of the identity 00:20:50.900 --> 00:20:56.630 matrix was one, for example, out of this formula? 00:20:56.630 --> 00:20:59.680 Why is -- if A is the identity matrix, 00:20:59.680 --> 00:21:04.080 how does this formula give me a plus one? 00:21:04.080 --> 00:21:05.160 You see it, right? 00:21:05.160 --> 00:21:10.440 Because, because almost all the terms are zeros. 00:21:10.440 --> 00:21:15.450 Which term isn't zero, if, if A is the identity matrix? 00:21:15.450 --> 00:21:18.501 Almost all the terms are zero because almost all the As are 00:21:18.501 --> 00:21:19.000 zero. 00:21:19.000 --> 00:21:21.070 It's only, the only time I'll get something 00:21:21.070 --> 00:21:25.270 is if it's a one one times a two two times a three three. 00:21:25.270 --> 00:21:28.590 Only, only the, only the permutation 00:21:28.590 --> 00:21:32.650 that's in the right order will, will give me something. 00:21:32.650 --> 00:21:34.400 It'll come with a plus sign. 00:21:34.400 --> 00:21:37.490 And the determinant of the identity is one. 00:21:37.490 --> 00:21:40.910 So, so we could go back from this formula and prove 00:21:40.910 --> 00:21:41.710 everything. 00:21:41.710 --> 00:21:45.460 We could even try to prove that the determinant of A B 00:21:45.460 --> 00:21:48.860 was the determinant of A times the determinant of B. 00:21:48.860 --> 00:21:51.070 But like next week we would still be working on it, 00:21:51.070 --> 00:21:54.600 because it's not -- 00:21:54.600 --> 00:21:56.500 clear from -- if I took A B, 00:21:56.500 --> 00:21:57.030 my God. 00:21:57.030 --> 00:21:57.530 You know --. 00:21:57.530 --> 00:22:02.210 The entries of A B would be all these pieces. 00:22:02.210 --> 00:22:06.630 Well, probably, it's probably -- historically it's been done, 00:22:06.630 --> 00:22:09.131 but it won't be repeated in eighteen oh six. 00:22:09.131 --> 00:22:09.630 OK. 00:22:09.630 --> 00:22:16.480 It would be possible probably to see, why the determinant of A 00:22:16.480 --> 00:22:18.190 equals the determinant of A transpose. 00:22:18.190 --> 00:22:21.154 That was another, like, miracle property at the end. 00:22:21.154 --> 00:22:22.820 That would, that would, that's an easier 00:22:22.820 --> 00:22:25.290 one, which we could find. 00:22:25.290 --> 00:22:26.200 OK. 00:22:26.200 --> 00:22:30.460 Is that all right for the big formula? 00:22:30.460 --> 00:22:33.160 I could take you then a, a typical -- 00:22:33.160 --> 00:22:36.070 let me do an example. 00:22:36.070 --> 00:22:39.280 Which I'll just create. 00:22:39.280 --> 00:22:42.410 I'll take a four by four matrix. 00:22:42.410 --> 00:22:46.820 I'll put some, I'll put some ones in and some zeros in. 00:22:46.820 --> 00:22:47.350 OK. 00:22:47.350 --> 00:22:48.530 Let me -- 00:22:48.530 --> 00:22:52.690 I don't know how many to put in, to tell the truth. 00:22:52.690 --> 00:22:54.100 I've never done this before. 00:22:58.840 --> 00:23:02.260 I don't know the determinant of that matrix. 00:23:02.260 --> 00:23:05.380 So like mathematics is being done for the first time 00:23:05.380 --> 00:23:07.772 in, in front of your eyes. 00:23:07.772 --> 00:23:08.730 What's the determinant? 00:23:12.090 --> 00:23:15.210 Well, a lot of -- there are twenty-four terms, 00:23:15.210 --> 00:23:17.700 because it's four by four. 00:23:17.700 --> 00:23:19.460 Many of them will be zero, because I've 00:23:19.460 --> 00:23:22.430 got all those zeros there. 00:23:22.430 --> 00:23:25.350 Maybe the whole determinant is zero. 00:23:25.350 --> 00:23:29.390 I mean, I -- is that a singular matrix? 00:23:29.390 --> 00:23:33.470 That possibility definitely exists. 00:23:33.470 --> 00:23:37.620 I could, I could, So one way to do it would be elimination. 00:23:37.620 --> 00:23:42.090 Actually, that would probably be a fairly reasonable way. 00:23:42.090 --> 00:23:44.440 I could use elimination, so I could use -- 00:23:44.440 --> 00:23:47.760 go back to those properties, that -- and use elimination, 00:23:47.760 --> 00:23:50.990 get down, eliminate it down, do I have a row of zeros 00:23:50.990 --> 00:23:53.020 at the end of elimination? 00:23:53.020 --> 00:23:54.230 The answer is zero. 00:23:54.230 --> 00:23:58.320 I was thinking, shall I try this big formula? 00:23:58.320 --> 00:23:59.390 OK. 00:23:59.390 --> 00:24:00.810 Let's try the big formula. 00:24:00.810 --> 00:24:08.190 How -- tell me one way I can go down the matrix, taking a one, 00:24:08.190 --> 00:24:13.250 taking a one from every row and column, and make it to the end? 00:24:13.250 --> 00:24:15.640 So it's -- I get something that isn't zero. 00:24:15.640 --> 00:24:18.280 Well, one way to do it, I could take that times that times 00:24:18.280 --> 00:24:20.360 that times that times that. 00:24:20.360 --> 00:24:23.510 That would be one and, and, and I just said, 00:24:23.510 --> 00:24:25.990 that comes in with what sign? 00:24:25.990 --> 00:24:26.820 Plus. 00:24:26.820 --> 00:24:28.690 That comes with a plus sign. 00:24:28.690 --> 00:24:32.600 Because, because that permutation -- 00:24:32.600 --> 00:24:35.160 I've just written the permutation 00:24:35.160 --> 00:24:38.530 about four three two one, and one exchange 00:24:38.530 --> 00:24:40.900 and a second exchange, two exchanges 00:24:40.900 --> 00:24:42.540 puts it in the correct order. 00:24:46.750 --> 00:24:51.620 Keep walking away, don't.... 00:24:51.620 --> 00:24:54.550 OK, we're executing a determinant formula here. 00:24:54.550 --> 00:25:07.900 Uh as long as it's not periodic, of course. 00:25:07.900 --> 00:25:11.290 If he comes back I'm in -- 00:25:11.290 --> 00:25:11.790 no. 00:25:11.790 --> 00:25:13.620 All right, all right. 00:25:13.620 --> 00:25:16.260 OK, so that would give me a plus one. 00:25:21.260 --> 00:25:22.700 All right. 00:25:22.700 --> 00:25:24.240 Are there any others? 00:25:24.240 --> 00:25:26.610 Well, of course we see another one here. 00:25:26.610 --> 00:25:29.840 This times this times this times this strikes us right 00:25:29.840 --> 00:25:30.340 away. 00:25:30.340 --> 00:25:34.470 So that's the order three, the order -- 00:25:34.470 --> 00:25:37.450 let me make a little different mark here. 00:25:37.450 --> 00:25:41.520 Three two one four. 00:25:41.520 --> 00:25:45.330 And is that a plus or a minus, three two one four? 00:25:48.030 --> 00:25:52.950 Is that, is that permutation a plus or a minus permutation? 00:25:52.950 --> 00:25:53.827 It's a minus. 00:25:53.827 --> 00:25:54.660 How do you see that? 00:25:54.660 --> 00:25:59.640 What exchange shall I do to get it in the right order? 00:25:59.640 --> 00:26:02.480 If I exchange the one and the three I'm in the right orders, 00:26:02.480 --> 00:26:05.120 took one exchange to do it, so that would be a plus -- 00:26:05.120 --> 00:26:07.040 that would be a minus one. 00:26:07.040 --> 00:26:09.700 And now I don't know if there're any more here. 00:26:09.700 --> 00:26:10.280 Let's see. 00:26:10.280 --> 00:26:15.670 Let me try again starting with this. 00:26:15.670 --> 00:26:18.360 Now I've got to pick somebody from -- oh yeah, see, 00:26:18.360 --> 00:26:20.400 you see what's happening. 00:26:20.400 --> 00:26:24.900 If I I start there, OK, column three is used. 00:26:24.900 --> 00:26:27.680 So then when I go to next row, I can't use that, 00:26:27.680 --> 00:26:28.850 I must use that. 00:26:28.850 --> 00:26:30.810 Now columns two and three are used. 00:26:30.810 --> 00:26:33.370 When I come to this row I must use that. 00:26:33.370 --> 00:26:34.700 And then I must use that. 00:26:34.700 --> 00:26:38.280 So if I start there, this is the only one I get. 00:26:38.280 --> 00:26:42.220 And similarly, if I start there, that's the only one I get. 00:26:42.220 --> 00:26:45.280 So what's the determinant? 00:26:45.280 --> 00:26:47.420 What's the determinant? 00:26:47.420 --> 00:26:47.920 Zero. 00:26:47.920 --> 00:26:51.940 The determinant is zero for that case. 00:26:51.940 --> 00:26:56.080 Because we, we were able to check the twenty-four terms. 00:26:56.080 --> 00:26:57.850 Twenty-two of them were zero. 00:26:57.850 --> 00:26:59.290 One of them was plus one. 00:26:59.290 --> 00:27:01.100 One of them was minus one. 00:27:01.100 --> 00:27:04.600 Add up the twenty-four terms, zero is the answer. 00:27:04.600 --> 00:27:05.120 OK. 00:27:05.120 --> 00:27:07.136 Well, I didn't know it would be zero, I -- 00:27:07.136 --> 00:27:08.760 because I wasn't, like, thinking ahead. 00:27:08.760 --> 00:27:10.700 I was a little scared, actually. 00:27:13.300 --> 00:27:18.360 I said, that, apparition went by. 00:27:18.360 --> 00:27:22.420 So and I don't know if the camera caught that. 00:27:22.420 --> 00:27:23.970 So whether the rest of the world will 00:27:23.970 --> 00:27:27.450 realize that I was in danger or not, we don't know. 00:27:27.450 --> 00:27:29.674 But anyway, I guess he just wanted 00:27:29.674 --> 00:27:31.590 to be sure that we got the right answer, which 00:27:31.590 --> 00:27:33.260 is determinant zero. 00:27:33.260 --> 00:27:36.060 And then that makes me think, OK, the matrix 00:27:36.060 --> 00:27:41.110 must be, the matrix must be singular. 00:27:41.110 --> 00:27:43.020 And then if the matrix is singular, 00:27:43.020 --> 00:27:46.000 maybe there's another way to see that it's singular, like find 00:27:46.000 --> 00:27:47.225 something in its null space. 00:27:50.690 --> 00:27:54.190 Or find a combination of the rows that gives zero. 00:27:54.190 --> 00:27:58.320 And like what d- what, what combination of those rows 00:27:58.320 --> 00:28:01.070 does give zero. 00:28:01.070 --> 00:28:05.980 Suppose I add rows one and rows three. 00:28:05.980 --> 00:28:08.660 If I add rows one and rows three, what do I get? 00:28:08.660 --> 00:28:10.990 I get a row of all ones. 00:28:10.990 --> 00:28:15.340 Then if I add rows two and rows four I get a row of all ones. 00:28:15.340 --> 00:28:19.370 So row one minus row two plus row three minus row four 00:28:19.370 --> 00:28:21.280 is probably the zero row. 00:28:21.280 --> 00:28:24.490 It's a singular matrix. 00:28:24.490 --> 00:28:27.330 And I could find something in its null space the same way. 00:28:27.330 --> 00:28:29.720 That would be a combination of columns that gives zero. 00:28:29.720 --> 00:28:32.040 OK, there's an example. 00:28:32.040 --> 00:28:32.950 All right. 00:28:32.950 --> 00:28:36.840 So that's, well, that shows two things. 00:28:36.840 --> 00:28:39.280 That shows how we get the twenty-four terms 00:28:39.280 --> 00:28:41.700 and it shows the great advantage of having 00:28:41.700 --> 00:28:43.810 a lot of zeros in there. 00:28:43.810 --> 00:28:45.830 OK. 00:28:45.830 --> 00:28:48.970 So we'll use this big formula, but I want to pick -- 00:28:48.970 --> 00:28:53.710 I want to go onward now to cofactors. 00:28:53.710 --> 00:28:56.030 Onward to cofactors. 00:28:56.030 --> 00:29:03.330 Cofactors is a way of breaking up this big formula that 00:29:03.330 --> 00:29:08.910 connects this n by n -- this is an n by n determinant that 00:29:08.910 --> 00:29:13.880 we've just have a formula for, the big formula. 00:29:13.880 --> 00:29:18.450 So cofactors is a way to connect this n by n determinant to, 00:29:18.450 --> 00:29:22.270 determinants one smaller. 00:29:22.270 --> 00:29:24.170 One smaller. 00:29:24.170 --> 00:29:29.430 And the way we want to do it is actually going to show up in 00:29:29.430 --> 00:29:30.260 this. 00:29:30.260 --> 00:29:34.470 Since the three by three is the one that we wrote out in full, 00:29:34.470 --> 00:29:38.060 let's, let me do this three by -- 00:29:38.060 --> 00:29:43.350 so I'm talking about cofactors, and I'm going to start again 00:29:43.350 --> 00:29:44.420 with three by three. 00:29:48.180 --> 00:29:51.220 And I'm going to take the, the exact formula, 00:29:51.220 --> 00:29:55.690 and I'm just going to write it as a one one -- 00:29:55.690 --> 00:29:59.540 this is the determinant I'm writing. 00:29:59.540 --> 00:30:03.360 I'm just going to say a one one times what? 00:30:03.360 --> 00:30:04.760 A one one times what? 00:30:04.760 --> 00:30:08.830 And it's a one one times a two two a three three 00:30:08.830 --> 00:30:12.230 minus a two three a three two. 00:30:15.410 --> 00:30:22.640 Then I've got the a one two stuff times something. 00:30:22.640 --> 00:30:26.910 And I've got the a one three stuff times something. 00:30:26.910 --> 00:30:29.500 Do you see what I'm doing? 00:30:29.500 --> 00:30:33.990 I'm taking our big formula and I'm saying, OK, 00:30:33.990 --> 00:30:37.320 choose column -- 00:30:37.320 --> 00:30:40.380 out of the first row, choose column one. 00:30:40.380 --> 00:30:43.520 And take all the possibilities. 00:30:43.520 --> 00:30:46.240 And those extra factors will be what 00:30:46.240 --> 00:30:51.800 we'll call the cofactor, co meaning going with a one one. 00:30:51.800 --> 00:30:55.890 So this in parenthesis are, these are in, 00:30:55.890 --> 00:30:57.930 the cofactors are in parens. 00:31:02.210 --> 00:31:05.660 A one one times something. 00:31:05.660 --> 00:31:09.820 And I figured out what that something was by just looking 00:31:09.820 --> 00:31:14.290 back -- if I can walk back here to the, to the a one one, 00:31:14.290 --> 00:31:17.440 the one that comes down the diagonal minus the one that 00:31:17.440 --> 00:31:19.830 comes that way. 00:31:19.830 --> 00:31:24.570 That's, those are the two, only two that used a one one. 00:31:24.570 --> 00:31:26.970 So there they are, one with a plus and one with a 00:31:26.970 --> 00:31:28.240 minus. 00:31:28.240 --> 00:31:30.800 And now I can write in the -- 00:31:30.800 --> 00:31:33.100 I could look back and see what used a one two 00:31:33.100 --> 00:31:34.740 and I can see what used a one three, 00:31:34.740 --> 00:31:36.770 and those will give me the cofactors 00:31:36.770 --> 00:31:38.920 of a one two and a one 00:31:38.920 --> 00:31:41.650 three. 00:31:41.650 --> 00:31:45.770 Before I do that, what's this number, what is this cofactor? 00:31:48.310 --> 00:31:52.080 What is it there that's multiplying a one one? 00:31:52.080 --> 00:31:55.360 Tell me what a two two a three three minus a two three a three 00:31:55.360 --> 00:31:59.310 two is, for this -- 00:31:59.310 --> 00:32:00.340 do you recognize that? 00:32:03.600 --> 00:32:06.500 Do you recognize -- 00:32:06.500 --> 00:32:08.780 let's see, I can -- and I'll put it here. 00:32:08.780 --> 00:32:11.520 There's the a one one. 00:32:11.520 --> 00:32:14.160 That's used column one. 00:32:14.160 --> 00:32:17.645 Then there's -- the other factors involved these other 00:32:17.645 --> 00:32:18.145 columns. 00:32:25.070 --> 00:32:27.360 This row is used. 00:32:27.360 --> 00:32:29.140 This column is used. 00:32:29.140 --> 00:32:32.870 So this the only things left to use are these. 00:32:32.870 --> 00:32:36.250 And this formula uses them, and what's 00:32:36.250 --> 00:32:39.860 the, what's the cofactor? 00:32:39.860 --> 00:32:42.420 Tell me what it is because you see it, and then -- 00:32:42.420 --> 00:32:47.150 I'll be happy you see what the idea of cofactors. 00:32:47.150 --> 00:32:50.010 It's the determinant of this smaller guy. 00:32:52.920 --> 00:32:55.400 A one one multiplies the determinant 00:32:55.400 --> 00:32:56.590 of this smaller guy. 00:32:56.590 --> 00:33:02.840 That gives me all the a one one part of the big formula. 00:33:02.840 --> 00:33:03.540 You see that? 00:33:03.540 --> 00:33:05.500 This, the determinant of this smaller guy 00:33:05.500 --> 00:33:10.540 is a two two a three three minus a two three a three two. 00:33:10.540 --> 00:33:14.050 In other words, once I've used column one and row 00:33:14.050 --> 00:33:20.850 one, what's left is all the ways to use the other n-1 00:33:20.850 --> 00:33:25.130 columns and n-1 rows, one of each. 00:33:25.130 --> 00:33:28.970 All the other -- and that's the determinant of the smaller guy 00:33:28.970 --> 00:33:30.880 of size n-1. 00:33:30.880 --> 00:33:33.710 So that's the whole idea of cofactors. 00:33:33.710 --> 00:33:37.150 And we just have to remember that with determinants we've 00:33:37.150 --> 00:33:39.900 got pluses and minus signs to keep straight. 00:33:39.900 --> 00:33:43.260 Can we keep this next one straight? 00:33:43.260 --> 00:33:45.090 Let's do the next one. 00:33:45.090 --> 00:33:51.640 OK, the next one will be when I use a one two. 00:33:51.640 --> 00:33:55.330 I'll have left -- so I can't use that column any more, 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 00:34:02.710 --> 00:34:06.330 one and a three three. 00:34:06.330 --> 00:34:08.850 So this one gave me a one times that determinant. 00:34:08.850 --> 00:34:13.760 This will give me a one two times this determinant, a two 00:34:13.760 --> 00:34:24.690 one a three three minus a two three a three one. 00:34:24.690 --> 00:34:27.969 So that's all the stuff involving a one two. 00:34:27.969 --> 00:34:32.360 But have I got the sign right? 00:34:32.360 --> 00:34:35.280 Is the determinant of that correctly given by that 00:34:35.280 --> 00:34:37.510 or is there a minus sign? 00:34:37.510 --> 00:34:39.520 There is a minus sign. 00:34:39.520 --> 00:34:40.929 I can follow one of these. 00:34:40.929 --> 00:34:43.080 If I do that times that times that, 00:34:43.080 --> 00:34:45.046 that was one that's showing up here, 00:34:45.046 --> 00:34:47.420 but it should have showed -- it should have been a minus. 00:34:52.969 --> 00:34:55.675 So I'm going to build that minus sign into the cofactor. 00:34:58.580 --> 00:35:01.390 So, so the cofactor -- so I'll put, 00:35:01.390 --> 00:35:04.110 put that minus sign in here. 00:35:04.110 --> 00:35:07.180 So because the cofactor is going to be strictly 00:35:07.180 --> 00:35:09.900 the thing that multiplies the, the factor. 00:35:09.900 --> 00:35:12.570 The factor is a one two, the cofactor is this, 00:35:12.570 --> 00:35:15.620 is the parens, the stuff in parentheses. 00:35:15.620 --> 00:35:18.590 So it's got the minus sign built in. 00:35:18.590 --> 00:35:23.960 And if I did -- if I went on to the third guy, there w- 00:35:23.960 --> 00:35:26.920 there'll be this and this, this and this. 00:35:26.920 --> 00:35:28.410 And it would take its determinant. 00:35:28.410 --> 00:35:31.790 It would come out plus the determinant. 00:35:31.790 --> 00:35:34.960 So now I'm ready to say what cofactors are. 00:35:34.960 --> 00:35:40.620 So this would be a plus and a one three times its cofactor. 00:35:40.620 --> 00:35:45.940 And over here we had plus a one one times this determinant. 00:35:45.940 --> 00:35:49.400 But and there we had the a one two times its cofactor, 00:35:49.400 --> 00:35:53.310 but the -- so the point is the cofactor is either plus 00:35:53.310 --> 00:35:56.320 or minus the determinant. 00:35:56.320 --> 00:35:57.990 So let me write that underneath them. 00:35:57.990 --> 00:36:00.750 What is the, what are cofactors? 00:36:00.750 --> 00:36:09.650 The cofactor if any number aij, let's say. 00:36:13.500 --> 00:36:18.630 This is, this is all the terms in the, in the big formula that 00:36:18.630 --> 00:36:20.260 involve aij. 00:36:20.260 --> 00:36:25.490 We're especially interested in a1j, the first row, that's 00:36:25.490 --> 00:36:28.790 what I've been talking about, but any row would be all right. 00:36:28.790 --> 00:36:30.650 All right, so -- 00:36:30.650 --> 00:36:32.900 what terms involve aij? 00:36:32.900 --> 00:36:42.563 So -- it's the determinant of the n minus one matrix -- 00:36:46.430 --> 00:36:51.470 with row i, column j erased. 00:36:56.460 --> 00:37:00.900 So it's the, it's a matrix of size n-1 with -- 00:37:00.900 --> 00:37:05.710 of course, because I can't use this row or this column again. 00:37:05.710 --> 00:37:08.120 So I have the matrix all there. 00:37:08.120 --> 00:37:11.400 But now it's multiplied by a plus or a minus. 00:37:11.400 --> 00:37:14.165 This is the cofactor, and I'm going to call that cij. 00:37:17.750 --> 00:37:19.990 Capital, I use capital c just to, 00:37:19.990 --> 00:37:22.570 just to emphasize that these are important 00:37:22.570 --> 00:37:25.850 and emphasize that they're, they're, they're 00:37:25.850 --> 00:37:29.591 different from the (a)s. 00:37:29.591 --> 00:37:30.090 OK. 00:37:30.090 --> 00:37:34.340 So now is it a plus or is it a minus? 00:37:34.340 --> 00:37:36.340 Because we see that in this case, 00:37:36.340 --> 00:37:41.260 for a one one it was a plus, for a one two I -- this is ij -- 00:37:41.260 --> 00:37:43.360 it was a minus. 00:37:43.360 --> 00:37:46.530 For this ij it was a plus. 00:37:46.530 --> 00:37:50.550 So any any guess on the rule for plus or minus 00:37:50.550 --> 00:37:55.650 when we see those examples, ij equal one one or one three 00:37:55.650 --> 00:37:58.070 was a plus? 00:37:58.070 --> 00:38:04.400 It sounds very like i+j odd or even. 00:38:04.400 --> 00:38:06.170 That, that's doesn't surprise us, 00:38:06.170 --> 00:38:07.600 and that's the right answer. 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. 00:38:24.450 --> 00:38:28.170 So if I go along row one and look at the cofactors, 00:38:28.170 --> 00:38:32.430 I just take those determinants, those one smaller determinants, 00:38:32.430 --> 00:38:36.830 and they come in order plus minus plus minus plus minus. 00:38:36.830 --> 00:38:42.120 But if I go along row two and, and, and take the cofactors 00:38:42.120 --> 00:38:46.310 of sub-determinants, they would start with a minus, 00:38:46.310 --> 00:38:52.520 because the two one entry, two plus one is odd, so the -- 00:38:52.520 --> 00:38:57.560 like there's a pattern plus minus plus minus plus if it was 00:38:57.560 --> 00:39:01.180 five by five, but then if I was doing a cofactor then this sign 00:39:01.180 --> 00:39:06.087 would be minus plus minus plus minus, plus minus plus -- 00:39:06.087 --> 00:39:07.170 it's sort of checkerboard. 00:39:12.540 --> 00:39:13.040 OK. 00:39:17.551 --> 00:39:18.050 OK. 00:39:18.050 --> 00:39:22.220 Those are the signs that, that are given by this rule, 00:39:22.220 --> 00:39:24.540 i+j even or odd. 00:39:24.540 --> 00:39:27.760 And those are built into the cofactors. 00:39:27.760 --> 00:39:31.030 The thing is called a minor without th- 00:39:31.030 --> 00:39:34.400 before you've built in the sign, but I don't care about those. 00:39:34.400 --> 00:39:39.700 Build in that sign and call it a cofactor. 00:39:39.700 --> 00:39:41.950 So what's the cofactor formula? 00:39:41.950 --> 00:39:42.450 OK. 00:39:42.450 --> 00:39:44.240 What's the cofactor formula then? 00:39:44.240 --> 00:39:49.110 Let me come back to this board and say, 00:39:49.110 --> 00:39:50.560 what's the cofactor formula? 00:39:58.150 --> 00:40:02.840 Determinant of A is -- 00:40:02.840 --> 00:40:04.670 let's go along the first row. 00:40:04.670 --> 00:40:11.480 It's a one one times its cofactor, 00:40:11.480 --> 00:40:15.730 and then the second guy is a one two times its cofactor, 00:40:15.730 --> 00:40:19.070 and you just keep going to the end of the row, 00:40:19.070 --> 00:40:22.750 a1n times its cofactor. 00:40:22.750 --> 00:40:24.840 So that's cofactor for -- 00:40:24.840 --> 00:40:30.780 along row one. 00:40:30.780 --> 00:40:39.030 And if I went along row I, I would -- those ones would be 00:40:39.030 --> 00:40:39.530 Is. 00:40:39.530 --> 00:40:43.520 That's worth putting a box over. 00:40:43.520 --> 00:40:47.470 That's the cofactor formula. 00:40:47.470 --> 00:40:51.180 Do you see that -- 00:40:51.180 --> 00:40:53.740 actually, this would give me another way 00:40:53.740 --> 00:41:00.150 I could have started the whole topic of determinants. 00:41:00.150 --> 00:41:02.000 And some, some people might do it this -- 00:41:02.000 --> 00:41:04.440 choose to do it this way. 00:41:04.440 --> 00:41:06.540 Because the cofactor formula would 00:41:06.540 --> 00:41:09.740 allow me to build up an n by n determinant out 00:41:09.740 --> 00:41:14.790 of n-1 sized determinants, build those out of n-2, and so on. 00:41:14.790 --> 00:41:17.740 I could boil all the way down to one by ones. 00:41:17.740 --> 00:41:21.470 So what's the cofactor formula for two by two matrices? 00:41:21.470 --> 00:41:23.300 Yeah, tell me that. 00:41:23.300 --> 00:41:24.580 What's the cofactor for us? 00:41:24.580 --> 00:41:28.620 Here is the, here is the world's smallest example, practically, 00:41:28.620 --> 00:41:32.580 of a cofactor formula. 00:41:32.580 --> 00:41:33.170 OK. 00:41:33.170 --> 00:41:35.570 Let's go along row one. 00:41:35.570 --> 00:41:39.500 I take this first guy times its cofactor. 00:41:39.500 --> 00:41:44.350 What's the cofactor of the one one entry? 00:41:44.350 --> 00:41:48.460 d, because you strike out the one one row and column 00:41:48.460 --> 00:41:50.460 and you're left with d. 00:41:50.460 --> 00:41:54.330 Then I take this guy, b, times its cofactor. 00:41:54.330 --> 00:41:57.740 What's the cofactor of b? 00:41:57.740 --> 00:42:00.070 Is it c or it's -- 00:42:00.070 --> 00:42:03.330 minus c, because I strike out this guy, 00:42:03.330 --> 00:42:08.360 I take that determinant, and then I follow the i+j rule 00:42:08.360 --> 00:42:11.730 and I get a minus, I get an odd. 00:42:11.730 --> 00:42:13.450 So it's b times minus c. 00:42:17.230 --> 00:42:18.120 OK, it worked. 00:42:18.120 --> 00:42:20.430 Of course it, it worked. 00:42:20.430 --> 00:42:23.100 And the three by three works. 00:42:23.100 --> 00:42:28.200 So that's the cofactor formula, and that is, that's an -- 00:42:28.200 --> 00:42:33.230 that's a good formula to know, and now I'm feeling like, wow, 00:42:33.230 --> 00:42:38.560 I'm giving you a lot of algebra to swallow here. 00:42:38.560 --> 00:42:41.955 Last lecture gave you ten properties. 00:42:44.900 --> 00:42:46.750 Now I'm giving you -- 00:42:46.750 --> 00:42:50.390 and by the way, those ten properties led us to a formula 00:42:50.390 --> 00:42:52.480 for the determinant which was very important, 00:42:52.480 --> 00:42:55.850 and I haven't repeated it till now. 00:42:55.850 --> 00:42:56.820 What was that? 00:42:56.820 --> 00:43:01.140 The, the determinant is the product of the pivots. 00:43:01.140 --> 00:43:03.980 So the pivot formula is, is very important. 00:43:03.980 --> 00:43:07.440 The pivots have all this complicated mess already 00:43:07.440 --> 00:43:08.810 built in. 00:43:08.810 --> 00:43:11.460 As you did elimination to get the pivots, 00:43:11.460 --> 00:43:17.670 you built in all this horrible stuff, quite efficiently. 00:43:17.670 --> 00:43:20.680 Then the big formula with the n factorial terms, 00:43:20.680 --> 00:43:24.180 that's got all the horrible stuff spread out. 00:43:24.180 --> 00:43:28.870 And the cofactor formula is like in between. 00:43:28.870 --> 00:43:35.460 It's got easy stuff times horrible stuff, basically. 00:43:35.460 --> 00:43:39.840 But it's, it shows you, how to get determinants 00:43:39.840 --> 00:43:42.780 from smaller determinants, and that's the application that I 00:43:42.780 --> 00:43:45.290 now want to make. 00:43:45.290 --> 00:43:51.030 So may I do one more example? 00:43:51.030 --> 00:43:54.670 So I remember the general idea. 00:43:54.670 --> 00:43:59.360 But I'm going to use this cofactor formula for a matrix 00:43:59.360 --> 00:44:01.230 -- 00:44:01.230 --> 00:44:03.640 so here is going to be my example. 00:44:03.640 --> 00:44:07.890 It's -- I promised in the, in the lecture, 00:44:07.890 --> 00:44:11.090 outline at the very beginning to do an example. 00:44:11.090 --> 00:44:12.650 And let me do -- 00:44:12.650 --> 00:44:18.280 I'm going to pick tri-diagonal matrix of ones. 00:44:21.700 --> 00:44:26.340 I could, I'm drawing here the four by four. 00:44:26.340 --> 00:44:27.880 So this will be the matrix. 00:44:27.880 --> 00:44:29.260 I could call that A4. 00:44:34.830 --> 00:44:40.590 But my real idea is to do n by n. 00:44:40.590 --> 00:44:43.100 To do them all. 00:44:43.100 --> 00:44:44.930 So A -- I could -- 00:44:44.930 --> 00:44:49.010 everybody understands what A1 and A2 are. 00:44:49.010 --> 00:44:49.510 Yeah. 00:44:49.510 --> 00:44:54.900 Maybe we should just do A1 and A2 and A3 just for -- 00:44:54.900 --> 00:44:55.400 so this is 00:44:55.400 --> 00:44:57.591 What's the determinant of A1? 00:44:57.591 --> 00:44:58.090 A4. 00:44:58.090 --> 00:45:01.120 What's the determinant of A1? 00:45:01.120 --> 00:45:04.470 So, so what's the matrix A1 in this formula? 00:45:04.470 --> 00:45:06.310 It's just got that. 00:45:06.310 --> 00:45:08.850 So the determinant is one. 00:45:08.850 --> 00:45:10.950 What's the determinant of A2? 00:45:10.950 --> 00:45:14.831 So it's just got this two by two, and its determinant is -- 00:45:17.780 --> 00:45:19.690 zero. 00:45:19.690 --> 00:45:22.350 And then the three by three. 00:45:22.350 --> 00:45:23.800 Can we see its determinant? 00:45:23.800 --> 00:45:27.740 Can you take the determinant of that three by three? 00:45:27.740 --> 00:45:32.910 Well, that's not quite so obvious, at least not to me. 00:45:32.910 --> 00:45:35.060 Being three by three, I don't know -- 00:45:35.060 --> 00:45:36.710 so here's a, here's a good example. 00:45:36.710 --> 00:45:39.590 How would you do that three by three determinant? 00:45:39.590 --> 00:45:43.160 We've got, like, n factorial different ways. 00:45:43.160 --> 00:45:44.080 Well, three factorial. 00:45:44.080 --> 00:45:45.260 So we've got six ways. 00:45:45.260 --> 00:45:46.220 OK. 00:45:46.220 --> 00:45:49.440 I mean, one way to do it -- 00:45:49.440 --> 00:45:51.040 actually the way I would probably 00:45:51.040 --> 00:45:52.990 do it, being three by three, I would use 00:45:52.990 --> 00:45:55.250 the complete the big formula. 00:45:55.250 --> 00:45:57.120 I would say, I've got a one from that, 00:45:57.120 --> 00:46:00.710 I've got a zero from that, I've got a zero from that, a zero 00:46:00.710 --> 00:46:03.140 from that, and this direction is a minus one, 00:46:03.140 --> 00:46:04.410 that direction's a minus one. 00:46:04.410 --> 00:46:06.200 I believe the answer is minus one. 00:46:06.200 --> 00:46:14.220 Would you do it another way? 00:46:14.220 --> 00:46:16.400 Here's another way to do it, look. 00:46:16.400 --> 00:46:18.065 Subtract row three from -- 00:46:18.065 --> 00:46:19.440 I'm just looking at this three by 00:46:19.440 --> 00:46:19.939 three. 00:46:19.939 --> 00:46:22.400 Everybody's looking at the three by three. 00:46:22.400 --> 00:46:25.630 Subtract row three from row two. 00:46:25.630 --> 00:46:26.930 Determinant doesn't change. 00:46:26.930 --> 00:46:29.710 So those become zeros. 00:46:29.710 --> 00:46:31.710 OK, now use the cofactor formula. 00:46:31.710 --> 00:46:33.160 How's that? 00:46:33.160 --> 00:46:36.310 How can, how -- if this was now zeros and I'm looking at this 00:46:36.310 --> 00:46:39.530 three by three, use the cofactor formula. 00:46:39.530 --> 00:46:42.875 Why not use the cofactor formula along that row? 00:46:45.890 --> 00:46:49.360 Because then I take that number times its cofactor, 00:46:49.360 --> 00:46:52.800 so I take this number -- let me put a box around it -- 00:46:52.800 --> 00:46:56.030 times its cofactor, which is the determinant of that and that, 00:46:56.030 --> 00:46:56.830 which is what? 00:47:02.310 --> 00:47:07.230 That two by two matrix has determinant one. 00:47:07.230 --> 00:47:08.625 So what's the cofactor? 00:47:11.410 --> 00:47:15.150 What's the cofactor of this guy here? 00:47:15.150 --> 00:47:17.220 Looking just at this three by three. 00:47:17.220 --> 00:47:21.470 The cofactor of that one is this determinant, 00:47:21.470 --> 00:47:26.490 which is one times negative. 00:47:26.490 --> 00:47:30.370 So that's why the answer came out minus one. 00:47:30.370 --> 00:47:31.280 OK. 00:47:31.280 --> 00:47:32.910 So I did the three by three. 00:47:32.910 --> 00:47:35.310 I don't know if we want to try the four by four. 00:47:35.310 --> 00:47:38.090 Yeah, let's -- I guess that was the point of my example, 00:47:38.090 --> 00:47:41.250 of course, so I have to try it. 00:47:41.250 --> 00:47:44.120 Sorry, I'm in a good mood today, so you have 00:47:44.120 --> 00:47:45.840 to stand for all the bad jokes. 00:47:45.840 --> 00:47:46.340 OK. 00:47:46.340 --> 00:47:47.200 OK. 00:47:47.200 --> 00:47:50.830 So what was the matrix? 00:47:50.830 --> 00:47:51.330 Ah. 00:47:55.580 --> 00:47:57.660 OK, now I'm ready for four by four. 00:47:57.660 --> 00:48:00.720 Who wants to -- who wants to guess the, the -- 00:48:00.720 --> 00:48:04.270 I don't know, frankly, this four by four, 00:48:04.270 --> 00:48:07.280 what's, what's the determinant. 00:48:07.280 --> 00:48:08.575 I plan to use cofactors. 00:48:12.640 --> 00:48:14.310 OK, let's use cofactors. 00:48:14.310 --> 00:48:17.960 The determinant of A4 is -- 00:48:17.960 --> 00:48:20.420 OK, let's use cofactors on the first row. 00:48:20.420 --> 00:48:21.630 Those are easy. 00:48:21.630 --> 00:48:25.720 So I multiply this number, which is a convenient one, 00:48:25.720 --> 00:48:28.240 times this determinant. 00:48:28.240 --> 00:48:31.790 So it's, it's one times the, this three by three 00:48:31.790 --> 00:48:32.380 determinant. 00:48:32.380 --> 00:48:36.600 Now what is -- do you recognize that matrix? 00:48:36.600 --> 00:48:37.900 It's A3. 00:48:37.900 --> 00:48:41.940 So it's one times the determinant of A3. 00:48:41.940 --> 00:48:46.830 Coming along this row is a one times this determinant, 00:48:46.830 --> 00:48:49.430 and it goes with a plus, right? 00:48:49.430 --> 00:48:50.665 And then we have this one. 00:48:53.170 --> 00:48:55.290 And what is its cofactor? 00:48:55.290 --> 00:48:58.780 Now I'm looking at, now I'm looking at this three 00:48:58.780 --> 00:49:00.990 by three, this three by three, so I'm 00:49:00.990 --> 00:49:04.270 looking at the three by three that I haven't X-ed out. 00:49:04.270 --> 00:49:08.630 What is that -- oh, now it, we did a plus or a -- 00:49:08.630 --> 00:49:10.880 is it plus this determinant, this three by three 00:49:10.880 --> 00:49:13.730 determinant, or minus it? 00:49:13.730 --> 00:49:16.150 It's minus it, right, because this is -- 00:49:16.150 --> 00:49:20.180 I'm starting in a one two position, and that's a minus. 00:49:20.180 --> 00:49:22.390 So I want minus this determinant. 00:49:22.390 --> 00:49:25.180 But these guys are X-ed out. 00:49:25.180 --> 00:49:25.680 OK. 00:49:25.680 --> 00:49:27.260 So I've got a three by three. 00:49:27.260 --> 00:49:31.170 Well, let's use cofactors again. 00:49:31.170 --> 00:49:33.740 Use cofactors of the column, because actually we 00:49:33.740 --> 00:49:35.620 could use cofactors of columns just 00:49:35.620 --> 00:49:39.930 as well as rows, because, because the transpose rule. 00:49:39.930 --> 00:49:43.750 So I'll take this one, which is now sitting in the plus 00:49:43.750 --> 00:49:46.440 position, times its determinant -- oh! 00:49:46.440 --> 00:49:47.773 Oh, hell. 00:49:50.870 --> 00:49:53.990 What -- oh yeah, I shouldn't have said hell, 00:49:53.990 --> 00:49:55.260 because it's all right. 00:49:55.260 --> 00:49:55.760 OK. 00:49:55.760 --> 00:49:58.230 One times the determinant. 00:49:58.230 --> 00:50:00.410 What is that matrix now that I'm taking 00:50:00.410 --> 00:50:01.920 the, this smaller one of? 00:50:01.920 --> 00:50:03.420 Oh, but there's a minus, right? 00:50:03.420 --> 00:50:05.630 The minus came from, from the fact 00:50:05.630 --> 00:50:10.570 that this was in the one two position and that's odd. 00:50:10.570 --> 00:50:14.750 So this is a minus one times -- and what's -- 00:50:14.750 --> 00:50:17.130 and then this one is the upper left, 00:50:17.130 --> 00:50:21.880 that's the one one position in its matrix, so plus. 00:50:21.880 --> 00:50:23.910 And what's this matrix here? 00:50:23.910 --> 00:50:26.460 Do you recognize that? 00:50:26.460 --> 00:50:28.840 That matrix is -- 00:50:28.840 --> 00:50:30.275 yes, please say it -- 00:50:30.275 --> 00:50:30.775 A2. 00:50:36.320 --> 00:50:39.110 And we -- that's our formula for any case. 00:50:39.110 --> 00:50:46.790 A of any size n is equal to the determinant of A n minus one, 00:50:46.790 --> 00:50:49.310 that's what came from taking the one in the upper corner, 00:50:49.310 --> 00:50:55.520 the first cofactor, minus the determinant of A n minus two. 00:50:55.520 --> 00:51:02.020 What we discovered there is true for all n. 00:51:02.020 --> 00:51:04.680 I didn't even mention it, but I stopped taking 00:51:04.680 --> 00:51:06.610 cofactors when I got this one. 00:51:06.610 --> 00:51:08.920 Why did I stop? 00:51:08.920 --> 00:51:12.660 Why didn't I take the cofactor of this guy? 00:51:12.660 --> 00:51:16.490 Because he's going to get multiplied by zero, and no, 00:51:16.490 --> 00:51:18.130 no use wasting time. 00:51:18.130 --> 00:51:20.260 Or this one too. 00:51:20.260 --> 00:51:22.570 The cofactor, her cofactor will be 00:51:22.570 --> 00:51:24.400 whatever that determinant is, but it'll 00:51:24.400 --> 00:51:26.980 be multiplied by zero, so I won't bother. 00:51:26.980 --> 00:51:30.010 OK, there is the formula. 00:51:30.010 --> 00:51:32.160 And that now tells us -- 00:51:32.160 --> 00:51:34.620 I could figure out what A4 is now. 00:51:34.620 --> 00:51:37.570 Oh yeah, finally I can get A4. 00:51:37.570 --> 00:51:43.770 Because it's A3, which is minus one, minus A2, which is zero, 00:51:43.770 --> 00:51:44.880 so it's minus one. 00:51:47.750 --> 00:51:50.090 You see how we're getting kind of numbers 00:51:50.090 --> 00:51:51.710 that you might not have guessed. 00:51:51.710 --> 00:51:55.800 So our sequence right now is one zero minus one minus one. 00:51:55.800 --> 00:52:00.760 What's the next one in the sequence, A5? 00:52:00.760 --> 00:52:06.300 A5 is this minus this, so it is zero. 00:52:06.300 --> 00:52:09.260 What's A6? 00:52:09.260 --> 00:52:15.140 A6 is this minus this, which is one. 00:52:15.140 --> 00:52:18.150 What's A7? 00:52:18.150 --> 00:52:21.640 I'm, I'm going to be stopped by either the time runs out 00:52:21.640 --> 00:52:23.230 or the board runs out. 00:52:23.230 --> 00:52:27.740 OK, A7 is this minus this, which is one. 00:52:27.740 --> 00:52:30.790 I'll stop here, because time is out, but let me tell you 00:52:30.790 --> 00:52:32.180 what we've got. 00:52:32.180 --> 00:52:35.850 What -- these determinants have this series, 00:52:35.850 --> 00:52:39.730 one zero minus one minus one zero one, 00:52:39.730 --> 00:52:42.760 and then it starts repeating. 00:52:42.760 --> 00:52:44.600 It's pretty fantastic. 00:52:44.600 --> 00:52:48.480 These determinants have period six. 00:52:48.480 --> 00:52:51.880 So the determinant of A sixty-one 00:52:51.880 --> 00:52:55.771 would be the determinant of A1, which would be one. 00:52:55.771 --> 00:52:56.270 OK. 00:52:56.270 --> 00:52:58.190 I hope you liked that example. 00:52:58.190 --> 00:53:04.640 A non-trivial example of a tri-diagonal determinant. 00:53:04.640 --> 00:53:05.520 Thanks. 00:53:05.520 --> 00:53:08.260 See you on Wednesday.