WEBVTT 00:00:07.730 --> 00:00:11.436 OK, this lecture is like the beginning 00:00:11.436 --> 00:00:13.060 of the second half of this is to prove. 00:00:13.060 --> 00:00:18.590 this course because up to now we paid a lot of attention 00:00:18.590 --> 00:00:22.240 to rectangular matrices. 00:00:22.240 --> 00:00:27.020 Now, concentrating on square matrices, 00:00:27.020 --> 00:00:29.150 so we're at two big topics. 00:00:29.150 --> 00:00:32.210 The determinant of a square matrix, 00:00:32.210 --> 00:00:35.040 so this is the first lecture in that new chapter 00:00:35.040 --> 00:00:38.940 on determinants, and the reason, the big reason 00:00:38.940 --> 00:00:43.030 we need the determinants is for the Eigen values. 00:00:43.030 --> 00:00:46.420 So this is really determinants and Eigen values, 00:00:46.420 --> 00:00:50.330 the next big, big chunk of 18.06. 00:00:50.330 --> 00:00:57.100 OK, so the determinant is a number associated 00:00:57.100 --> 00:01:01.360 with every square matrix, so every square matrix 00:01:01.360 --> 00:01:07.160 has this number associated with called the, its determinant. 00:01:07.160 --> 00:01:15.720 I'll often write it as D E T A or often also I'll write it as, 00:01:15.720 --> 00:01:19.440 A with vertical bars, so that's going to mean 00:01:19.440 --> 00:01:21.030 the determinant of the matrix. 00:01:21.030 --> 00:01:26.400 That's going to mean this one, like, magic number. 00:01:26.400 --> 00:01:32.420 Well, one number can't tell you what the whole matrix was. 00:01:32.420 --> 00:01:36.960 But this one number, just packs in as much information 00:01:36.960 --> 00:01:39.310 as possible into a single number, 00:01:39.310 --> 00:01:43.250 and of course the one fact that you've seen before 00:01:43.250 --> 00:01:48.280 and we have to see it again is the matrix 00:01:48.280 --> 00:01:53.680 is invertible when the determinant is not zero. 00:01:53.680 --> 00:01:58.260 The matrix is singular when the determinant is zero. 00:01:58.260 --> 00:02:03.590 So the determinant will be a test for invertibility, 00:02:03.590 --> 00:02:07.430 but the determinant's got a lot more to it than that, 00:02:07.430 --> 00:02:08.889 so let me start. 00:02:08.889 --> 00:02:12.260 OK, now the question is how to start. 00:02:12.260 --> 00:02:14.850 Do I give you a big formula for the determinant, 00:02:14.850 --> 00:02:16.600 all in one gulp? 00:02:16.600 --> 00:02:18.190 I don't think so! 00:02:18.190 --> 00:02:21.410 That big formula has got too much packed in it. 00:02:21.410 --> 00:02:28.900 I would rather start with three properties of the determinant, 00:02:28.900 --> 00:02:31.150 three properties that it has. 00:02:31.150 --> 00:02:33.990 And let me tell you property one. 00:02:33.990 --> 00:02:39.380 The determinant of the identity is one. 00:02:39.380 --> 00:02:40.670 OK. 00:02:40.670 --> 00:02:41.710 I... 00:02:41.710 --> 00:02:43.540 I wish the other two properties were 00:02:43.540 --> 00:02:46.810 as easy to tell you as that. 00:02:46.810 --> 00:02:51.490 But actually the second property is pretty straightforward too, 00:02:51.490 --> 00:02:56.050 and then once we get the third we will actually 00:02:56.050 --> 00:02:57.780 have the determinant. 00:02:57.780 --> 00:03:02.240 Those three properties define the determinant and we can -- 00:03:02.240 --> 00:03:13.610 then we can figure out, well, what is the determinant? 00:03:13.610 --> 00:03:16.880 What's a formula for it? 00:03:16.880 --> 00:03:22.000 Now, the second property is what happens if you 00:03:22.000 --> 00:03:24.620 exchange two rows of a matrix. 00:03:24.620 --> 00:03:27.040 What happens to the determinant? 00:03:27.040 --> 00:03:31.000 So, property two is exchange rows, 00:03:31.000 --> 00:03:46.390 reverse the sign of the determinant. 00:03:51.470 --> 00:03:53.360 A lot of plus and minus signs. 00:03:53.360 --> 00:03:56.150 I even wrote here, "plus and minus signs," 00:03:56.150 --> 00:03:57.880 because this is, like, that's what 00:03:57.880 --> 00:04:00.570 you have to pay attention to in the formulas 00:04:00.570 --> 00:04:03.080 and properties of determinants. 00:04:03.080 --> 00:04:08.200 So that -- you see what I mean by a property here? 00:04:08.200 --> 00:04:10.890 I haven't yet told you what the determinant is, 00:04:10.890 --> 00:04:13.890 but whatever it is, if I exchange 00:04:13.890 --> 00:04:17.910 two rows to get a different matrix that reverses 00:04:17.910 --> 00:04:20.950 the sign of the determinant. 00:04:20.950 --> 00:04:25.180 And so now, actually, what matrices 00:04:25.180 --> 00:04:28.140 do we now know the determinant of? 00:04:28.140 --> 00:04:32.260 From one and two, I now know the determinant. 00:04:32.260 --> 00:04:34.640 Well, I certainly know the determinant of the identity 00:04:34.640 --> 00:04:37.210 matrix and now I know the determinant 00:04:37.210 --> 00:04:41.970 of every other matrix that comes from row exchanges 00:04:41.970 --> 00:04:44.540 from the identities still. 00:04:44.540 --> 00:04:48.460 So what matrices have I gotten at this point? 00:04:48.460 --> 00:04:50.500 The permutations, right. 00:04:50.500 --> 00:04:55.020 At this point I know every permutation matrix, 00:04:55.020 --> 00:04:58.730 so now I'm saying the determinant of a permutation 00:04:58.730 --> 00:05:03.690 matrix is one or minus one. 00:05:03.690 --> 00:05:09.320 One or minus one, depending whether the number of exchanges 00:05:09.320 --> 00:05:16.720 was even or the number of exchanges was odd. 00:05:16.720 --> 00:05:19.100 So this is the determinant of a permutation. 00:05:19.100 --> 00:05:22.720 Now, P is back to standing for permutation. 00:05:22.720 --> 00:05:23.560 OK. 00:05:23.560 --> 00:05:28.180 if I could carry on this board, I could, like, 00:05:28.180 --> 00:05:29.940 do the two-by-two's. 00:05:29.940 --> 00:05:34.790 So, property one tells me that this two-by-two matrix. 00:05:34.790 --> 00:05:37.560 Oh, I better write absolute -- 00:05:37.560 --> 00:05:41.150 I mean, I'd better write vertical bars, not brackets, 00:05:41.150 --> 00:05:43.770 for that determinant. 00:05:43.770 --> 00:05:46.890 Property one said, in the two-by-two case, 00:05:46.890 --> 00:05:51.700 that this matrix has determinant one. 00:05:51.700 --> 00:05:59.130 Property two tells me that this matrix has determinant -- 00:05:59.130 --> 00:06:00.500 what? 00:06:00.500 --> 00:06:02.740 Negative one. 00:06:02.740 --> 00:06:04.530 This is, like, two-by-twos. 00:06:04.530 --> 00:06:08.090 Now, I finally want to get -- 00:06:08.090 --> 00:06:10.210 well, ultimately I want to get to, 00:06:10.210 --> 00:06:13.130 the formula that we all know. 00:06:13.130 --> 00:06:16.600 Let me put that way over here, that the determinant 00:06:16.600 --> 00:06:23.190 of a general two-by-two is ad-bc. 00:06:23.190 --> 00:06:23.690 OK. 00:06:23.690 --> 00:06:24.070 I'm going to leave that up, like, as the two by two case 00:06:24.070 --> 00:06:24.190 I'm down to the product of the diagonal and if I transpose, 00:06:24.190 --> 00:06:26.065 that we already know, so that every property, 00:06:26.065 --> 00:06:44.980 I can, like, check that it's correct for two-by-twos. 00:06:44.980 --> 00:06:47.960 But the whole point of these properties 00:06:47.960 --> 00:06:51.850 is that they're going to give me a formula for n-by-n. 00:06:51.850 --> 00:06:53.760 That's the whole point. 00:06:53.760 --> 00:06:57.060 They're going to give me this number that's 00:06:57.060 --> 00:07:00.460 a test for invertibility and other great properties 00:07:00.460 --> 00:07:02.240 for any size matrix. 00:07:02.240 --> 00:07:08.500 OK, now you see I'm like, slowing down because property 00:07:08.500 --> 00:07:13.220 three is the key property. 00:07:13.220 --> 00:07:17.570 Property three is the key property and can I somehow 00:07:17.570 --> 00:07:22.500 describe it -- maybe I'll separate it into 3A I said that 00:07:22.500 --> 00:07:26.610 if you do a row exchange, the determinant and 3B. 00:07:26.610 --> 00:07:30.750 Property 3A says that if I multiply one of the rows, 00:07:30.750 --> 00:07:40.020 say the first row, by a number T -- 00:07:40.020 --> 00:07:41.590 I'm going to erase that. 00:07:41.590 --> 00:07:46.150 That's, like, what I'm headed for but I'm not there yet. 00:07:46.150 --> 00:07:49.890 It's the one we know and you'll see that it's 00:07:49.890 --> 00:07:54.990 checked out by each property. 00:07:54.990 --> 00:08:00.220 OK, so this is for any matrix. 00:08:00.220 --> 00:08:03.990 For any matrix, if I multiply one row by T 00:08:03.990 --> 00:08:08.740 and leave the other row or other n-1 rows alone, 00:08:08.740 --> 00:08:11.220 what happens to the determinant? 00:08:11.220 --> 00:08:15.610 The factor T comes out. 00:08:15.610 --> 00:08:17.260 It's T times this determinant. 00:08:22.200 --> 00:08:23.230 That's not hard. 00:08:23.230 --> 00:08:25.560 I shouldn't have made a big deal out of property 3A, 00:08:25.560 --> 00:08:29.630 and 3B is that, if is, is if I keep -- 00:08:29.630 --> 00:08:33.640 I'm always keeping this second row the same, 00:08:33.640 --> 00:08:37.929 or that last n-1 rows are all staying the same. 00:08:37.929 --> 00:08:39.460 I'm just working -- 00:08:39.460 --> 00:08:43.679 I'm just looking inside the first row and if I have 00:08:43.679 --> 00:08:53.770 an a+a' there and a b+b' there -- 00:08:53.770 --> 00:08:54.480 sorry, I didn't. 00:08:54.480 --> 00:08:55.150 Ahh. 00:08:55.150 --> 00:08:59.690 Why don't -- I'll use an eraser, do it right. 00:08:59.690 --> 00:09:00.190 b+b' there. 00:09:00.190 --> 00:09:02.390 You see what I'm doing? 00:09:02.390 --> 00:09:06.730 This property and this property are about linear combinations, 00:09:06.730 --> 00:09:13.720 of the first row only, leaving the other rows unchanged. 00:09:13.720 --> 00:09:14.680 They'll copy along. 00:09:14.680 --> 00:09:20.320 Then, then I get the sum -- this breaks up into the sum of this 00:09:20.320 --> 00:09:23.500 determinant and this one. 00:09:32.740 --> 00:09:34.120 I'm putting up formulas. 00:09:34.120 --> 00:09:36.850 Maybe I can try to say it in words. 00:09:36.850 --> 00:09:41.110 The determinant is a linear function. 00:09:41.110 --> 00:09:48.170 It behaves like a linear function of first row 00:09:48.170 --> 00:09:51.540 if all the other rows stay the same. 00:09:51.540 --> 00:09:54.340 I not saying that -- 00:09:54.340 --> 00:09:55.640 let me emphasize. 00:09:55.640 --> 00:10:00.860 I not saying that the determinant of A plus B 00:10:00.860 --> 00:10:07.870 is determinant of A plus determinant of B. 00:10:07.870 --> 00:10:09.180 I not saying that. 00:10:09.180 --> 00:10:11.490 I'd better -- can I -- 00:10:11.490 --> 00:10:15.620 how do I get it onto tape that I'm not saying that? 00:10:15.620 --> 00:10:22.350 You see, this would allow all the rows -- you know, 00:10:22.350 --> 00:10:24.830 A to have a bunch of rows, B to have a bunch of rows. 00:10:24.830 --> 00:10:29.510 That's not the linearity I'm after. 00:10:29.510 --> 00:10:32.840 I'm only after linearity in each row. 00:10:32.840 --> 00:10:38.030 Linear for each row. 00:10:40.840 --> 00:10:44.990 Well, you may say I only talked about the first row, 00:10:44.990 --> 00:10:48.610 but I claim it's also linear in the second row, 00:10:48.610 --> 00:10:53.520 if I had this -- but not, I can't, I can't have 00:10:53.520 --> 00:10:56.320 a combination in both first and second. 00:10:56.320 --> 00:10:58.830 If I had a combination in the second row, 00:10:58.830 --> 00:11:04.230 then I could use rule two to put it up in the first row, 00:11:04.230 --> 00:11:09.560 use my property and then use rule two again to put it back, 00:11:09.560 --> 00:11:14.340 so each row is OK, not only the first row, 00:11:14.340 --> 00:11:17.370 but each row separately. 00:11:17.370 --> 00:11:19.760 OK, those are the three properties, 00:11:19.760 --> 00:11:23.030 and from those properties, so that's 00:11:23.030 --> 00:11:26.660 properties one, two, three. 00:11:26.660 --> 00:11:31.584 From those, I want to get all -- 00:11:31.584 --> 00:11:33.750 I'm going to learn a lot more about the determinant. 00:11:36.490 --> 00:11:38.030 Let me take an example. 00:11:38.030 --> 00:11:39.990 What would I like to learn? 00:11:39.990 --> 00:11:43.280 I would like to learn that -- so here's our property four. 00:11:43.280 --> 00:11:47.310 Let me use the same numbering as here. 00:11:47.310 --> 00:11:58.720 Property four is if two rows are equal, the determinant is zero. 00:11:58.720 --> 00:12:02.720 OK, so property four. 00:12:02.720 --> 00:12:13.570 Two equal rows lead to determinant equals zero. 00:12:13.570 --> 00:12:14.280 Right. 00:12:14.280 --> 00:12:19.360 Now, of course I can -- in the two-by-two case I can check, 00:12:19.360 --> 00:12:23.980 sure, the determinant of ab ab comes out zero. 00:12:23.980 --> 00:12:28.950 But I want to see why it's true for n-by-n. 00:12:28.950 --> 00:12:36.100 Suppose row one equals row three for a seven-by-seven matrix. 00:12:36.100 --> 00:12:39.070 So two rows are the same in a big matrix. 00:12:39.070 --> 00:12:43.350 And all I have to work with is these properties. 00:12:43.350 --> 00:12:47.540 The exchange property, which flips the sign, 00:12:47.540 --> 00:12:53.580 and the linearity property which works in each row separately. 00:12:53.580 --> 00:12:57.170 OK, can you see the reason? 00:12:57.170 --> 00:13:02.200 How do I get this one out of properties one, two, three? 00:13:02.200 --> 00:13:04.680 Because -- that's all I have to work with. 00:13:04.680 --> 00:13:07.770 Everything has to come from properties one, two, three. 00:13:07.770 --> 00:13:12.040 OK, so suppose I have a matrix, and two rows are even. 00:13:14.770 --> 00:13:16.310 How do I see that its determinant 00:13:16.310 --> 00:13:22.120 has to be zero from these properties? 00:13:22.120 --> 00:13:24.490 I do an exchange. 00:13:24.490 --> 00:13:27.190 Property two is just set up for this. 00:13:27.190 --> 00:13:28.610 Use property two. 00:13:28.610 --> 00:13:33.090 Use exchange -- exchange rows. 00:13:33.090 --> 00:13:40.720 Exchange those rows, and I get the same matrix. 00:13:40.720 --> 00:13:42.700 Of course, because those rows were equal. 00:13:47.870 --> 00:13:50.580 So the determinant didn't change. 00:13:50.580 --> 00:13:52.360 But on the other hand, property two 00:13:52.360 --> 00:13:56.170 says that the sign did change. 00:13:56.170 --> 00:13:59.350 So the -- so I, I have a determinant whose sign 00:13:59.350 --> 00:14:03.390 doesn't change and does change, and the only possibility then 00:14:03.390 --> 00:14:06.520 is that the determinant is zero. 00:14:06.520 --> 00:14:08.780 You see the reasoning there? 00:14:08.780 --> 00:14:09.550 Straightforward. 00:14:09.550 --> 00:14:15.250 Property two just told us, hey, if we've got two equal rows we. 00:14:15.250 --> 00:14:19.210 we've got a zero determinant. 00:14:19.210 --> 00:14:22.140 And of course in our minds, that matches the fact 00:14:22.140 --> 00:14:26.550 that if I have two equal rows the matrix isn't invertible. 00:14:26.550 --> 00:14:29.430 If I have two equal rows, I know that the rank 00:14:29.430 --> 00:14:31.190 is less changes sign. than n. 00:14:31.190 --> 00:14:34.540 OK, ready for property five. 00:14:34.540 --> 00:14:38.900 Now, property five you'll recognize as P. 00:14:38.900 --> 00:14:45.710 It says that the elimination step that I'm always doing, 00:14:45.710 --> 00:14:51.850 or U and U transposed, when they're triangular,4 subtract 00:14:51.850 --> 00:15:00.560 a multiple, subtract some multiple l times row one from 00:15:00.560 --> 00:15:04.800 another row, row k, let's say. 00:15:07.950 --> 00:15:10.760 You remember why I did that. 00:15:10.760 --> 00:15:14.620 In elimination I'm always choosing this multiplier so as 00:15:14.620 --> 00:15:17.190 to produce zero in that position. 00:15:17.190 --> 00:15:21.050 What I -- way, way back in property two,4 00:15:21.050 --> 00:15:24.230 Or row I from row k, maybe I should just 00:15:24.230 --> 00:15:28.250 make very clear that there's nothing special about row one 00:15:28.250 --> 00:15:30.530 here. 00:15:30.530 --> 00:15:34.400 OK, so that, you can see why I want that who cares? 00:15:34.400 --> 00:15:37.350 one, because that will allow me to start 00:15:37.350 --> 00:15:40.890 with this full matrix whose determinant I don't know, 00:15:40.890 --> 00:15:45.590 and I can do elimination and clean it out. 00:15:45.590 --> 00:15:47.830 I can get zeroes below the diagonal 00:15:47.830 --> 00:15:50.970 by these elimination steps and the point 00:15:50.970 --> 00:15:58.420 is that the determinant, the determinant doesn't change. 00:16:10.210 --> 00:16:12.450 So all those steps of elimination 00:16:12.450 --> 00:16:15.070 are OK not changing the determinant. 00:16:15.070 --> 00:16:18.580 In our language in the early chapter the determinant of A is 00:16:18.580 --> 00:16:20.900 So if I do seven row exchanges, the determinant 00:16:20.900 --> 00:16:23.989 changes sign, going to be the same as the determinant of U, 00:16:23.989 --> 00:16:25.030 the upper triangular one. 00:16:25.030 --> 00:16:27.090 It just has the pivots on the diagonal. 00:16:27.090 --> 00:16:29.100 That's why we'll want this property. 00:16:29.100 --> 00:16:31.250 OK, do you see where that property's coming from? 00:16:34.180 --> 00:16:37.110 Let me do the two-by-two case. 00:16:37.110 --> 00:16:39.490 Let me do the two-by-two case here. 00:16:39.490 --> 00:16:44.340 So, I'll keep property five going along. 00:16:44.340 --> 00:16:45.300 So what I doing? 00:16:45.300 --> 00:16:46.890 I'm going to keep -- 00:16:46.890 --> 00:16:52.910 I'm going to have ab cd, but I'm going 00:16:52.910 --> 00:16:57.460 to subtract l times the first row from the second row. 00:16:57.460 --> 00:17:03.182 And that's the determinant and of 00:17:03.182 --> 00:17:03.890 OK, that's not -- 00:17:03.890 --> 00:17:06.260 I didn't put in every comma and, course 00:17:06.260 --> 00:17:10.900 I can multiply that out and figure out, sure enough, ad-bc 00:17:10.900 --> 00:17:17.849 is there and this minus ALB plus ALB cancels out, 00:17:17.849 --> 00:17:19.569 but I just cheated, 00:17:19.569 --> 00:17:20.190 right? 00:17:20.190 --> 00:17:21.760 I've got to use the properties. 00:17:21.760 --> 00:17:22.510 So what property? 00:17:22.510 --> 00:17:24.599 Well, of course, this is a com -- 00:17:24.599 --> 00:17:28.420 I'm keeping the first row the same and the second row 00:17:28.420 --> 00:17:31.690 has a c and a d, and then there's 00:17:31.690 --> 00:17:36.190 the determinant of the A and the B, and the minus LA, 00:17:36.190 --> 00:17:37.240 and the minus LB. 00:17:41.840 --> 00:17:44.150 So what property was that? 00:17:44.150 --> 00:17:46.420 3B. 00:17:46.420 --> 00:17:49.340 I kept one row the same and I had 00:17:49.340 --> 00:17:52.690 a combination in the second, in the other row, 00:17:52.690 --> 00:17:56.520 and I just separated it out. 00:17:56.520 --> 00:17:59.300 OK, so that's property 3. 00:17:59.300 --> 00:18:03.350 That's by number 3, 3B if you like. 00:18:03.350 --> 00:18:04.980 OK, now use 3A. 00:18:04.980 --> 00:18:10.480 How do you use 3A, which says I can factor out an l, 00:18:10.480 --> 00:18:13.110 I can factor out a minus l here. 00:18:13.110 --> 00:18:17.140 I can factor a minus l out from this row, no problem. 00:18:17.140 --> 00:18:19.000 That was 3A. 00:18:19.000 --> 00:18:25.390 So now I've used property three and now I'm ready for the kill. 00:18:25.390 --> 00:18:32.070 Property four says that this determinant is zero, 00:18:32.070 --> 00:18:34.810 has two equal rows. 00:18:34.810 --> 00:18:37.100 You see how that would always work? 00:18:37.100 --> 00:18:40.280 I subtract a multiple of one row from another one. 00:18:40.280 --> 00:18:46.860 It gives me a combination in row k of the old row and l times 00:18:46.860 --> 00:18:51.310 this copy of the higher row, and then if -- 00:18:51.310 --> 00:18:53.550 since I have two equal rows, that's zero, 00:18:53.550 --> 00:18:56.990 so the determinant after elimination is the same 00:18:56.990 --> 00:18:58.280 as before. 00:18:58.280 --> 00:18:59.430 Good. 00:18:59.430 --> 00:19:00.310 OK. 00:19:00.310 --> 00:19:04.170 Now, let's see -- if I rescue my glasses, 00:19:04.170 --> 00:19:07.140 I can see what's property six. 00:19:07.140 --> 00:19:11.700 Oh, six is easy, plenty of space. 00:19:11.700 --> 00:19:22.450 Row of zeroes leads to determinant of A equals zero. 00:19:26.840 --> 00:19:28.380 A complete row of zeroes. 00:19:28.380 --> 00:19:32.550 So I'm again, this is like, something 00:19:32.550 --> 00:19:34.880 I'll use in the singular case. 00:19:34.880 --> 00:19:39.380 Actually, you can look ahead to why I need these properties. 00:19:39.380 --> 00:19:42.220 So I'm going to use property five, the elimination, 00:19:42.220 --> 00:19:45.980 use this stuff to say that this determinant is 00:19:45.980 --> 00:19:49.600 the same as that determinant and I'll produce a zero there. 00:19:49.600 --> 00:19:51.910 But what if I also produce a zero there? 00:19:51.910 --> 00:19:54.620 What if elimination gives a row of zeroes? 00:19:54.620 --> 00:19:59.120 That, that used to be my test for, mmm, singular, 00:19:59.120 --> 00:20:03.220 not invertible, rank two -- rank less than N, 00:20:03.220 --> 00:20:07.120 and now I'm seeing it's also gives determinant zero. 00:20:07.120 --> 00:20:12.600 How do I get that one from the previous properties? 00:20:12.600 --> 00:20:15.006 'Cause I -- this is not a new law, 00:20:15.006 --> 00:20:16.630 this has got to come from the old ones. 00:20:16.630 --> 00:20:20.635 So what shall I do? 00:20:23.210 --> 00:20:24.690 Yeah, use -- that's brilliant. 00:20:24.690 --> 00:20:26.430 If you use 3A with T equals zero. 00:20:26.430 --> 00:20:27.350 Right. 00:20:27.350 --> 00:20:32.760 So I have this zero zero cd, and I'm 00:20:32.760 --> 00:20:35.850 trying to show that that determinant is zero. triangular 00:20:35.850 --> 00:20:37.560 matrices, l and l transposed, 00:20:37.560 --> 00:20:41.000 OK, so the zero is the same is -- five, 00:20:41.000 --> 00:20:45.900 can I take T equals five, just to, like, pin it down? 00:20:45.900 --> 00:20:48.530 I'll multiply this row by five. 00:20:48.530 --> 00:20:52.780 Five, well then, five of this should -- if I, 00:20:52.780 --> 00:21:01.300 if there's a factor five in that, you see what -- 00:21:01.300 --> 00:21:05.500 so this is property 3A, with taking T as five. 00:21:05.500 --> 00:21:08.660 If I multiply a row by five, out comes a five. 00:21:08.660 --> 00:21:14.450 So why I doing this? 00:21:14.450 --> 00:21:19.320 Oh, because that's still zero zero, right? 00:21:19.320 --> 00:21:21.000 So that's this determinant equals 00:21:21.000 --> 00:21:28.780 five times this determinant, and the determinant has to be zero. 00:21:28.780 --> 00:21:34.100 I think I didn't do that the very best way. 00:21:34.100 --> 00:21:36.670 You were, yeah, you had the idea better. 00:21:36.670 --> 00:21:40.620 Multiply, yeah, take T equals zero. 00:21:40.620 --> 00:21:44.170 Was that your idea? 00:21:44.170 --> 00:21:46.840 Take T equals zero in rule 3B. 00:21:46.840 --> 00:21:51.990 If T is zero in rule 3B, and I bring the camera back to rule 00:21:51.990 --> 00:21:52.720 3B -- 00:21:52.720 --> 00:21:55.260 sorry. 00:21:55.260 --> 00:22:01.300 If T is zero, then I have a zero zero there 00:22:01.300 --> 00:22:03.330 and the determinant is zero. 00:22:03.330 --> 00:22:08.710 OK, one way or another, a row of zeroes means zero determinant. 00:22:08.710 --> 00:22:14.930 OK, now I have to get serious. 00:22:14.930 --> 00:22:20.500 The next properties are the ones that we're building up to. 00:22:20.500 --> 00:22:23.750 OK, so I can do elimination. 00:22:23.750 --> 00:22:26.540 I can reduce to a triangular matrix 00:22:26.540 --> 00:22:30.120 and now what's the determinant of that triangular matrix? 00:22:30.120 --> 00:22:34.100 OK, so they had to wait until the last minute. 00:22:34.100 --> 00:22:37.280 Suppose, suppose I -- all right, rule seven. 00:22:37.280 --> 00:22:42.430 So suppose my matrix is now triangular. 00:22:42.430 --> 00:22:44.660 So it's got -- 00:22:44.660 --> 00:22:48.870 so I even give these the names of the pivots, d1, d2, to dn, 00:22:48.870 --> 00:22:54.710 and stuff is up here, I don't know what that is, 00:22:54.710 --> 00:22:57.970 but what I do know is this is all zeroes. 00:22:57.970 --> 00:23:04.750 That's all zeroes, and I would like to know the determinant, 00:23:04.750 --> 00:23:08.370 because elimination will get me to this. 00:23:08.370 --> 00:23:11.610 So once I'm here, what's the determinant then? 00:23:11.610 --> 00:23:16.560 Let me use an eraser to get those, get that vertical bar 00:23:16.560 --> 00:23:22.350 again, so that I'm taking the determinant of U so that, so, 00:23:22.350 --> 00:23:28.510 what is the determinant of an upper triangular matrix? 00:23:28.510 --> 00:23:33.215 Do you know the answer? 00:23:36.090 --> 00:23:40.220 It's just the product of the d's. for it. 00:23:40.220 --> 00:23:44.740 The -- these things that I don't even put in letters 00:23:44.740 --> 00:23:53.020 for, because they don't matter, the determinant is d1 times d2 00:23:53.020 --> 00:23:54.197 times dn. 00:23:57.120 --> 00:24:02.100 If I have a triangular matrix, then the diagonal 00:24:02.100 --> 00:24:04.770 is all I have to work with. 00:24:04.770 --> 00:24:06.550 And that's, that's telling us then. 00:24:06.550 --> 00:24:15.120 We now have the way that MATLAB, any reasonable software, 00:24:15.120 --> 00:24:17.250 would compute a determinant. 00:24:17.250 --> 00:24:20.990 If I have a matrix of size a hundred, 00:24:20.990 --> 00:24:24.817 the way I would actually compute its determinant would be 00:24:24.817 --> 00:24:27.400 elimination, make it triangular, multiply the pivots together, 00:24:27.400 --> 00:24:29.940 but it -- would it be possible t- 00:24:29.940 --> 00:24:33.110 to produce the same matrix the product of the pivots, 00:24:33.110 --> 00:24:34.080 the product of pivots. 00:24:34.080 --> 00:24:39.080 Now, there's always in determinants a plus or minus 00:24:39.080 --> 00:24:44.630 and cross every T in that proof, but that's really 00:24:44.630 --> 00:24:46.850 the sign to remember. 00:24:46.850 --> 00:24:51.840 Where -- where does that come into this rule? 00:24:51.840 --> 00:24:54.500 Could it be, could the determinant 00:24:54.500 --> 00:24:58.210 be minus the product of the pivots? 00:24:58.210 --> 00:25:00.740 The determinant is d1, d2, to dn. 00:25:00.740 --> 00:25:02.380 No doubt about that. 00:25:02.380 --> 00:25:05.370 But to get to this triangular form, 00:25:05.370 --> 00:25:12.660 we may have had to do a row exchange, so, so this -- 00:25:12.660 --> 00:25:15.890 it's the product of the pivots if there were no row exchanges. 00:25:15.890 --> 00:25:19.370 If, if SLU code, the simple LU code, 00:25:19.370 --> 00:25:21.770 the square LU went right through. 00:25:21.770 --> 00:25:24.070 If we had to do some row exchanges, 00:25:24.070 --> 00:25:26.840 then we've got to watch plus or minus. 00:25:26.840 --> 00:25:31.680 OK, but though -- this law is simply that. 00:25:31.680 --> 00:25:33.180 OK, how do I prove that? 00:25:37.420 --> 00:25:42.730 Let's see, let me suppose that d's are not zeroes. 00:25:42.730 --> 00:25:44.620 The pivots are not zeroes. 00:25:44.620 --> 00:25:50.550 And tell me, how do I show that none of this upper stuff 00:25:50.550 --> 00:25:53.840 makes any difference? 00:25:53.840 --> 00:25:56.550 How do I get zeroes there? 00:25:56.550 --> 00:25:58.630 By elimination, right? 00:25:58.630 --> 00:26:01.940 I just multiply this row by the right number, 00:26:01.940 --> 00:26:05.950 subtract from that row, kills that. 00:26:05.950 --> 00:26:09.030 I multiply this row by the right number, kills that, 00:26:09.030 --> 00:26:11.010 by the right number, kills that. 00:26:11.010 --> 00:26:15.850 Now, you kill every one of these off-diagonal terms, no problem 00:26:15.850 --> 00:26:17.450 and I'm just left with the diagonal. 00:26:20.690 --> 00:26:22.940 Well, strictly speaking, I still have 00:26:22.940 --> 00:26:26.680 to figure out why is, for a diagonal matrix 00:26:26.680 --> 00:26:28.745 now, why is that the right determinant? 00:26:31.540 --> 00:26:37.260 I mean, we sure hope it is, but why? 00:26:37.260 --> 00:26:41.270 I have to go back to properties one, two, three. 00:26:41.270 --> 00:26:46.710 Why is -- now that the matrix is suddenly diagonal, 00:26:46.710 --> 00:26:48.950 how do I know that the determinant is just 00:26:48.950 --> 00:26:51.070 a product of That's my proof, really, 00:26:51.070 --> 00:26:53.200 that once I've got those diagonal entries? 00:26:53.200 --> 00:26:55.030 Well, what I going to use? 00:26:55.030 --> 00:26:57.760 I'm going to use property 3A, is that right? 00:26:57.760 --> 00:27:01.250 I'll factor this, I'll factor this, 00:27:01.250 --> 00:27:05.380 I'll factor that d1 out and have one and have 00:27:05.380 --> 00:27:07.280 the first row will be that. 00:27:07.280 --> 00:27:09.710 And then I'll factor out the d2, shall I shall 00:27:09.710 --> 00:27:13.160 I put the d2 here, and the second row 00:27:13.160 --> 00:27:16.110 will look like that, and so on. 00:27:16.110 --> 00:27:20.150 So I've factored out all the d's and what I left with? 00:27:20.150 --> 00:27:21.370 The identity. 00:27:21.370 --> 00:27:24.970 And what rule do I finally get to use? 00:27:24.970 --> 00:27:26.030 Rule one. 00:27:26.030 --> 00:27:29.900 Finally, this is the point where rule one finally chips 00:27:29.900 --> 00:27:33.480 in and says that this determinant is one, 00:27:33.480 --> 00:27:35.520 so it's the product of the d's. 00:27:35.520 --> 00:27:40.950 So this was rules five, to do elimination, 00:27:40.950 --> 00:27:48.680 3A to factor out the D's, and, and our best friend, rule one. 00:27:48.680 --> 00:27:52.240 And possibly rule two, the exchanges 00:27:52.240 --> 00:27:53.710 may have been needed also. 00:27:53.710 --> 00:27:54.210 OK. 00:27:56.940 --> 00:28:01.350 Now I guess I have to consider also the case if some d is 00:28:01.350 --> 00:28:06.680 zero, because I was assuming I could use the d's to clean out 00:28:06.680 --> 00:28:08.490 and make a diagonal, but what if -- 00:28:08.490 --> 00:28:13.390 what if one of those diagonal entries is zero? 00:28:13.390 --> 00:28:16.380 Well, then with elimination we know 00:28:16.380 --> 00:28:21.440 that we can get a row of zeroes, and for a row 00:28:21.440 --> 00:28:25.200 of zeroes I'm using rule six, the determinant is zero, 00:28:25.200 --> 00:28:26.020 and that's right. 00:28:26.020 --> 00:28:28.690 So I can check the singular case. 00:28:28.690 --> 00:28:36.250 In fact, I can now get to the key point that determinant of A 00:28:36.250 --> 00:28:44.920 is zero, exactly when, exactly when A is singular. 00:28:48.790 --> 00:28:52.880 And otherwise is not singular, so that the determinant 00:28:52.880 --> 00:28:58.800 is a fair test for invertibility or non-invertibility. 00:28:58.800 --> 00:29:03.610 So, I must be close to that because I can take any matrix 00:29:03.610 --> 00:29:05.290 and get there. 00:29:05.290 --> 00:29:06.870 Do I -- did I have anything to say? 00:29:09.570 --> 00:29:12.970 The, the proofs, it starts by saying by elimination 00:29:12.970 --> 00:29:14.680 go from A to U. 00:29:14.680 --> 00:29:15.290 Oh, yeah. 00:29:15.290 --> 00:29:17.940 Actually looks to me like I don't -- 00:29:17.940 --> 00:29:22.450 haven't said anything brand-new here, that, that really, 00:29:22.450 --> 00:29:28.950 I've got this, because let's just remember the 00:29:28.950 --> 00:29:37.980 By elimination, I can go from the original A to reason. 00:29:37.980 --> 00:29:43.440 Well, OK, now suppose the matrix is U. singular. 00:29:43.440 --> 00:29:46.630 If the matrix is singular, what happens? 00:29:46.630 --> 00:29:50.480 Then by elimination I get a row of zeroes 00:29:50.480 --> 00:29:55.240 and therefore the determinant is zero. 00:29:55.240 --> 00:29:59.170 And if the matrix is not singular, I don't get zero, 00:29:59.170 --> 00:30:02.220 so maybe -- do you want me to put this, like, in two parts? 00:30:02.220 --> 00:30:10.100 The determinant of A is not zero when A is invertible. 00:30:15.480 --> 00:30:18.750 Because I've already -- 00:30:18.750 --> 00:30:23.240 in chapter two we figured out when is the matrix invertible. 00:30:23.240 --> 00:30:27.770 It's invertible when elimination produces a full set of pivots 00:30:27.770 --> 00:30:31.420 and now, and we now, we know the determinant is the product 00:30:31.420 --> 00:30:34.160 of those non-zero numbers. 00:30:34.160 --> 00:30:36.410 So those are the two cases. 00:30:36.410 --> 00:30:39.180 If it's singular, I go to a row of zeroes. 00:30:43.370 --> 00:30:49.680 If it's invertible, I go to U and then to the diagonal D, 00:30:49.680 --> 00:30:57.150 and then which -- and then to d1, d2, up to dn. 00:30:57.150 --> 00:31:00.050 As the formula -- so we have a formula now. 00:31:02.700 --> 00:31:05.110 We have a formula for the determinant 00:31:05.110 --> 00:31:08.520 and it's actually a very much more practical 00:31:08.520 --> 00:31:12.100 formula than the but they didn't matter anyway. ad-bc formula. 00:31:12.100 --> 00:31:18.050 Is it correct, maybe you should just -- let's just check that. 00:31:18.050 --> 00:31:18.970 Two-by-two. 00:31:18.970 --> 00:31:23.850 What are the pivots of a two-by-two matrix? 00:31:23.850 --> 00:31:28.270 What does elimination do with a two-by-two matrix? 00:31:28.270 --> 00:31:30.370 It -- there's the first pivot, fine. 00:31:30.370 --> 00:31:33.880 What's the second pivot? 00:31:33.880 --> 00:31:38.970 We subtract, so I'm putting it in this upper triangular form. 00:31:38.970 --> 00:31:44.270 What do I -- my multiplier is c over a, right? 00:31:44.270 --> 00:31:46.790 I multiply that row by c over a and I 00:31:46.790 --> 00:31:50.590 subtract to get that zero, and here I 00:31:50.590 --> 00:31:54.470 have d minus c over a times b. 00:31:58.500 --> 00:32:01.870 That's the elimination on a two-by-two. 00:32:01.870 --> 00:32:07.440 So I've finally discovered that the determinant of this matrix 00:32:07.440 --> 00:32:07.940 -- 00:32:07.940 --> 00:32:12.240 I've got it from the properties, not by knowing the answer 00:32:12.240 --> 00:32:18.830 from last year, that the determinant of this two-by-two 00:32:18.830 --> 00:32:22.730 is the product of A times that, and of course 00:32:22.730 --> 00:32:26.180 when I multiply A by that, the product of that and that 00:32:26.180 --> 00:32:30.485 is ad minus, the a is canceled, 00:32:30.485 --> 00:32:30.985 bc. 00:32:34.270 --> 00:32:36.320 So that's great, provided a isn't zero. 00:32:36.320 --> 00:32:38.960 because all math professors watching this will be waiting 00:32:38.960 --> 00:32:42.190 If a was zero, that step wasn't allowed, with seven row 00:32:42.190 --> 00:32:45.130 exchanges and with ten row exchanges? zero wasn't a pivot. 00:32:45.130 --> 00:32:46.600 So that's what I -- 00:32:46.600 --> 00:32:48.850 I've covered all the bases. 00:32:48.850 --> 00:32:53.090 I have to -- if a is zero, then I have to do the exchange, 00:32:53.090 --> 00:32:56.570 and if the exchange doesn't work, it's because a is proof. 00:32:56.570 --> 00:32:57.730 singular. 00:32:57.730 --> 00:33:03.130 OK, those are -- 00:33:03.130 --> 00:33:06.520 those are the direct properties of the determinant. 00:33:06.520 --> 00:33:11.120 And now, finally, I've got two more, nine and ten. 00:33:11.120 --> 00:33:13.920 And that's -- 00:33:13.920 --> 00:33:15.030 I think you can... 00:33:15.030 --> 00:33:25.210 Like, the ones we've got here are 00:33:25.210 --> 00:33:28.870 totally connected with our elimination process 00:33:28.870 --> 00:33:35.340 and whether pivots are available and whether we 00:33:35.340 --> 00:33:36.820 get a row of zeroes. 00:33:36.820 --> 00:33:40.360 I think all that you can swallow in one shot. 00:33:40.360 --> 00:33:43.730 Let me tell you properties nine and ten. 00:33:46.990 --> 00:33:50.000 They're quick to write down. 00:33:50.000 --> 00:33:53.940 They're very, very useful. 00:33:53.940 --> 00:33:56.260 So I'll just write them down and use them. 00:33:56.260 --> 00:34:01.410 Property nine says that the determinant of a product -- 00:34:01.410 --> 00:34:05.610 if I That's the, like, concrete proof that, 00:34:05.610 --> 00:34:07.180 multiply two matrices. 00:34:07.180 --> 00:34:11.909 So if I multiply two matrices, A and B, 00:34:11.909 --> 00:34:14.050 that the determinant of the product 00:34:14.050 --> 00:34:26.730 is determinant of A times determinant of B, and for me 00:34:26.730 --> 00:34:31.750 that one is like, that's a very valuable property, 00:34:31.750 --> 00:34:34.449 and it's sort of like partly coming out of the blue, 00:34:34.449 --> 00:34:37.050 because we haven't been multiplying matrices 00:34:37.050 --> 00:34:41.159 and here suddenly this determinant 00:34:41.159 --> 00:34:44.590 has this multiplying property. 00:34:44.590 --> 00:34:46.750 Remember, it didn't have the linear property, 00:34:46.750 --> 00:34:48.810 it didn't have the adding property. 00:34:48.810 --> 00:34:52.389 The determinant of A plus B is not 00:34:52.389 --> 00:34:57.210 the sum of the determinants, but the determinant of A times B 00:34:57.210 --> 00:35:01.220 is the product, is the product of the determinants. 00:35:01.220 --> 00:35:06.955 OK, so for example, what's the determinant of A inverse? 00:35:12.560 --> 00:35:14.240 Using that property nine. 00:35:19.360 --> 00:35:21.230 Let me, let me put that under here 00:35:21.230 --> 00:35:27.800 because the camera is happier if it can focus on both at once. 00:35:27.800 --> 00:35:29.140 So let me put it underneath. 00:35:29.140 --> 00:35:34.510 The determinant of A inverse, because property ten 00:35:34.510 --> 00:35:40.420 will come in that space. 00:35:40.420 --> 00:35:44.530 What does this tell me about A inverse, its determinant? 00:35:44.530 --> 00:35:47.730 OK, well, what do I know about A inverse? 00:35:47.730 --> 00:35:54.980 I know that A inverse times A is odd. 00:35:54.980 --> 00:35:55.960 So what I going to do? 00:35:59.200 --> 00:36:02.450 I'm going to take determinants of both sides. 00:36:02.450 --> 00:36:06.290 The determinant of I is one, and what's 00:36:06.290 --> 00:36:10.390 the determinant of A inverse A? 00:36:10.390 --> 00:36:13.670 That's a product of two matrices, A and B. 00:36:13.670 --> 00:36:15.510 So it's the product of the determinant, 00:36:15.510 --> 00:36:16.830 so what I learning? 00:36:16.830 --> 00:36:18.840 I'm learning that the determinant 00:36:18.840 --> 00:36:24.070 of A inverse times the determinant of A 00:36:24.070 --> 00:36:27.800 is the determinant of I, that's this one. 00:36:27.800 --> 00:36:31.870 Again, I happily use property one. 00:36:31.870 --> 00:36:35.950 OK, so that tells me that the determinant of A inverse 00:36:35.950 --> 00:36:37.270 is one over. 00:36:37.270 --> 00:36:40.000 Here's my -- here's my conclusion -- 00:36:40.000 --> 00:36:53.910 is one over the determinant of A. 00:36:53.910 --> 00:36:55.810 I guess that that -- 00:36:55.810 --> 00:36:59.620 I, I always try to think, well, do we know some cases of that? 00:36:59.620 --> 00:37:04.450 Of course, we know it's right already if A is diagonal. 00:37:04.450 --> 00:37:09.000 If A is a diagonal matrix, then its determinant 00:37:09.000 --> 00:37:10.480 is just a product of those numbers. 00:37:10.480 --> 00:37:14.890 So if A is, for example, two-three, 00:37:14.890 --> 00:37:20.740 then we know that A-inverse is one-half one-third, 00:37:20.740 --> 00:37:26.440 and sure enough, that has determinant six, 00:37:26.440 --> 00:37:29.430 and that has determinant one-sixth. 00:37:29.430 --> 00:37:32.360 And our rule checks. 00:37:32.360 --> 00:37:39.490 So somehow this proof, this property has to -- 00:37:39.490 --> 00:37:41.960 somehow the proof of that property -- 00:37:41.960 --> 00:37:45.950 if we can boil it down to diagonal matrices then we can 00:37:45.950 --> 00:37:49.140 read it off, whether it's A and A-inverse, 00:37:49.140 --> 00:37:52.660 or two different diagonal matrices A and B. 00:37:52.660 --> 00:37:54.430 For diagonal -- so what I saying? 00:37:54.430 --> 00:37:59.050 I'm saying for a diagonal matrices, check. 00:37:59.050 --> 00:38:02.380 But we'd have to do elimination steps, 00:38:02.380 --> 00:38:08.510 we'd have to patiently do the, the, argument 00:38:08.510 --> 00:38:11.810 if we want to use these previous properties to get it 00:38:11.810 --> 00:38:12.980 for other matrices. 00:38:12.980 --> 00:38:18.094 And it also tells me -- what, just let's, see what else 00:38:18.094 --> 00:38:18.760 it's telling me. 00:38:18.760 --> 00:38:21.900 What's the determinant of, of A-squared? 00:38:21.900 --> 00:38:26.750 If I take a matrix and square it? 00:38:26.750 --> 00:38:30.460 Then the determinant just got squared. 00:38:30.460 --> 00:38:31.140 Right? 00:38:31.140 --> 00:38:34.180 The determinant of A-squared is the determinant 00:38:34.180 --> 00:38:35.864 of A times the determinant of A. 00:38:35.864 --> 00:38:38.030 So if I square the matrix, I square the determinant. 00:38:38.030 --> 00:38:43.180 If I double the matrix, what do I do to the non-zeroes flipped 00:38:43.180 --> 00:38:46.350 to the other side of the diagonal, determinant? 00:38:46.350 --> 00:38:47.930 Think about that one. 00:38:47.930 --> 00:38:52.690 If I double the matrix, what -- so the determinant of A, 00:38:52.690 --> 00:38:56.150 since I'm writing down, like, facts that follow, 00:38:56.150 --> 00:39:02.500 the determinant of A-squared is the determinant of A, 00:39:02.500 --> 00:39:04.740 all squared. 00:39:04.740 --> 00:39:09.580 The determinant of 2A is what? 00:39:12.250 --> 00:39:16.380 That's A plus A. 00:39:16.380 --> 00:39:19.910 But wait, er, I don't want the answer 00:39:19.910 --> 00:39:22.410 to determinant of A here. 00:39:22.410 --> 00:39:23.150 That's wrong. 00:39:23.150 --> 00:39:25.860 It's not two determinant of A, What is it? 00:39:25.860 --> 00:39:28.880 OK, one more coming, which I I have to make, 00:39:28.880 --> 00:39:32.190 what's the number that I have to multiply determinant of A 00:39:32.190 --> 00:39:34.900 by if I double the whole matrix, if I 00:39:34.900 --> 00:39:36.710 double every entry in the matrix? 00:39:36.710 --> 00:39:38.220 What happens to the determinant? 00:39:38.220 --> 00:39:41.230 If that were possible, that would be a bad thing, 00:39:41.230 --> 00:39:43.640 Supposed it's an n-by-n matrix. that gets -- 00:39:43.640 --> 00:39:44.850 get down to triangular 00:39:44.850 --> 00:39:46.690 Two to the n, right. 00:39:46.690 --> 00:39:48.120 Two to the nth. 00:39:48.120 --> 00:39:50.650 Now, why is it two to the nth, and not just two? 00:39:54.700 --> 00:39:58.070 So why is it two to the nth? 00:39:58.070 --> 00:40:02.230 Because I'm factoring out two from every row. 00:40:02.230 --> 00:40:05.610 There's a factor -- this has a factor two in every row, 00:40:05.610 --> 00:40:08.640 so I can factor two out of the first row. 00:40:08.640 --> 00:40:12.120 I factor a different two out of the second row, a different two 00:40:12.120 --> 00:40:15.640 out of the nth row, so I've got all those twos coming out. 00:40:15.640 --> 00:40:20.290 So it's like volume, really, and that's 00:40:20.290 --> 00:40:23.570 one of the great applications of determinants. 00:40:23.570 --> 00:40:30.860 If I -- if I have a box and I double all the sides, 00:40:30.860 --> 00:40:35.800 I multiply the volume by two to the nth. 00:40:35.800 --> 00:40:38.440 If it's a box in three dimensions, 00:40:38.440 --> 00:40:40.970 I multiply the volume by 8. 00:40:43.710 --> 00:40:47.190 So this is rule 3A here. 00:40:47.190 --> 00:40:49.480 This is rule nine. 00:40:49.480 --> 00:40:55.100 And notice the way this rule sort of checks out with 00:40:55.100 --> 00:41:02.550 the singular/non-singular stuff, that if A is invertible, 00:41:02.550 --> 00:41:05.650 what does that mean about its determinant? 00:41:05.650 --> 00:41:08.020 It's not zero, and therefore this makes sense. 00:41:10.770 --> 00:41:12.940 The case when determinant of A is 00:41:12.940 --> 00:41:19.870 zero, that's the case where my formula doesn't work anymore. 00:41:19.870 --> 00:41:24.030 If determinant of A is zero, this is ridiculous, 00:41:24.030 --> 00:41:25.800 and that's ridiculous. 00:41:25.800 --> 00:41:31.300 A-inverse doesn't exist, and one over zero doesn't make sense. 00:41:31.300 --> 00:41:36.090 So don't miss this property. 00:41:36.090 --> 00:41:38.560 It's sort of, like, amazing that it can... 00:41:38.560 --> 00:41:44.660 And the tenth property is equally simple to state, 00:41:44.660 --> 00:41:47.720 that the determinant of A transposed 00:41:47.720 --> 00:41:57.010 equals the determinant of A. 00:41:57.010 --> 00:42:03.030 And of course, let's just check it on the ab cd guy. 00:42:03.030 --> 00:42:07.075 We could check that sure enough, that's ab cd, it works. 00:42:09.710 --> 00:42:14.350 It's ad - bc, it's ad - bc, sure enough. 00:42:14.350 --> 00:42:19.140 So that transposing did not change the determinant. 00:42:19.140 --> 00:42:24.790 But what it does change is -- 00:42:24.790 --> 00:42:28.400 well, what it does is it lists, so all -- 00:42:28.400 --> 00:42:31.340 I've been working with rows. 00:42:31.340 --> 00:42:35.690 If a row is all zeroes, the determinant is zero. 00:42:35.690 --> 00:42:40.360 But now, with rule ten, I know what to do 00:42:40.360 --> 00:42:42.750 is a column is all zero. 00:42:42.750 --> 00:42:46.550 If a column is all zero, what's the determinant? 00:42:46.550 --> 00:42:48.260 Zero, again. 00:42:48.260 --> 00:42:53.250 So, like all those properties about rows, exchanging two rows 00:42:53.250 --> 00:42:55.080 reverses the sign. 00:42:55.080 --> 00:42:58.210 Now, exchanging two columns reverses 00:42:58.210 --> 00:43:00.990 the sign, because I can always, if I 00:43:00.990 --> 00:43:03.690 want to see why, I can transpose, 00:43:03.690 --> 00:43:08.680 those columns become rows, I do the exchange, I transpose back. 00:43:08.680 --> 00:43:11.760 And I've done a column operation. 00:43:11.760 --> 00:43:17.530 So, in, in conclusion, there was nothing special about row one, 00:43:17.530 --> 00:43:20.610 'cause I could exchange rows, and now there's 00:43:20.610 --> 00:43:25.060 nothing special about rows that isn't equally true for columns 00:43:25.060 --> 00:43:26.980 because this is the same. 00:43:26.980 --> 00:43:27.580 OK. 00:43:27.580 --> 00:43:32.080 And again, maybe I won't -- 00:43:32.080 --> 00:43:33.320 oh, let's see. 00:43:33.320 --> 00:43:33.820 Do we...? 00:43:33.820 --> 00:43:37.930 Maybe it's worth seeing a quick proof of this number ten, 00:43:37.930 --> 00:43:44.620 quick, quick, er, proof of number ten. 00:43:44.620 --> 00:43:48.970 Er, let me take the -- this is number ten. 00:43:48.970 --> 00:43:51.380 A transposed equals A. 00:43:51.380 --> 00:43:56.480 Determinate of A transposed equals determinate of A. 00:43:56.480 --> 00:43:58.110 That's what I should have said. 00:43:58.110 --> 00:43:59.280 OK. 00:43:59.280 --> 00:44:07.820 So, let's just, er. 00:44:07.820 --> 00:44:11.450 A typical matrix A, if I use elimination, 00:44:11.450 --> 00:44:16.100 this factors into LU. 00:44:16.100 --> 00:44:21.710 And the transpose is U transpose, l transpose. 00:44:25.430 --> 00:44:26.370 Er... let me. 00:44:29.150 --> 00:44:36.870 So this is proof, this is proof number ten, using -- 00:44:36.870 --> 00:44:39.820 well, I don't know which ones I'll use, so I'll put 00:44:39.820 --> 00:44:42.840 'em all in, one to nine. 00:44:42.840 --> 00:44:43.650 OK. 00:44:43.650 --> 00:44:47.400 I'm going to prove number ten by using one to nine. 00:44:47.400 --> 00:44:50.910 I won't cover every case, but I'll cover almost every 00:44:50.910 --> 00:44:51.550 case. 00:44:51.550 --> 00:44:55.400 So in almost every case, A can factor into LU, 00:44:55.400 --> 00:44:57.710 and A transposed can factor into that. 00:44:57.710 --> 00:45:00.070 Now, what do I do next? 00:45:00.070 --> 00:45:03.910 So I want to prove that these are the same. 00:45:03.910 --> 00:45:06.610 I see a product here. 00:45:06.610 --> 00:45:09.560 So I use rule nine. 00:45:09.560 --> 00:45:14.860 So, now what I want to prove is, so now I know that this is LU, 00:45:14.860 --> 00:45:19.810 this is U transposed and l transposed. 00:45:19.810 --> 00:45:24.010 Now, just for a practice, what are all those determinants? 00:45:24.010 --> 00:45:28.710 So this is, this is, this is prove this, prove this, prove 00:45:28.710 --> 00:45:32.460 this, and now I'm ready to do it. 00:45:32.460 --> 00:45:34.460 What's the determinant of l? 00:45:34.460 --> 00:45:40.240 You remember what l is, it's this lower triangular matrix 00:45:40.240 --> 00:45:43.085 with ones on the diagonals. 00:45:43.085 --> 00:45:44.710 So what is the determinant of that guy? 00:45:44.710 --> 00:45:45.210 I- It's one. 00:45:48.640 --> 00:45:53.000 Any time I have this triangular matrix, 00:45:53.000 --> 00:46:00.390 I can get rid of all the non-zeroes, 00:46:00.390 --> 00:46:07.920 down to the diagonal case. 00:46:07.920 --> 00:46:12.750 The determinate of l is one. 00:46:12.750 --> 00:46:21.810 How about the determinant of l transposed? 00:46:21.810 --> 00:46:25.410 That's triangular also, right? 00:46:25.410 --> 00:46:28.050 Still got those ones on the diagonal, 00:46:28.050 --> 00:46:59.050 it's just the matrices and then get down to diagonal matrices. 00:46:59.050 --> 00:47:00.600 right? 00:47:00.600 --> 00:47:09.290 If If I could -- why would it be bad? 00:47:09.290 --> 00:47:11.800 My whole lecture would die, right? 00:47:11.800 --> 00:47:37.360 Because rule two said that if you do seven row exchanges, 00:47:37.360 --> 00:47:47.700 then the sign of the determinant reverses. 00:47:47.700 --> 00:47:56.820 But if you do ten row exchanges, the sign of the determinant 00:47:56.820 --> 00:48:02.520 stays the same, because minus one ten times is plus one. 00:48:02.520 --> 00:48:16.630 OK, so there's a hidden fact here, that every -- 00:48:16.630 --> 00:48:19.150 like, every permutation, the permutations 00:48:19.150 --> 00:48:23.310 are either odd or even. 00:48:23.310 --> 00:48:26.360 I could get the permutation with seven row exchanges, 00:48:26.360 --> 00:48:27.770 then I could probably get it with 00:48:27.770 --> 00:48:31.430 twenty-one, or twenty-three, or a hundred and one, 00:48:31.430 --> 00:48:33.430 if it's an odd one. 00:48:33.430 --> 00:48:35.980 Or an even number of permutations, so, 00:48:35.980 --> 00:48:39.070 but that's the key fact that just takes 00:48:39.070 --> 00:48:43.760 another little algebraic trick to see, 00:48:43.760 --> 00:48:45.947 and that means that the determinant is well-defined 00:48:45.947 --> 00:48:47.530 by properties one, two, three and it's 00:48:47.530 --> 00:48:47.580 got properties four to ten. 00:48:47.580 --> 00:48:47.660 OK, that's today and I'll try to get 00:48:47.660 --> 00:48:47.890 the homework for next Wednesday onto the web this afternoon. 00:48:47.890 --> 00:48:49.440 Thanks.