WEBVTT 00:00:00.000 --> 00:00:10.250 OK, this is lecture twenty. 00:00:10.250 --> 00:00:15.680 And this is the final lecture on determinants. 00:00:15.680 --> 00:00:18.390 And it's about the applications. 00:00:18.390 --> 00:00:22.730 So we worked hard in the last two lectures 00:00:22.730 --> 00:00:25.900 to get a formula for the determinant 00:00:25.900 --> 00:00:27.980 and the properties of the determinant. 00:00:27.980 --> 00:00:35.250 Now to use the determinant and, and always this determinant 00:00:35.250 --> 00:00:40.710 packs all this information into a single number. 00:00:40.710 --> 00:00:45.800 And that number can give us formulas 00:00:45.800 --> 00:00:50.510 for all sorts of, things that we've been calculating already 00:00:50.510 --> 00:00:51.550 without formulas. 00:00:51.550 --> 00:00:54.110 Now what was A inverse? 00:00:54.110 --> 00:00:57.690 So, so I'm beginning with the formula for A inverse. 00:01:00.400 --> 00:01:02.450 Two, two by two formula we know, 00:01:02.450 --> 00:01:03.070 right? 00:01:03.070 --> 00:01:09.700 The two by two formula for A inverse, the inverse of a b c d 00:01:09.700 --> 00:01:22.160 inverse is one over the determinant times d a -b -c. 00:01:25.630 --> 00:01:31.140 Somehow I want to see what's going on for three by three 00:01:31.140 --> 00:01:33.140 and n by n. 00:01:33.140 --> 00:01:37.200 And actually maybe you can see what's going on from this two 00:01:37.200 --> 00:01:38.870 by two case. 00:01:38.870 --> 00:01:41.060 So there's a formula for the inverse, 00:01:41.060 --> 00:01:44.390 and what did I divide by? 00:01:44.390 --> 00:01:45.880 The determinant. 00:01:45.880 --> 00:01:50.220 So what I'm looking for is a formula 00:01:50.220 --> 00:01:53.590 where it has one over the determinant 00:01:53.590 --> 00:01:56.690 and, and you remember why that makes good sense, 00:01:56.690 --> 00:02:02.150 because then that's perfect as long as the determinant isn't 00:02:02.150 --> 00:02:06.330 zero, and that's exactly when there is an inverse. 00:02:06.330 --> 00:02:11.640 But now I have to ask can we recognize any of this stuff? 00:02:11.640 --> 00:02:16.410 Do you recognize what that number d is from the past? 00:02:16.410 --> 00:02:19.730 From last, from the last lecture? 00:02:19.730 --> 00:02:24.200 My hint is think cofactors. 00:02:24.200 --> 00:02:28.480 Because my formula is going to be, my formula for the inverse 00:02:28.480 --> 00:02:30.760 is going to be one over the determinant 00:02:30.760 --> 00:02:33.660 times a matrix of cofactors. 00:02:33.660 --> 00:02:35.290 So you remember that D? 00:02:35.290 --> 00:02:37.700 What's that the cofactor of? 00:02:37.700 --> 00:02:39.230 Remember cofactors? 00:02:39.230 --> 00:02:41.530 If -- that's the one one cofactor, 00:02:41.530 --> 00:02:47.570 because if I strike out row and column one, I'm left with d. 00:02:47.570 --> 00:02:51.080 And what's minus b? 00:02:51.080 --> 00:02:51.930 OK. 00:02:51.930 --> 00:02:53.780 Which cofactor is that one? 00:02:53.780 --> 00:03:00.900 Oh, minus b is the cofactor of c, right? 00:03:00.900 --> 00:03:04.670 If I strike out the c, I'm left with a b there. 00:03:04.670 --> 00:03:07.400 And why the minus sign? 00:03:07.400 --> 00:03:12.310 Because this c was in a two one position, and two plus one is 00:03:12.310 --> 00:03:12.910 odd. 00:03:12.910 --> 00:03:17.000 So there was a minus went into the cofactor, and that's it. 00:03:17.000 --> 00:03:17.780 OK. 00:03:17.780 --> 00:03:21.710 I'll write down next what my formula is. 00:03:21.710 --> 00:03:26.900 Here's the big formula for the A -- for A inverse. 00:03:26.900 --> 00:03:35.800 It's one over the determinant of A and then some matrix. 00:03:35.800 --> 00:03:40.280 And that matrix is the matrix of cofactors, 00:03:40.280 --> 00:03:41.370 c. 00:03:41.370 --> 00:03:45.330 Only one thing, it turns -- you'll see, 00:03:45.330 --> 00:03:47.920 I have to, I transpose. 00:03:47.920 --> 00:03:50.760 So this is the matrix of cofactors, the -- 00:03:50.760 --> 00:03:52.090 what I'll just -- 00:03:52.090 --> 00:03:55.710 but why don't we just call it the cofactor matrix. 00:03:55.710 --> 00:04:02.400 So the one one entry of, of c is the cof- is the one one 00:04:02.400 --> 00:04:05.840 cofactor, the thing that we get by throwing away row and column 00:04:05.840 --> 00:04:06.340 one. 00:04:06.340 --> 00:04:09.770 It's the d. 00:04:09.770 --> 00:04:15.420 And, because of the transpose, what I see up here 00:04:15.420 --> 00:04:19.140 is the cofactor of this guy down here, right? 00:04:19.140 --> 00:04:22.450 That's where the transpose came in. 00:04:22.450 --> 00:04:27.420 What I see here, this is the cofactor not of this one, 00:04:27.420 --> 00:04:28.690 because I've transposed. 00:04:28.690 --> 00:04:30.920 This is the cofactor of the b. 00:04:30.920 --> 00:04:35.350 When I throw away the b, the b row and the b column, 00:04:35.350 --> 00:04:38.240 I'm left with c, and then I have that minus sign 00:04:38.240 --> 00:04:38.940 again. 00:04:38.940 --> 00:04:44.580 And of course the two two entry is the cofactor 00:04:44.580 --> 00:04:48.660 of d, and that's this a. 00:04:48.660 --> 00:04:49.930 So there's the formula. 00:04:49.930 --> 00:04:50.790 OK. 00:04:50.790 --> 00:04:53.080 But we got to think why. 00:04:53.080 --> 00:04:56.680 I mean, it worked in this two by two case, 00:04:56.680 --> 00:04:59.590 but a lot of other formulas would have worked just as well. 00:04:59.590 --> 00:05:02.200 We, we have to see why that's true. 00:05:02.200 --> 00:05:05.790 In other words, why is it -- 00:05:05.790 --> 00:05:09.530 so this is what I aim to find. 00:05:09.530 --> 00:05:12.570 And, and let's just sort of look to see what is that telling 00:05:12.570 --> 00:05:13.210 us. 00:05:13.210 --> 00:05:17.400 That tells us that the -- the expression for A inverse -- 00:05:17.400 --> 00:05:20.670 let's look at a three by three. 00:05:20.670 --> 00:05:27.240 Can I just write down a a b c d e f g h i? 00:05:27.240 --> 00:05:30.780 And I'm looking for its inverse. 00:05:30.780 --> 00:05:33.340 And what kind of a formula -- do I see there? 00:05:33.340 --> 00:05:34.720 I mean, what -- 00:05:34.720 --> 00:05:38.630 the determinant is a bunch of products of three factors, 00:05:38.630 --> 00:05:39.370 right? 00:05:39.370 --> 00:05:43.190 The determinant of this matrix'll involve a e i, 00:05:43.190 --> 00:05:49.160 and b f times g, and c times d times h, and minus c 00:05:49.160 --> 00:05:50.900 e g, and so on. 00:05:50.900 --> 00:05:54.810 So things with three factors go in here. 00:05:54.810 --> 00:06:03.140 Things with how many factors do things in the cofactor matrix 00:06:03.140 --> 00:06:03.640 have? 00:06:03.640 --> 00:06:06.030 What's a typical cofactor? 00:06:06.030 --> 00:06:09.000 What's the cofactor of a? 00:06:09.000 --> 00:06:12.800 The cofactor of a, the one one entry up here in the inverse 00:06:12.800 --> 00:06:14.700 is? 00:06:14.700 --> 00:06:17.690 I throw away the row and column containing a 00:06:17.690 --> 00:06:19.740 and I take the determinant of what's left, 00:06:19.740 --> 00:06:21.230 that's the cofactor. 00:06:21.230 --> 00:06:24.390 And that's e i minus f h. 00:06:24.390 --> 00:06:25.890 Products of two things. 00:06:25.890 --> 00:06:29.560 Now, I'm just making the observation 00:06:29.560 --> 00:06:36.900 that the determinant of A involves products of n entries. 00:06:41.200 --> 00:06:51.310 And the cofactor matrix involves products of n minus 1 entries. 00:06:51.310 --> 00:06:56.620 And, like, we never noticed any of this stuff 00:06:56.620 --> 00:06:58.780 when we were computing the inverse 00:06:58.780 --> 00:07:01.090 by the Gauss-Jordan method or whatever. 00:07:01.090 --> 00:07:02.380 You remember how we did it? 00:07:02.380 --> 00:07:06.620 We took the matrix A, we tucked the identity next to it, 00:07:06.620 --> 00:07:10.740 we did elimination till A became the identity. 00:07:10.740 --> 00:07:13.980 And then that, the identity suddenly was A inverse. 00:07:13.980 --> 00:07:16.870 Well, that was great numerically. 00:07:16.870 --> 00:07:21.300 But we never knew what was going on, basically. 00:07:21.300 --> 00:07:25.000 And now we see what the formula is 00:07:25.000 --> 00:07:27.830 in terms of letters, what's the algebra instead 00:07:27.830 --> 00:07:29.530 of the algorithm. 00:07:29.530 --> 00:07:30.140 OK. 00:07:30.140 --> 00:07:33.780 But I have to say why this is right, right? 00:07:33.780 --> 00:07:37.940 I still -- that's a pretty magic formula. 00:07:37.940 --> 00:07:40.150 Where does it come from? 00:07:40.150 --> 00:07:41.320 Well, I'll just check it. 00:07:41.320 --> 00:07:44.200 Having, having got it up there, let me -- 00:07:44.200 --> 00:07:48.510 I'll say, how can we check -- 00:07:48.510 --> 00:07:50.510 what do I want to check? 00:07:50.510 --> 00:07:55.150 I want to check that A times its inverse gives the identity. 00:07:55.150 --> 00:07:58.130 So I want, I want to check that A times this thing, 00:07:58.130 --> 00:08:02.000 A times this -- now I'm going to write in the inverse -- 00:08:02.000 --> 00:08:03.220 gives the identity. 00:08:03.220 --> 00:08:06.830 So I check that A times C transpose -- 00:08:06.830 --> 00:08:09.340 let me bring the determinant up here. 00:08:09.340 --> 00:08:14.490 Determinant of A times the identity. 00:08:14.490 --> 00:08:16.900 That's my job. 00:08:16.900 --> 00:08:24.340 That's it, that if this is true, and it is, then, then I've 00:08:24.340 --> 00:08:28.760 correctly identified A inverse as C transpose divided 00:08:28.760 --> 00:08:30.810 by the determinant. 00:08:30.810 --> 00:08:31.310 OK. 00:08:31.310 --> 00:08:34.260 But why is this true? 00:08:34.260 --> 00:08:35.220 Why is that true? 00:08:35.220 --> 00:08:38.059 Let me, let me put down what I'm doing here. 00:08:38.059 --> 00:08:40.220 I have A -- 00:08:40.220 --> 00:08:43.630 here, here's A, here's a11 -- 00:08:43.630 --> 00:08:48.150 I'm doing this multiplication -- along to a1n. 00:08:48.150 --> 00:08:53.210 And then down in this last row will be an an1 along to ann. 00:08:53.210 --> 00:08:59.660 And I'm multiplying by the cofactor matrix transposed. 00:08:59.660 --> 00:09:07.820 So when I transpose, it'll be c11 c12 down to c1n. 00:09:07.820 --> 00:09:11.340 Notice usually that one coming first 00:09:11.340 --> 00:09:15.000 would mean I'm in row one, but I've transposed, 00:09:15.000 --> 00:09:18.200 so that's, those are the cofactors. 00:09:18.200 --> 00:09:21.700 This first column are the cofactors from row one. 00:09:21.700 --> 00:09:28.670 And then the last column would be the cofactors from row n. 00:09:28.670 --> 00:09:33.960 And why should that come out to be anything good? 00:09:33.960 --> 00:09:37.790 In fact, why should it come out to be as good as this? 00:09:37.790 --> 00:09:43.701 Well, you can tell me what the one one entry in the product 00:09:43.701 --> 00:09:44.200 is. 00:09:44.200 --> 00:09:47.630 This is like you're seeing the main point if you just 00:09:47.630 --> 00:09:49.750 tell me one entry. 00:09:49.750 --> 00:09:54.470 What do I get up there in the one one entry 00:09:54.470 --> 00:10:00.600 when I do this row of this row from A times 00:10:00.600 --> 00:10:04.100 this column of cofactors? 00:10:04.100 --> 00:10:06.150 What, what will I get there? 00:10:06.150 --> 00:10:08.460 Because we have seen this. 00:10:08.460 --> 00:10:10.590 I mean, we're, right, building exactly 00:10:10.590 --> 00:10:15.290 on what the last lecture reached. 00:10:15.290 --> 00:10:21.130 So this is a11 times c11, a12 times c12, a1n times c1n. 00:10:21.130 --> 00:10:25.700 What does that what does that sum up to? 00:10:25.700 --> 00:10:32.900 That's the cofactor formula for the determinant. 00:10:32.900 --> 00:10:35.240 That's the, this cofactor formula, 00:10:35.240 --> 00:10:39.110 which I wrote, which we got last time. 00:10:39.110 --> 00:10:42.700 The determinant of A is, if I use row one, let, 00:10:42.700 --> 00:10:47.450 let I equal one, then I have a11 times 00:10:47.450 --> 00:10:51.600 its cofactor, a12 times its cofactor, and so on. 00:10:51.600 --> 00:10:54.650 And that gives me the determinant. 00:10:54.650 --> 00:10:59.470 And it worked in this, in this case. 00:10:59.470 --> 00:11:06.020 This row times this thing is, sure enough, ad minus bc. 00:11:06.020 --> 00:11:08.800 But this formula says it always works. 00:11:08.800 --> 00:11:12.830 So up here in this, in this position, 00:11:12.830 --> 00:11:15.470 I'm getting determinant of A. 00:11:15.470 --> 00:11:18.130 What about in the two two position? 00:11:18.130 --> 00:11:23.460 Row two times column two there, what, what is that? 00:11:23.460 --> 00:11:26.640 That's just the cofactors, that's just row two 00:11:26.640 --> 00:11:28.960 times its cofactors. 00:11:28.960 --> 00:11:32.980 So of course I get the determinant again. 00:11:32.980 --> 00:11:36.310 And in the last here, this is the last row 00:11:36.310 --> 00:11:37.780 times its cofactors. 00:11:37.780 --> 00:11:41.540 It's exactly -- you see, we're realizing that the cofactor 00:11:41.540 --> 00:11:44.780 formula is just this sum of products, 00:11:44.780 --> 00:11:47.160 so of course we think, hey, we've got a matrix 00:11:47.160 --> 00:11:48.510 multiplication there. 00:11:48.510 --> 00:11:57.870 And we get determinant of A. 00:11:57.870 --> 00:12:00.920 But there's one more idea here, right? 00:12:00.920 --> 00:12:01.990 Great. 00:12:01.990 --> 00:12:05.410 What else, what have I not -- so I haven't got that formula 00:12:05.410 --> 00:12:10.520 completely proved yet, because I've still got to do all 00:12:10.520 --> 00:12:14.130 the off-diagonal stuff, which I want to be zero, 00:12:14.130 --> 00:12:15.120 right? 00:12:15.120 --> 00:12:17.560 I just want this to be determinant 00:12:17.560 --> 00:12:22.850 of A times the identity, and then I'm, I'm a happy person. 00:12:22.850 --> 00:12:24.860 So why should that be? 00:12:24.860 --> 00:12:30.660 Why should it be that one row times the cofactors 00:12:30.660 --> 00:12:34.920 from a different row, which become a column because I 00:12:34.920 --> 00:12:39.360 transpose, give zero? 00:12:39.360 --> 00:12:44.210 In other words, the cofactor formula gives the determinant 00:12:44.210 --> 00:12:48.440 if the row and the, and the cofactors -- you know, 00:12:48.440 --> 00:12:54.050 if the entries of A and the cofactors are for the same row. 00:12:54.050 --> 00:12:58.470 But for some reason, if I take the cofactors from the -- 00:12:58.470 --> 00:13:01.920 entries from the first row and the cofactors from the second 00:13:01.920 --> 00:13:04.290 row, for some reason I automatically 00:13:04.290 --> 00:13:05.270 get zero. 00:13:05.270 --> 00:13:08.140 And it's sort of like interesting 00:13:08.140 --> 00:13:10.540 to say, why does that happen? 00:13:10.540 --> 00:13:14.620 And can I just check that -- of course, we know it happens, 00:13:14.620 --> 00:13:16.410 in this case. 00:13:16.410 --> 00:13:19.060 Here are the numbers from row one 00:13:19.060 --> 00:13:24.600 and here are the cofactors from row two, right? 00:13:24.600 --> 00:13:26.820 Those are the numbers in row one. 00:13:26.820 --> 00:13:29.510 And th- these are the cofactors from row two, 00:13:29.510 --> 00:13:32.110 because the cofactor of c is minus b 00:13:32.110 --> 00:13:33.530 and the cofactor of d is 00:13:33.530 --> 00:13:34.030 a. 00:13:34.030 --> 00:13:40.380 And sure enough, that row times this column gives -- 00:13:40.380 --> 00:13:41.840 please say it. 00:13:41.840 --> 00:13:43.370 Zero, right. 00:13:43.370 --> 00:13:43.870 OK. 00:13:43.870 --> 00:13:47.330 So now how come? 00:13:47.330 --> 00:13:48.250 How come? 00:13:48.250 --> 00:13:51.190 Can we even see it in this two by two case? 00:13:51.190 --> 00:13:54.460 Why did -- well, I mean, I guess we, you know, 00:13:54.460 --> 00:13:57.352 in one way we certainly do see it, because it's right here. 00:13:57.352 --> 00:13:59.310 I mean, do we just do it, and then we get zero. 00:13:59.310 --> 00:14:01.870 But we want to think of some reason 00:14:01.870 --> 00:14:07.280 why the answer's zero, some reason that we can use in the n 00:14:07.280 --> 00:14:08.820 by n case. 00:14:08.820 --> 00:14:10.590 So let -- here, here is my thinking. 00:14:13.450 --> 00:14:16.310 We must be, if we're getting the answer's zero, 00:14:16.310 --> 00:14:20.330 we suspect that what we're doing somehow, 00:14:20.330 --> 00:14:24.020 we're taking the determinant of some matrix that 00:14:24.020 --> 00:14:26.660 has two equal rows. 00:14:26.660 --> 00:14:31.040 So I believe that if we multiply these by the cofactors from 00:14:31.040 --> 00:14:35.770 some other row, we're taking the determinant -- ye, 00:14:35.770 --> 00:14:38.660 what matrix are we taking the determinant of? 00:14:38.660 --> 00:14:40.380 Here it's, this is it. 00:14:40.380 --> 00:14:44.900 We're, when we do that times this, we're really taking -- 00:14:44.900 --> 00:14:48.510 can I put this in little letters down here? 00:14:48.510 --> 00:14:57.220 I'm taking -- let me look at the matrix a b a b. 00:14:57.220 --> 00:15:03.610 Let me call that matrix AS, A screwed up. 00:15:03.610 --> 00:15:04.290 OK. 00:15:04.290 --> 00:15:05.240 All right. 00:15:05.240 --> 00:15:09.260 So now that matrix is certainly singular. 00:15:09.260 --> 00:15:12.380 So if we find its determinant, we're going to get zero. 00:15:12.380 --> 00:15:16.810 But I claim that if we find its determinant by the cofactor 00:15:16.810 --> 00:15:19.960 rule, go along the first row, we would 00:15:19.960 --> 00:15:23.810 take a times the cofactor of a. 00:15:23.810 --> 00:15:27.180 And what is the -- 00:15:27.180 --> 00:15:32.440 see, how -- oh no -- let me go along the second row. 00:15:32.440 --> 00:15:33.810 OK. 00:15:33.810 --> 00:15:36.320 So let's see, which -- 00:15:36.320 --> 00:15:37.190 if I take -- 00:15:37.190 --> 00:15:40.510 I know I've got a singular matrix here. 00:15:40.510 --> 00:15:46.310 And I believe that when I do this multiplication, what 00:15:46.310 --> 00:15:51.110 I'm doing is using the cofactor formula for the determinant. 00:15:51.110 --> 00:15:52.700 And I know I'm going to get zero. 00:15:52.700 --> 00:15:54.530 Let me try this again. 00:15:54.530 --> 00:15:56.920 So the cofactor formula for the determinant 00:15:56.920 --> 00:16:04.170 says I should take a times its cofactor, which is this b, 00:16:04.170 --> 00:16:09.260 plus b times its cofactor, which is this minus a. 00:16:09.260 --> 00:16:10.040 OK. 00:16:10.040 --> 00:16:15.260 That's what we're doing, apart from a sign here. 00:16:15.260 --> 00:16:20.980 Oh yeah, so you know, there might be a minus multiplying 00:16:20.980 --> 00:16:21.950 everything. 00:16:21.950 --> 00:16:24.280 So if I take this determinant, it's A -- 00:16:24.280 --> 00:16:29.000 the determinant of this, the determinant of A screwed up is 00:16:29.000 --> 00:16:33.050 a times its cofactor, which is b, 00:16:33.050 --> 00:16:38.980 plus the second guy times its cofactor, which is minus a. 00:16:38.980 --> 00:16:41.790 And of course I get the answer zero, 00:16:41.790 --> 00:16:45.790 and this is exactly what's happening in that, in that, 00:16:45.790 --> 00:16:47.650 row times this wrong column. 00:16:47.650 --> 00:16:48.150 OK. 00:16:52.740 --> 00:16:56.460 That's the two by two picture, and it's just the same here. 00:16:56.460 --> 00:17:01.870 That the reason I get a zero up in there is, the reason 00:17:01.870 --> 00:17:09.319 I get a zero is that when I multiply the first row of A 00:17:09.319 --> 00:17:12.880 and the last row of the cofactor matrix, 00:17:12.880 --> 00:17:16.450 it's as if I'm taking the determinant of this screwed up 00:17:16.450 --> 00:17:19.420 matrix that has first and last rows identical. 00:17:22.270 --> 00:17:26.150 The book pins this down more specific -- 00:17:26.150 --> 00:17:29.410 and more carefully than I can do in the lecture. 00:17:29.410 --> 00:17:31.200 I hope you're seeing the point. 00:17:31.200 --> 00:17:34.130 That this is an identity. 00:17:34.130 --> 00:17:37.310 That it's a beautiful identity and it tells us what 00:17:37.310 --> 00:17:40.690 the inverse of the matrix is. 00:17:40.690 --> 00:17:43.820 So it gives us the inverse, the formula for the inverse. 00:17:43.820 --> 00:17:44.510 OK. 00:17:44.510 --> 00:17:49.180 So that's the first goal of my lecture, was to find this 00:17:49.180 --> 00:17:49.960 formula. 00:17:49.960 --> 00:17:51.490 It's done. 00:17:51.490 --> 00:17:53.800 OK. 00:17:53.800 --> 00:17:58.180 And of course I could invert, now, I can, 00:17:58.180 --> 00:18:00.900 I sort of like I can see what -- 00:18:00.900 --> 00:18:02.850 I can answer questions like this. 00:18:02.850 --> 00:18:08.510 Suppose I have a matrix, and let me move the one one 00:18:08.510 --> 00:18:09.740 entry. 00:18:09.740 --> 00:18:13.600 What happens to the inverse? 00:18:13.600 --> 00:18:15.780 Just, just think about that question. 00:18:15.780 --> 00:18:18.030 Suppose I have some matrix, I just write down 00:18:18.030 --> 00:18:21.300 some nice, non-singular matrix that's got an inverse, 00:18:21.300 --> 00:18:25.170 and then I move the one one entry a little bit. 00:18:25.170 --> 00:18:27.880 I add one to it, for example. 00:18:27.880 --> 00:18:30.730 What happens to the inverse matrix? 00:18:30.730 --> 00:18:33.160 Well, this formula should tell me. 00:18:33.160 --> 00:18:36.110 I have to look to see what happens to the determinant 00:18:36.110 --> 00:18:39.400 and what happens to all the cofactors. 00:18:39.400 --> 00:18:44.300 And, the picture, it's all there. 00:18:44.300 --> 00:18:44.960 It's all there. 00:18:44.960 --> 00:18:49.420 We can really understand how the inverse changes 00:18:49.420 --> 00:18:51.250 when the matrix changes. 00:18:51.250 --> 00:18:52.330 OK. 00:18:52.330 --> 00:18:58.910 Now my second application is to -- let me put that over here -- 00:18:58.910 --> 00:18:59.630 is to Ax=b. 00:19:03.650 --> 00:19:09.060 Well, the -- course, the solution is A inverse b. 00:19:09.060 --> 00:19:11.820 But now I have a formula for A inverse. 00:19:11.820 --> 00:19:17.170 A inverse is one over the determinant 00:19:17.170 --> 00:19:22.030 times this C transpose times B. 00:19:22.030 --> 00:19:24.510 I now know what A inverse is. 00:19:24.510 --> 00:19:27.270 So now I just have to say, what have I got here? 00:19:27.270 --> 00:19:33.600 Is there any way to, to make this formula, this answer, 00:19:33.600 --> 00:19:36.550 which is the one and only answer -- 00:19:36.550 --> 00:19:40.980 it's the very same answer we got on the first day of the class 00:19:40.980 --> 00:19:42.620 by elimination. 00:19:42.620 --> 00:19:47.290 Now I'm -- now I've got a formula for the answer. 00:19:47.290 --> 00:19:51.890 Can I play with it further to see what's going on? 00:19:51.890 --> 00:20:01.640 And Cramer's, this Cramer's Rule is exactly, that -- 00:20:01.640 --> 00:20:07.010 a way of looking at this formula. 00:20:07.010 --> 00:20:07.510 OK. 00:20:07.510 --> 00:20:11.620 So this is a formula for x. 00:20:11.620 --> 00:20:13.620 Here's my formula. 00:20:13.620 --> 00:20:14.350 Well, of course. 00:20:14.350 --> 00:20:16.050 The first thing I see from the formula 00:20:16.050 --> 00:20:21.100 is that the answer x always has that in the determinant. 00:20:21.100 --> 00:20:22.590 I'm not surprised. 00:20:22.590 --> 00:20:25.240 There's a division by the determinant. 00:20:25.240 --> 00:20:29.190 But then I have to say a little more carefully what's going on 00:20:29.190 --> 00:20:31.610 And let me tell you what Cramer's Rule is. up here. 00:20:31.610 --> 00:20:35.610 Let, let me take x1, the first component. 00:20:35.610 --> 00:20:37.740 So this is the first component of the answer. 00:20:37.740 --> 00:20:40.870 Then there'll be a second component and a, 00:20:40.870 --> 00:20:42.900 all the other components. 00:20:42.900 --> 00:20:46.310 Can I take just the first component of this formula? 00:20:46.310 --> 00:20:52.040 Well, I certainly have determinant of A down under. 00:20:52.040 --> 00:20:56.400 And what the heck is the first -- 00:20:56.400 --> 00:21:00.300 so what do I get in C transpose b? 00:21:00.300 --> 00:21:03.440 What's the first entry of C transpose b? 00:21:03.440 --> 00:21:06.060 That's what I have to answer myself. 00:21:06.060 --> 00:21:10.420 Well, what's the first entry of C transpose b? 00:21:13.820 --> 00:21:17.670 This B is -- let me tell you what it is. 00:21:17.670 --> 00:21:18.170 OK. 00:21:18.170 --> 00:21:22.290 Somehow I'm multiplying cofactors 00:21:22.290 --> 00:21:26.970 by the entries of B, right, in this product. 00:21:26.970 --> 00:21:30.430 Cofactors from the matrix times entries of b. 00:21:30.430 --> 00:21:33.650 So any time I'm multiplying cofactors by numbers, 00:21:33.650 --> 00:21:36.980 I think, I'm getting the determinant of something. 00:21:36.980 --> 00:21:39.990 And let me call that something B1. 00:21:39.990 --> 00:21:46.730 So this is a matrix, the matrix whose determinant is coming out 00:21:46.730 --> 00:21:47.230 of that. 00:21:47.230 --> 00:21:49.530 And we'll, we'll see what it is. 00:21:49.530 --> 00:21:54.550 x2 will be the determinant of some other matrix B2, also 00:21:54.550 --> 00:21:57.650 divided by determinant of A. 00:21:57.650 --> 00:21:59.190 So now I just -- 00:21:59.190 --> 00:22:00.970 Cramer just had a good idea. 00:22:00.970 --> 00:22:06.920 He realized what matrix it was, what these B1 and B2 and B3 00:22:06.920 --> 00:22:08.470 and so on matrices were. 00:22:08.470 --> 00:22:10.520 Let me write them on the board underneath. 00:22:14.610 --> 00:22:15.260 OK. 00:22:15.260 --> 00:22:17.470 So what is this B1? 00:22:17.470 --> 00:22:24.430 This B1 is the matrix that has b in its first column 00:22:24.430 --> 00:22:27.900 and otherwise the rest of it is A. 00:22:27.900 --> 00:22:38.740 So it otherwise it has the rest, the, the n-1 columns of A. 00:22:38.740 --> 00:22:42.880 It's the matrix with -- 00:22:42.880 --> 00:22:51.410 it's just the matrix A with column one replaced 00:22:51.410 --> 00:22:58.520 by the right-hand side, by the right-hand side b. 00:22:58.520 --> 00:23:04.490 Because somehow when I take the determinant of this guy, 00:23:04.490 --> 00:23:09.140 it's giving me this matrix multiplication. 00:23:09.140 --> 00:23:10.570 Well, how could that be? 00:23:13.180 --> 00:23:16.490 How could -- so what's, what's the determinant formula 00:23:16.490 --> 00:23:18.050 I'll use here? 00:23:18.050 --> 00:23:21.910 I'll use cofactors, of course. 00:23:21.910 --> 00:23:25.240 And I might as well use cofactors down column one. 00:23:25.240 --> 00:23:27.910 So when I use cofactors down column one, 00:23:27.910 --> 00:23:32.840 I'm taking the first entry of b times what? 00:23:32.840 --> 00:23:35.650 Times the cofactor c11. 00:23:35.650 --> 00:23:38.410 Do you see that? 00:23:38.410 --> 00:23:40.830 When I, when I use cofactors here, 00:23:40.830 --> 00:23:43.530 I take the first entry here, B one 00:23:43.530 --> 00:23:47.120 let's call it, times the cofactor there. 00:23:47.120 --> 00:23:51.390 But what's the cofactor in -- my little hand-waving is meant 00:23:51.390 --> 00:23:54.650 to indicate that it's a matrix of one size smaller, 00:23:54.650 --> 00:23:56.030 the cofactor. 00:23:56.030 --> 00:23:58.730 And it's exactly c11. 00:23:58.730 --> 00:24:01.130 Well, that's just what we wanted. 00:24:01.130 --> 00:24:06.200 This first entry is c11 times b1. 00:24:06.200 --> 00:24:13.970 And then the next entry is whatever, is c21 times b2, 00:24:13.970 --> 00:24:14.860 and so on. 00:24:14.860 --> 00:24:17.310 And sure enough, if I look here, when 00:24:17.310 --> 00:24:19.720 I'm, when I do the cofactor expansion, 00:24:19.720 --> 00:24:22.790 b2 is getting multiplied by the right thing, and so on. 00:24:25.450 --> 00:24:27.400 So there's Cramer's Rule. 00:24:27.400 --> 00:24:31.560 And the book gives another kind of cute proof 00:24:31.560 --> 00:24:36.960 without, without building so much on, on cofactors. 00:24:36.960 --> 00:24:38.950 But here we've got cofactors, so I thought 00:24:38.950 --> 00:24:40.650 I'd just give you this proof. 00:24:40.650 --> 00:24:42.700 So what is B -- 00:24:42.700 --> 00:24:45.430 in general, what is Bj? 00:24:45.430 --> 00:24:58.970 This is the, this is A with column j replaced by, by b. 00:25:02.920 --> 00:25:08.870 So that's -- the determinant of that matrix that you divide 00:25:08.870 --> 00:25:11.700 by determinant of A to get xj. 00:25:11.700 --> 00:25:14.980 So x -- let me change this general formula. 00:25:14.980 --> 00:25:18.190 xj, the j-th one, is the determinant 00:25:18.190 --> 00:25:22.510 of Bj divided by the determinant of A. 00:25:22.510 --> 00:25:24.360 And now we've said what Bj is. 00:25:30.160 --> 00:25:33.720 Well, so Cramer found a rule. 00:25:33.720 --> 00:25:39.320 And we could ask him, OK, great, good work, Cramer. 00:25:39.320 --> 00:25:44.180 But is your rule any good in practice? 00:25:44.180 --> 00:25:49.960 So he says, well, you couldn't ask about a rule in mine, 00:25:49.960 --> 00:25:52.220 right, because it's just -- 00:25:52.220 --> 00:25:56.850 all you have to do is find the determinant of A and these 00:25:56.850 --> 00:26:01.220 other determinants, so I guess -- oh, he just says, well, 00:26:01.220 --> 00:26:04.140 all you have to do is find n+1 determinants, 00:26:04.140 --> 00:26:06.170 the, the n Bs and the A. 00:26:06.170 --> 00:26:17.530 And actually, I remember reading -- there was a book, 00:26:17.530 --> 00:26:22.530 popular book that, that kids interested in math read when I 00:26:22.530 --> 00:26:25.980 was a kid interested in math called Mathematics 00:26:25.980 --> 00:26:29.840 for the Million or something, by a guy named Bell. 00:26:29.840 --> 00:26:34.830 And it had a little page about linear algebra. 00:26:34.830 --> 00:26:39.000 And it said,-- so it explained elimination 00:26:39.000 --> 00:26:41.250 in a very complicated way. 00:26:41.250 --> 00:26:43.200 I certainly didn't understand it. 00:26:43.200 --> 00:26:47.950 And, and it made it, you know, it sort of said, well, 00:26:47.950 --> 00:26:51.120 there is this formula for elimination, 00:26:51.120 --> 00:26:55.770 but look at this great formula, Cramer's Rule. 00:26:55.770 --> 00:27:00.410 So it certainly said Cramer's Rule was the way to go. 00:27:00.410 --> 00:27:05.180 But actually, Cramer's Rule is a disastrous way to go, 00:27:05.180 --> 00:27:07.920 because to compute these determinants, 00:27:07.920 --> 00:27:12.060 it takes, like, approximately forever. 00:27:12.060 --> 00:27:17.020 So actually I now think of that book title 00:27:17.020 --> 00:27:18.980 as being Mathematics for the Millionaire, 00:27:18.980 --> 00:27:22.160 because you'd have to be able to pay for, 00:27:22.160 --> 00:27:26.570 a hopelessly long calculation where elimination, of course, 00:27:26.570 --> 00:27:30.450 produced the x-s, in an instant. 00:27:30.450 --> 00:27:36.060 But having a formula allows you to, with, with letters, you 00:27:36.060 --> 00:27:40.010 know, allows you to do algebra instead of, algorithms. 00:27:40.010 --> 00:27:44.220 So the, there's some value in the Cramer's Rule formula 00:27:44.220 --> 00:27:52.000 for x and in the explicit formula for, for A inverse. 00:27:52.000 --> 00:27:54.890 They're nice formulas, but I just 00:27:54.890 --> 00:27:57.710 don't want you to use them. 00:27:57.710 --> 00:27:59.230 That'ss what it comes to. 00:27:59.230 --> 00:28:02.960 If you had to -- and Matlab would never, never do it. 00:28:02.960 --> 00:28:05.480 I mean, it would use elimination. 00:28:05.480 --> 00:28:06.720 OK. 00:28:06.720 --> 00:28:11.380 Now I'm ready for number three in today's list 00:28:11.380 --> 00:28:17.790 of amazing connections coming through the determinant. 00:28:17.790 --> 00:28:21.900 And that number three is the fact that the determinant gives 00:28:21.900 --> 00:28:22.950 a volume. 00:28:22.950 --> 00:28:24.480 OK. 00:28:24.480 --> 00:28:28.860 So now -- so that's my final topic for -- 00:28:28.860 --> 00:28:33.070 among these -- this my number three application, 00:28:33.070 --> 00:28:37.700 that the determinant is actually equals the volume of something. 00:28:37.700 --> 00:28:42.720 Can I use this little space to consider a special case, 00:28:42.720 --> 00:28:46.050 and then I'll use the far board to think 00:28:46.050 --> 00:28:47.840 about the general rule. 00:28:47.840 --> 00:28:50.400 So what I going to prove? 00:28:50.400 --> 00:28:51.970 Or claim. 00:28:51.970 --> 00:28:56.950 I claim that the determinant of the matrix 00:28:56.950 --> 00:29:02.800 is the volume of a box. 00:29:02.800 --> 00:29:05.930 OK, and you say, which box? 00:29:05.930 --> 00:29:06.970 Fair enough. 00:29:06.970 --> 00:29:08.570 OK. 00:29:08.570 --> 00:29:11.670 So let's see. 00:29:11.670 --> 00:29:16.170 I'm in -- shall we say we're in, say three by three? 00:29:16.170 --> 00:29:18.780 Shall we suppose -- let's, let's say three by three. 00:29:18.780 --> 00:29:22.680 So, so we can really -- we're, we're talking about boxes 00:29:22.680 --> 00:29:26.400 in three dimensions, and three by three matrices. 00:29:26.400 --> 00:29:30.990 And so all I do -- you could guess what the box is. 00:29:30.990 --> 00:29:34.700 Here is, here is, three dimensions. 00:29:34.700 --> 00:29:35.450 OK. 00:29:35.450 --> 00:29:41.880 Now I take the first row of the matrix, a11, a22, A -- 00:29:41.880 --> 00:29:44.360 sorry. a11, a12, a13. 00:29:44.360 --> 00:29:48.090 That row is a vector. 00:29:48.090 --> 00:29:50.190 It goes to some point. 00:29:50.190 --> 00:29:53.220 That point will be -- and that edge going to it, 00:29:53.220 --> 00:29:56.650 will be an edge of the box, and that point will be a corner 00:29:56.650 --> 00:29:57.740 of the box. 00:29:57.740 --> 00:30:01.010 So here is zero zero zero, of course. 00:30:01.010 --> 00:30:09.500 And here's the first row of the matrix: a11, a12, a13. 00:30:09.500 --> 00:30:14.840 So that's one edge of the box. 00:30:14.840 --> 00:30:19.700 Another edge of the box is to the second row 00:30:19.700 --> 00:30:21.820 of the matrix, row two. 00:30:21.820 --> 00:30:24.510 Can I just call it there row two? 00:30:24.510 --> 00:30:28.760 And a third row of the box will be to -- 00:30:28.760 --> 00:30:32.200 a third row -- a third edge of the box will be given by row 00:30:32.200 --> 00:30:33.300 three. 00:30:33.300 --> 00:30:36.470 So, so there's row three. 00:30:36.470 --> 00:30:38.780 That, the coordinates, what are the coordinates 00:30:38.780 --> 00:30:41.030 of that corner of the box? 00:30:41.030 --> 00:30:47.650 a31, a32, a33. 00:30:47.650 --> 00:30:51.220 So I've got that edge of the box, that edge of the box, 00:30:51.220 --> 00:30:53.830 that edge of the box, and that's all I need. 00:30:53.830 --> 00:30:59.820 Now I just finish out the box, right? 00:30:59.820 --> 00:31:03.650 I just -- the proper word, of course, is parallelepiped. 00:31:03.650 --> 00:31:08.920 But for obvious reasons, I wrote box. 00:31:08.920 --> 00:31:09.550 OK. 00:31:09.550 --> 00:31:11.360 So, OK. 00:31:11.360 --> 00:31:14.550 So there's the, there's the bottom of the box. 00:31:14.550 --> 00:31:19.660 There're the four edge sides of the box. 00:31:19.660 --> 00:31:23.570 There's the top of the box. 00:31:23.570 --> 00:31:24.540 Cute, right? 00:31:24.540 --> 00:31:28.470 It's the box that has these three edges 00:31:28.470 --> 00:31:33.080 and then it's completed to a, to a, each, you know, 00:31:33.080 --> 00:31:35.080 each side is a, is a parallelogram. 00:31:37.730 --> 00:31:43.050 And it's that box whose volume is given by the determinant. 00:31:45.610 --> 00:31:50.650 That's -- now it's -- possible that the determinant is 00:31:50.650 --> 00:31:52.920 negative. 00:31:52.920 --> 00:31:56.110 So we have to just say what's going on in that case. 00:31:56.110 --> 00:32:01.840 If the determinant is negative, then the volume, we, 00:32:01.840 --> 00:32:04.030 we should take the absolute value really. 00:32:04.030 --> 00:32:07.030 So the volume, if we, if we think of volume as positive, 00:32:07.030 --> 00:32:11.310 we should take the absolute value of the determinant. 00:32:11.310 --> 00:32:14.400 But the, the sign, what does the sign of the determinant -- 00:32:14.400 --> 00:32:16.560 it always must tell us something. 00:32:16.560 --> 00:32:20.380 And somehow it, it will tell us whether these three is a -- 00:32:20.380 --> 00:32:23.270 whether it's a right-handed box or a left-handed box. 00:32:23.270 --> 00:32:30.300 If we, if we reversed two of the edges, 00:32:30.300 --> 00:32:32.250 we would go between a right-handed box 00:32:32.250 --> 00:32:33.390 and a left-handed box. 00:32:33.390 --> 00:32:35.440 We wouldn't change the volume, but we would 00:32:35.440 --> 00:32:40.500 change the, the cyclic, order. 00:32:40.500 --> 00:32:42.900 So I won't worry about that. 00:32:42.900 --> 00:32:47.020 And, so one special case is what? 00:32:47.020 --> 00:32:49.930 A equal identity matrix. 00:32:49.930 --> 00:32:52.070 So let's take that special case. 00:32:52.070 --> 00:32:55.520 A equal identity matrix. 00:32:55.520 --> 00:32:59.550 Is the formula determinant of identity matrix, 00:32:59.550 --> 00:33:01.230 does that equal the volume of the box? 00:33:04.300 --> 00:33:06.460 Well, what is the box? 00:33:06.460 --> 00:33:07.370 What's the box? 00:33:07.370 --> 00:33:12.230 If A is the identity matrix, then these three rows are 00:33:12.230 --> 00:33:17.460 the three coordinate vectors, and the box is -- 00:33:17.460 --> 00:33:18.920 it's a cube. 00:33:18.920 --> 00:33:21.200 It's the unit cube. 00:33:21.200 --> 00:33:23.970 So if, if A is the identity matrix, of course 00:33:23.970 --> 00:33:26.080 our formula is 00:33:26.080 --> 00:33:29.850 Well, actually that proves property one -- 00:33:29.850 --> 00:33:31.740 that the volume right. has property one. 00:33:31.740 --> 00:33:35.440 Actually, we could, we could, we could get this thing if we -- 00:33:35.440 --> 00:33:39.210 if we can show that the box volume has the same three 00:33:39.210 --> 00:33:42.010 properties that define the determinant, 00:33:42.010 --> 00:33:44.060 then it must be the determinant. 00:33:46.670 --> 00:33:51.080 So that's like the, the, the elegant way to prove this. 00:33:51.080 --> 00:33:53.860 To prove this amazing fact that the determinant equals 00:33:53.860 --> 00:33:58.540 the volume, first we'll check it for the identity matrix. 00:33:58.540 --> 00:34:00.470 That's fine. 00:34:00.470 --> 00:34:03.190 The box is a cube and its volume is one 00:34:03.190 --> 00:34:07.180 and the determinant is one and, and one agrees with one. 00:34:07.180 --> 00:34:11.590 Now let me take one -- let me go up one level to an orthogonal 00:34:11.590 --> 00:34:12.889 matrix. 00:34:12.889 --> 00:34:16.060 Because I'd like to take this chance to bring in chapter -- 00:34:16.060 --> 00:34:18.750 the, the previous chapter. 00:34:18.750 --> 00:34:21.100 Suppose I have an orthogonal matrix. 00:34:21.100 --> 00:34:22.110 What did that mean? 00:34:22.110 --> 00:34:25.500 I always called those things Q. 00:34:25.500 --> 00:34:27.909 What was the point of -- suppose I have, 00:34:27.909 --> 00:34:32.670 suppose instead of the identity matrix I'm now going to take A 00:34:32.670 --> 00:34:36.910 equal Q, an orthogonal matrix. 00:34:42.940 --> 00:34:46.510 What was Q then? 00:34:46.510 --> 00:34:52.179 That was a matrix whose columns were orthonormal, right? 00:34:52.179 --> 00:34:54.610 Those were its columns were unit vectors, 00:34:54.610 --> 00:34:56.059 perpendicular unit vectors. 00:34:59.220 --> 00:35:03.090 So what kind of a box have we got now? 00:35:03.090 --> 00:35:06.520 What kind of a box comes from the rows or the columns, 00:35:06.520 --> 00:35:08.530 I don't mind, because the determinant 00:35:08.530 --> 00:35:10.420 is the determinant of the transpose, 00:35:10.420 --> 00:35:11.850 so I'm never worried about 00:35:11.850 --> 00:35:12.730 that. 00:35:12.730 --> 00:35:15.850 What kind of a box, what shape box have we 00:35:15.850 --> 00:35:20.380 got if the matrix is an orthogonal matrix? 00:35:20.380 --> 00:35:23.040 It's another cube. 00:35:23.040 --> 00:35:24.710 It's a cube again. 00:35:24.710 --> 00:35:27.900 How is it different from the identity cube? 00:35:30.510 --> 00:35:33.140 It's just rotated. 00:35:33.140 --> 00:35:36.830 It's just the orthogonal matrix Q doesn't 00:35:36.830 --> 00:35:38.860 have to be the identity matrix. 00:35:38.860 --> 00:35:43.240 It's just the unit cube but turned in space. 00:35:43.240 --> 00:35:48.490 So sure enough, it's the unit cube, and its volume is one. 00:35:48.490 --> 00:35:53.410 Now is the determinant one? 00:35:53.410 --> 00:35:55.410 What's the determinant of Q? 00:35:55.410 --> 00:35:58.980 We believe that the determinant of Q better be one or minus 00:35:58.980 --> 00:36:03.100 one, so that our formula is -- checks out in that -- 00:36:03.100 --> 00:36:06.800 if we can't check it in these easy cases where we got a cube, 00:36:06.800 --> 00:36:11.020 we're not going to get it in the general case. 00:36:11.020 --> 00:36:18.290 So why is the determinant of Q equal one or minus one? 00:36:18.290 --> 00:36:19.810 What do we know about Q? 00:36:19.810 --> 00:36:25.050 What's the one matrix statement of the properties of Q? 00:36:25.050 --> 00:36:29.020 A matrix with orthonormal columns has -- 00:36:29.020 --> 00:36:31.860 satisfies a certain equation. 00:36:31.860 --> 00:36:33.400 What, what is that? 00:36:33.400 --> 00:36:38.930 It's if we have this orthogonal matrix, then the fact -- 00:36:38.930 --> 00:36:45.550 the way to say what it, what its properties are is this. 00:36:45.550 --> 00:36:50.940 Q prime, u- u- Q transpose Q equals I. 00:36:50.940 --> 00:36:52.790 Right? 00:36:52.790 --> 00:36:57.450 That's what -- those are the matrices that get the name Q, 00:36:57.450 --> 00:37:01.241 the matrices that Q transpose Q is I. 00:37:01.241 --> 00:37:01.740 OK. 00:37:01.740 --> 00:37:07.180 Now from that, tell me why is the determinant one 00:37:07.180 --> 00:37:09.500 or minus one. 00:37:09.500 --> 00:37:12.930 How do I, out of this fact -- 00:37:12.930 --> 00:37:14.525 this may even be a homework problem. 00:37:17.240 --> 00:37:20.890 It's there in the, in the list of exercises in the book, 00:37:20.890 --> 00:37:23.100 and let's just do it. 00:37:23.100 --> 00:37:27.340 How do I get, how do I discover that the determinant of Q 00:37:27.340 --> 00:37:31.530 is one or maybe minus one? 00:37:31.530 --> 00:37:34.180 I take determinants of both sides, everybody says, 00:37:34.180 --> 00:37:36.870 so I won't -- 00:37:36.870 --> 00:37:38.610 I take determinants of both sides. 00:37:38.610 --> 00:37:41.250 On the right-hand side -- so I, when I take determinants 00:37:41.250 --> 00:37:46.240 of both sides, let me just do it. 00:37:46.240 --> 00:37:48.315 Take the determinant of -- take determinants. 00:37:51.680 --> 00:37:53.970 Determinant of the identity is one. 00:37:53.970 --> 00:37:55.840 What's the determinant of that product? 00:37:58.690 --> 00:38:02.400 Rule nine is paying off now. 00:38:02.400 --> 00:38:07.920 The determinant of a product is the determinant of this guy -- 00:38:07.920 --> 00:38:11.110 maybe I'll put it, I'll use that symbol for determinant. 00:38:11.110 --> 00:38:14.152 It's the determinant of that guy times the determinant 00:38:14.152 --> 00:38:14.860 of the other guy. 00:38:17.460 --> 00:38:21.400 And then what's the determinant of Q transpose? 00:38:21.400 --> 00:38:22.990 It's the same as the determinant of Q. 00:38:22.990 --> 00:38:24.810 Rule ten pays off. 00:38:24.810 --> 00:38:28.330 So this is just this thing squared. 00:38:28.330 --> 00:38:33.740 So that determinant squared is one and sure enough it's one 00:38:33.740 --> 00:38:35.290 or minus one. 00:38:35.290 --> 00:38:35.790 Great. 00:38:38.370 --> 00:38:42.000 So in these special cases of cubes, 00:38:42.000 --> 00:38:48.650 we really do have determinant equals volume. 00:38:48.650 --> 00:38:54.530 Now can I just push that to non-cubes. 00:38:54.530 --> 00:39:02.930 Let me push it first to rectangles, rectangular boxes, 00:39:02.930 --> 00:39:07.820 where I'm just multiplying the e- the edges are -- 00:39:07.820 --> 00:39:10.440 let me keep all the ninety degree angles, 00:39:10.440 --> 00:39:13.510 because those are -- that, that makes my life easy. 00:39:13.510 --> 00:39:16.910 And just stretch the edges. 00:39:16.910 --> 00:39:21.910 Suppose I stretch that first edge, suppose this first edge 00:39:21.910 --> 00:39:23.230 I double. 00:39:23.230 --> 00:39:27.990 Suppose I double that first edge, 00:39:27.990 --> 00:39:31.060 keeping the other edges the same. 00:39:31.060 --> 00:39:34.650 What happens to the volume? 00:39:34.650 --> 00:39:36.640 It doubles, right? 00:39:36.640 --> 00:39:39.420 We know that the volume of a cube doubles. 00:39:39.420 --> 00:39:42.370 In fact, because we know that the new cube would sit right 00:39:42.370 --> 00:39:43.400 on top -- 00:39:43.400 --> 00:39:46.210 I mean, the new, the added cube would sit right on -- 00:39:46.210 --> 00:39:47.620 would fit -- 00:39:47.620 --> 00:39:50.120 probably a geometer would say congruent or something -- 00:39:50.120 --> 00:39:51.810 would go right in, in the other. 00:39:51.810 --> 00:39:52.830 We'd have two. 00:39:52.830 --> 00:39:54.540 We have two identical cubes. 00:39:54.540 --> 00:39:58.421 Total volume is now two. 00:39:58.421 --> 00:39:58.920 OK. 00:39:58.920 --> 00:40:02.392 So I want -- if I double an edge, the volume doubles. 00:40:02.392 --> 00:40:03.725 What happens to the determinant? 00:40:07.580 --> 00:40:14.880 If I double, the first row of a matrix, what ch- 00:40:14.880 --> 00:40:18.010 ch- what's the effect on the determinant? 00:40:18.010 --> 00:40:21.320 It also doubles, right? 00:40:21.320 --> 00:40:25.910 And that was rule number 3a. 00:40:25.910 --> 00:40:29.500 Remember rule 3a was that if I, I could, 00:40:29.500 --> 00:40:36.780 if I had a factor in, in row one, T, I could factor it out. 00:40:36.780 --> 00:40:41.020 So if, if I have a factor two in that row one, 00:40:41.020 --> 00:40:43.340 I can factor it out of the determinant. 00:40:43.340 --> 00:40:48.860 It agrees with the -- the volume of the box has that factor two. 00:40:48.860 --> 00:40:53.140 So, so volume satisfies this property 3a. 00:40:53.140 --> 00:40:59.040 And now I really close, but I -- but to get to the very end 00:40:59.040 --> 00:41:01.980 of this proof, I have to get away from right 00:41:01.980 --> 00:41:02.650 angles. 00:41:02.650 --> 00:41:10.330 I have to allow the possibility of, other angles. 00:41:10.330 --> 00:41:13.090 And -- or what's saying the same thing, 00:41:13.090 --> 00:41:18.600 I have to check that the volume also satisfies 3b. 00:41:18.600 --> 00:41:20.600 So can I -- 00:41:20.600 --> 00:41:24.740 This is end of proof that the -- so I'm -- 00:41:24.740 --> 00:41:34.380 determinant of A equals volume of box, and where I right now? 00:41:34.380 --> 00:41:40.820 This volume has properties, properties one, no problem. 00:41:40.820 --> 00:41:44.030 If the box is the cube, everything is -- 00:41:44.030 --> 00:41:49.000 if the box is the unit cube, its volume is one. 00:41:49.000 --> 00:41:54.680 Property two was if I reverse two rows, 00:41:54.680 --> 00:41:57.960 but that doesn't change the box. 00:41:57.960 --> 00:42:00.520 And it doesn't change the absolute value, so no problem 00:42:00.520 --> 00:42:01.200 there. 00:42:01.200 --> 00:42:06.890 Property 3a was if I mul- you remember what 3a was? 00:42:06.890 --> 00:42:10.390 So property one was about the identity matrix. 00:42:10.390 --> 00:42:12.900 Property two was about a plus or minus sign 00:42:12.900 --> 00:42:14.560 that I don't care about. 00:42:14.560 --> 00:42:18.490 Property 3a was a factor T in a row. 00:42:18.490 --> 00:42:24.080 But now I've got property three B to deal with. 00:42:24.080 --> 00:42:25.520 What was property 3b? 00:42:25.520 --> 00:42:30.270 This is a great way to review these, properties. 00:42:30.270 --> 00:42:34.860 So that 3b, the property 3b said -- let's do, 00:42:34.860 --> 00:42:36.910 let's do two by two. 00:42:36.910 --> 00:42:42.620 So said that if I had a+a', b+b', c, 00:42:42.620 --> 00:42:47.590 d that this equaled what? 00:42:47.590 --> 00:42:49.120 So this is property 3b. 00:42:49.120 --> 00:42:53.840 This is the linearity in row one by itself. 00:42:53.840 --> 00:42:59.060 So c d is staying the same, and I can split this into a b 00:42:59.060 --> 00:43:01.340 and a' b'. 00:43:06.160 --> 00:43:12.890 That's property 3b, at least in the two by two case. 00:43:12.890 --> 00:43:15.340 And what I -- 00:43:15.340 --> 00:43:20.540 I wanted now to show that the volume, which 00:43:20.540 --> 00:43:25.040 two, two by two, that means area, has this, 00:43:25.040 --> 00:43:25.790 has this property. 00:43:28.920 --> 00:43:32.540 Let me just emphasize that we have got -- we're getting -- 00:43:32.540 --> 00:43:37.150 this is a formula, then, for the area of a parallelogram. 00:43:37.150 --> 00:43:40.310 The area of this parallelogram -- can I just draw it? 00:43:40.310 --> 00:43:42.160 OK, here's the, here's the parallelogram. 00:43:42.160 --> 00:43:45.370 I have the row a b. 00:43:45.370 --> 00:43:47.000 That's the first row. 00:43:47.000 --> 00:43:49.260 That's the point a b. 00:43:49.260 --> 00:43:54.060 And I tack on c d. 00:43:54.060 --> 00:43:57.690 c d, coming out of here. 00:43:57.690 --> 00:43:59.065 And I complete the parallelogram. 00:44:02.560 --> 00:44:03.750 So this is -- 00:44:03.750 --> 00:44:08.140 well, I better make it look right. 00:44:08.140 --> 00:44:12.530 It's really this one that has coordinates c d and this has 00:44:12.530 --> 00:44:17.040 coordinates -- well, whatever the sum is. 00:44:17.040 --> 00:44:18.820 And of course starting at zero zero. 00:44:22.240 --> 00:44:26.026 So we all know, this is a+c, b+d. 00:44:29.140 --> 00:44:31.120 Rather than -- 00:44:31.120 --> 00:44:33.460 I'm pausing on that proof for a minute 00:44:33.460 --> 00:44:37.320 just to going back to our formula. 00:44:37.320 --> 00:44:40.650 Because I want you to see that unlike Cramer's Rule, 00:44:40.650 --> 00:44:44.060 that I wasn't that impressed by, I'm 00:44:44.060 --> 00:44:46.640 very impressed by this formula for the area 00:44:46.640 --> 00:44:48.540 of a parallelogram. 00:44:48.540 --> 00:44:50.930 And what's our formula? 00:44:50.930 --> 00:44:54.890 What, what's the area of that parallelogram? 00:44:54.890 --> 00:45:00.250 If I had asked you that last year, 00:45:00.250 --> 00:45:03.950 you would have said OK, the area of a parallelogram 00:45:03.950 --> 00:45:06.060 is the base times the height, 00:45:06.060 --> 00:45:07.190 right? 00:45:07.190 --> 00:45:10.890 So you would have figured out what this base, the -- 00:45:10.890 --> 00:45:12.850 how long that base was. 00:45:12.850 --> 00:45:16.182 It's like the square root of A squared plus b squared. 00:45:16.182 --> 00:45:17.640 And then you would have figured out 00:45:17.640 --> 00:45:20.360 how much is this height, whatever it is. 00:45:20.360 --> 00:45:21.920 It's horrible. 00:45:21.920 --> 00:45:28.330 This, I mean, we got square roots, and in that height 00:45:28.330 --> 00:45:32.040 there would be other revolting stuff. 00:45:32.040 --> 00:45:35.800 But now what's the formula that we now know for the area? 00:45:42.290 --> 00:45:45.740 It's the determinant of our little matrix. 00:45:48.540 --> 00:45:51.050 It's just ad-bc. 00:45:56.990 --> 00:45:59.110 No square roots. 00:45:59.110 --> 00:46:03.000 Totally rememberable, because it's exactly a formula 00:46:03.000 --> 00:46:06.960 that we've been studying the whole, for three lectures. 00:46:06.960 --> 00:46:07.460 OK. 00:46:11.090 --> 00:46:13.960 That's, you know, that's the most important point 00:46:13.960 --> 00:46:15.190 I'm making here. 00:46:15.190 --> 00:46:20.990 Is that if you know the coordinates of a box, 00:46:20.990 --> 00:46:24.750 of the corners, then you have a great formula 00:46:24.750 --> 00:46:28.620 for the volume, area or volume, that 00:46:28.620 --> 00:46:33.550 doesn't involve any lengths or any angles or any heights, 00:46:33.550 --> 00:46:37.850 but just involves the coordinates that you've got. 00:46:37.850 --> 00:46:40.390 And similarly, what's the area of this triangle? 00:46:40.390 --> 00:46:43.530 Suppose I chop that off and say what about -- 00:46:43.530 --> 00:46:45.810 because you might often be interested in a triangle 00:46:45.810 --> 00:46:47.330 instead of a parallelogram. 00:46:47.330 --> 00:46:48.725 What's the area of this triangle? 00:46:52.040 --> 00:46:53.950 Now there again, everybody would have 00:46:53.950 --> 00:46:58.110 said the area of a triangle is half the base times the height. 00:47:01.200 --> 00:47:04.300 And in some cases, if you know the base that a, that's -- 00:47:04.300 --> 00:47:06.300 and the height, that's fine. 00:47:06.300 --> 00:47:10.280 But here, we, what we know is the coordinates of the corners. 00:47:10.280 --> 00:47:11.980 We know the vertices. 00:47:11.980 --> 00:47:14.690 And so what's the area of that triangle? 00:47:17.830 --> 00:47:22.550 If I know these, if I know a b, c d, and zero zero, 00:47:22.550 --> 00:47:25.060 what's the area? 00:47:25.060 --> 00:47:29.670 It's just half, so it's just half of this. 00:47:29.670 --> 00:47:33.040 So this is, this is a- a b -- a d - 00:47:33.040 --> 00:47:38.870 b c for the parallelogram and one half of that, 00:47:38.870 --> 00:47:43.585 one half of ad-bc for the triangle. 00:47:47.140 --> 00:47:50.900 So I mean, this is a totally trivial remark, to say, well, 00:47:50.900 --> 00:47:52.630 divide by two. 00:47:52.630 --> 00:47:56.910 But it's just that you more often see triangles, 00:47:56.910 --> 00:48:01.530 and you feel you know the formula for the area 00:48:01.530 --> 00:48:04.950 but the good formula for the area is this one. 00:48:04.950 --> 00:48:06.760 And I'm just going to -- 00:48:06.760 --> 00:48:08.260 I'm just going to say one more thing 00:48:08.260 --> 00:48:09.750 about the area of a triangle. 00:48:09.750 --> 00:48:11.220 It's just because it's -- you know, 00:48:11.220 --> 00:48:15.760 it's so great to have a good formula for something. 00:48:15.760 --> 00:48:20.990 What if our triangle did not start at zero zero? 00:48:20.990 --> 00:48:25.630 What if our triangle, what if we had this -- 00:48:25.630 --> 00:48:28.390 what if we had -- so I'm coming back to triangles again. 00:48:32.600 --> 00:48:40.320 But let me, let me put this triangle somewhere, it's -- 00:48:40.320 --> 00:48:43.800 I'm staying with triangles, I'm just in two dimensions, 00:48:43.800 --> 00:48:53.280 but I'm going to allow you to give me any three corners. 00:48:57.420 --> 00:49:01.430 And in -- those six numbers must determine the area. 00:49:01.430 --> 00:49:03.750 And what's the formula? 00:49:03.750 --> 00:49:05.280 The area is going to be, it's going 00:49:05.280 --> 00:49:09.530 to be, there'll be that half of a parallelogram. 00:49:09.530 --> 00:49:13.580 I mean, basically this can't be completely new, right? 00:49:13.580 --> 00:49:17.230 We've got the area when -- we, we know the area when this is 00:49:17.230 --> 00:49:20.400 zero zero. 00:49:20.400 --> 00:49:24.810 Now we just want to lift our sight slightly and get the area 00:49:24.810 --> 00:49:27.820 when all th- so let me write down what it, what it 00:49:27.820 --> 00:49:29.190 comes out to be. 00:49:29.190 --> 00:49:37.390 It turns out that if you do this, x1 y1 and a 1, x2 y2 00:49:37.390 --> 00:49:44.150 and a 1, x3 y3 and a 1, that that works. 00:49:44.150 --> 00:49:47.320 That the determinant symbol, of course. 00:49:47.320 --> 00:49:51.640 It's just -- if I gave you that determinant to find, 00:49:51.640 --> 00:49:53.890 you might subtract this row from this. 00:49:53.890 --> 00:49:55.780 It would kill that one. 00:49:55.780 --> 00:49:59.170 Subtract this row from this, it would kill that one. 00:49:59.170 --> 00:50:02.470 Then you'd have a simple determinant to do with 00:50:02.470 --> 00:50:06.300 differences, and it would -- 00:50:06.300 --> 00:50:08.960 this little subtraction, what I did 00:50:08.960 --> 00:50:12.580 was equivalent to moving the triangle 00:50:12.580 --> 00:50:16.550 to start at the origin. 00:50:16.550 --> 00:50:19.850 I did it fast, because time is up. 00:50:19.850 --> 00:50:24.530 And I didn't complete that proof of 3b. 00:50:24.530 --> 00:50:28.930 I'll leave -- the book has a carefully drawn figure to show 00:50:28.930 --> 00:50:30.870 why that works. 00:50:30.870 --> 00:50:34.110 But I hope you saw the main point is 00:50:34.110 --> 00:50:36.760 that for area and volume, determinant 00:50:36.760 --> 00:50:39.540 gives a great formula. 00:50:39.540 --> 00:50:40.100 OK. 00:50:40.100 --> 00:50:44.410 And next lectures are about eigenvalues, 00:50:44.410 --> 00:50:47.780 so we're really into the big stuff. 00:50:47.780 --> 00:50:49.330 Thanks.