WEBVTT 00:00:08.160 --> 00:00:13.730 I've been multiplying matrices already, 00:00:13.730 --> 00:00:17.640 but certainly time for me to discuss the rules 00:00:17.640 --> 00:00:19.390 for matrix multiplication. 00:00:19.390 --> 00:00:24.980 And the interesting part is the many ways you can do it, 00:00:24.980 --> 00:00:27.970 and they all give the same answer. 00:00:27.970 --> 00:00:29.780 And they're all important. 00:00:29.780 --> 00:00:33.840 So matrix multiplication, and then, come inverses. 00:00:33.840 --> 00:00:38.520 So we mentioned the inverse of a matrix. 00:00:38.520 --> 00:00:39.660 That's a big deal. 00:00:39.660 --> 00:00:43.400 Lots to do about inverses and how to find them. 00:00:43.400 --> 00:00:48.420 Okay, so I'll begin with how to multiply two matrices. 00:00:48.420 --> 00:00:57.330 First way, okay, so suppose I have a matrix A multiplying 00:00:57.330 --> 00:01:03.500 a matrix B and -- giving me a result -- 00:01:03.500 --> 00:01:05.990 well, I could call it C. 00:01:05.990 --> 00:01:07.870 A times B. 00:01:07.870 --> 00:01:10.340 Okay. 00:01:10.340 --> 00:01:18.360 So, let me just review the rule for this entry. 00:01:18.360 --> 00:01:22.000 That's the entry in row i and column j. 00:01:22.000 --> 00:01:24.340 So that's the i j entry. 00:01:24.340 --> 00:01:27.350 Right there is C i j. 00:01:27.350 --> 00:01:30.900 We always write the row number and then the column number. 00:01:30.900 --> 00:01:31.670 So I might -- 00:01:31.670 --> 00:01:37.370 I might -- maybe I take it C 3 4, just to make it specific. 00:01:37.370 --> 00:01:39.700 So instead of i j, let me use numbers. 00:01:39.700 --> 00:01:41.290 C 3 4. 00:01:41.290 --> 00:01:45.210 So where does that come from, the three four entry? 00:01:45.210 --> 00:01:53.750 It comes from row three, here, row three and column four, 00:01:53.750 --> 00:01:55.620 as you know. 00:01:55.620 --> 00:01:56.710 Column four. 00:01:56.710 --> 00:01:58.750 And can I just write down, or can we 00:01:58.750 --> 00:02:05.430 write down the formula for it? 00:02:05.430 --> 00:02:08.539 If we look at the whole row and the whole column, 00:02:08.539 --> 00:02:15.240 the quick way for me to say it is row three of A -- 00:02:15.240 --> 00:02:17.470 I could use a dot for dot product. 00:02:17.470 --> 00:02:19.740 I won't often use that, actually. 00:02:19.740 --> 00:02:25.790 Dot column four of B. 00:02:25.790 --> 00:02:28.370 But this gives us a chance to just, like, 00:02:28.370 --> 00:02:31.460 use a little matrix notation. 00:02:31.460 --> 00:02:32.500 What are the entries? 00:02:32.500 --> 00:02:38.440 What's this first entry in row three? 00:02:38.440 --> 00:02:42.160 That number that's sitting right there is... 00:02:42.160 --> 00:02:46.690 A, so it's got two indices and what are they? 00:02:46.690 --> 00:02:47.550 3 1. 00:02:47.550 --> 00:02:51.540 So there's an a 3 1 there. 00:02:51.540 --> 00:02:56.120 Now what's the first guy at the top of column four? 00:02:56.120 --> 00:02:58.430 So what's sitting up there? 00:02:58.430 --> 00:03:01.970 B 1 4, right. 00:03:01.970 --> 00:03:09.670 So that this dot product starts with A 3 1 times B 1 4. 00:03:09.670 --> 00:03:13.870 And then what's the next -- so this is like I'm accumulating 00:03:13.870 --> 00:03:20.710 this sum, then comes the next guy, A 3 2, second column, 00:03:20.710 --> 00:03:24.270 times B 2 4, second row. 00:03:24.270 --> 00:03:31.120 So it's b A 3 2, B 2 4 and so on. 00:03:31.120 --> 00:03:33.060 Just practice with indices. 00:03:33.060 --> 00:03:39.100 Oh, let me even practice with a summation formula. 00:03:39.100 --> 00:03:41.890 So this is -- 00:03:41.890 --> 00:03:44.910 most of the course, I use whole vectors. 00:03:44.910 --> 00:03:49.680 I very seldom, get down to the details 00:03:49.680 --> 00:03:53.070 of these particular entries, but here we'd better do it. 00:03:53.070 --> 00:03:56.710 So it's some kind of a sum, right? 00:03:56.710 --> 00:04:03.930 Of things in row three, column K shall I say? 00:04:03.930 --> 00:04:09.670 Times things in row K, column four. 00:04:09.670 --> 00:04:12.680 Do you see that that's what we're seeing here? 00:04:12.680 --> 00:04:17.470 This is K is one, here K is two, on along -- 00:04:17.470 --> 00:04:22.070 so the sum goes all the way along the row and down 00:04:22.070 --> 00:04:25.640 the column, say, one to N. 00:04:25.640 --> 00:04:29.990 So that's what the C three four entry looks like. 00:04:29.990 --> 00:04:34.280 A sum of a three K b K four. 00:04:34.280 --> 00:04:38.500 Just takes a little practice to do that. 00:04:38.500 --> 00:04:39.220 Okay. 00:04:39.220 --> 00:04:42.540 And -- well, maybe I should say -- 00:04:42.540 --> 00:04:45.480 when are we allowed to multiply these matrices? 00:04:45.480 --> 00:04:48.160 What are the shapes of these things? 00:04:48.160 --> 00:04:50.100 The shapes are -- 00:04:50.100 --> 00:04:54.895 if we allow them to be not necessarily square matrices. 00:04:54.895 --> 00:04:56.770 If they're square, they've got to be the same 00:04:56.770 --> 00:04:57.850 size. 00:04:57.850 --> 00:05:01.580 If they're rectangular, they're not the same size. 00:05:01.580 --> 00:05:04.560 If they're rectangular, this might be -- well, 00:05:04.560 --> 00:05:08.490 I always think of A as m by n. 00:05:08.490 --> 00:05:11.460 m rows, n columns. 00:05:11.460 --> 00:05:13.500 So that sum goes to n. 00:05:13.500 --> 00:05:19.220 Now what's the point -- how many rows does B have to have? 00:05:19.220 --> 00:05:19.780 n. 00:05:19.780 --> 00:05:22.930 The number of rows in B, the number of guys 00:05:22.930 --> 00:05:25.710 that we meet coming down has to match the number of ones 00:05:25.710 --> 00:05:26.610 across. 00:05:26.610 --> 00:05:30.750 So B will have to be n by something. 00:05:30.750 --> 00:05:31.360 Whatever. 00:05:31.360 --> 00:05:39.760 P. So the number of columns here has to match the number of rows 00:05:39.760 --> 00:05:41.610 there, and then what's the result? 00:05:41.610 --> 00:05:43.580 What's the shape of the result? 00:05:43.580 --> 00:05:45.770 What's the shape of C, the output? 00:05:45.770 --> 00:05:51.410 Well, it's got these same m rows -- it's got m rows. 00:05:51.410 --> 00:05:54.000 And how many columns? 00:05:54.000 --> 00:05:58.560 P. m by P. Okay. 00:05:58.560 --> 00:06:02.610 So there are m times P little numbers in there, entries, 00:06:02.610 --> 00:06:05.750 and each one, looks like that. 00:06:05.750 --> 00:06:06.850 Okay. 00:06:06.850 --> 00:06:08.990 So that's the standard rule. 00:06:08.990 --> 00:06:11.765 That's the way people think of multiplying matrices. 00:06:15.010 --> 00:06:17.810 I do it too. 00:06:17.810 --> 00:06:21.990 But I want to talk about other ways 00:06:21.990 --> 00:06:24.540 to look at that same calculation, 00:06:24.540 --> 00:06:30.170 looking at whole columns and whole rows. 00:06:30.170 --> 00:06:30.890 Okay. 00:06:30.890 --> 00:06:34.030 So can I do A B C again? 00:06:34.030 --> 00:06:35.670 A B equaling C again? 00:06:35.670 --> 00:06:38.490 But now, tell me about... 00:06:42.190 --> 00:06:43.040 I'll put it up here. 00:06:45.770 --> 00:06:58.560 So here goes A, again, times B producing C. 00:06:58.560 --> 00:07:01.560 And again, this is m by n. 00:07:01.560 --> 00:07:06.520 This is n by P and this is m by P. Okay. 00:07:06.520 --> 00:07:09.290 Now I want to look at whole columns. 00:07:09.290 --> 00:07:12.590 I want to look at the columns of -- 00:07:12.590 --> 00:07:16.480 here's the second way to multiply matrices. 00:07:16.480 --> 00:07:19.450 Because I'm going to build on what I know already. 00:07:19.450 --> 00:07:24.150 How do I multiply a matrix by a column? 00:07:24.150 --> 00:07:27.500 I know how to multiply this matrix by that column. 00:07:30.270 --> 00:07:33.250 Shall I call that column one? 00:07:33.250 --> 00:07:35.430 That tells me column one of the answer. 00:07:38.460 --> 00:07:42.890 The matrix times the first column is that first column. 00:07:42.890 --> 00:07:45.140 Because none of this stuff entered 00:07:45.140 --> 00:07:46.910 that part of the answer. 00:07:46.910 --> 00:07:49.550 The matrix times the second column 00:07:49.550 --> 00:07:51.770 is the second column of the answer. 00:07:51.770 --> 00:07:53.780 Do you see what I'm saying? 00:07:53.780 --> 00:07:57.270 That I could think of multiplying a matrix 00:07:57.270 --> 00:08:00.470 by a vector, which I already knew how to do, 00:08:00.470 --> 00:08:08.860 and I can think of just P columns sitting side by side, 00:08:08.860 --> 00:08:12.000 just like resting next to each other. 00:08:12.000 --> 00:08:15.800 And I multiply A times each one of those. 00:08:15.800 --> 00:08:19.650 And I get the P columns of the answer. 00:08:19.650 --> 00:08:21.970 Do you see this as -- this is quite nice, 00:08:21.970 --> 00:08:27.980 to be able to think, okay, matrix multiplication works 00:08:27.980 --> 00:08:31.710 so that I can just think of having several columns, 00:08:31.710 --> 00:08:35.090 multiplying by A and getting the columns of the answer. 00:08:35.090 --> 00:08:43.169 So, like, here's column one shall I call that column one? 00:08:43.169 --> 00:08:47.540 And what's going in there is A times column one. 00:08:53.310 --> 00:08:54.680 Okay. 00:08:54.680 --> 00:08:57.220 So that's the picture a column at a time. 00:08:57.220 --> 00:08:58.700 So what does that tell me? 00:08:58.700 --> 00:09:01.400 What does that tell me about these columns? 00:09:01.400 --> 00:09:09.580 These columns of C are combinations, 00:09:09.580 --> 00:09:23.410 because we've seen that before, of columns of A. 00:09:23.410 --> 00:09:27.890 Every one of these comes from A times this, 00:09:27.890 --> 00:09:30.720 and A times a vector is a combination 00:09:30.720 --> 00:09:34.390 of the columns of A. 00:09:34.390 --> 00:09:40.640 And it makes sense, because the columns of A have length m 00:09:40.640 --> 00:09:43.380 and the columns of C have length m. 00:09:43.380 --> 00:09:48.260 And every column of C is some combination 00:09:48.260 --> 00:09:49.440 of the columns of A. 00:09:49.440 --> 00:09:51.410 And it's these numbers in here that 00:09:51.410 --> 00:09:53.670 tell me what combination it is. 00:09:53.670 --> 00:09:57.760 Do you see that? 00:09:57.760 --> 00:10:01.970 That in that answer, C, I'm seeing stuff that's 00:10:01.970 --> 00:10:04.740 combinations of these columns. 00:10:04.740 --> 00:10:09.390 Now, suppose I look at it -- that's two ways now. 00:10:09.390 --> 00:10:12.950 The third way is look at it by rows. 00:10:12.950 --> 00:10:17.420 So now let me change to rows. 00:10:17.420 --> 00:10:18.590 Okay. 00:10:18.590 --> 00:10:23.320 So now I can think of a row of A -- 00:10:23.320 --> 00:10:29.210 a row of A multiplying all these rows here and producing a row 00:10:29.210 --> 00:10:32.210 of the product. 00:10:32.210 --> 00:10:36.160 So this row takes a combination of these rows 00:10:36.160 --> 00:10:37.840 and that's the answer. 00:10:37.840 --> 00:10:50.680 So these rows of C are combinations of what? 00:10:50.680 --> 00:10:54.870 Tell me how to finish that. 00:10:54.870 --> 00:11:01.000 The rows of C, when I have a matrix B, it's got its rows 00:11:01.000 --> 00:11:04.840 and I multiply by A, and what does that do? 00:11:04.840 --> 00:11:06.750 It mixes the rows up. 00:11:06.750 --> 00:11:13.660 It creates combinations of the rows of B, thanks. 00:11:13.660 --> 00:11:17.730 Rows of B. 00:11:17.730 --> 00:11:24.990 That's what I wanted to see, that this answer -- 00:11:24.990 --> 00:11:27.820 I can see where the pieces are coming from. 00:11:27.820 --> 00:11:31.330 The rows in the answer are coming as combinations of these 00:11:31.330 --> 00:11:32.090 rows. 00:11:32.090 --> 00:11:37.020 The columns in the answer are coming as combinations of those 00:11:37.020 --> 00:11:37.750 columns. 00:11:37.750 --> 00:11:39.920 And so that's three ways. 00:11:39.920 --> 00:11:44.620 Now you can say, okay, what's the fourth way? 00:11:44.620 --> 00:11:48.450 The fourth way -- 00:11:48.450 --> 00:11:52.440 so that's -- now we've got, like, the regular way, 00:11:52.440 --> 00:11:58.360 the column way, the row way and -- 00:11:58.360 --> 00:11:59.680 what's left? 00:11:59.680 --> 00:12:06.450 The one that I can -- 00:12:06.450 --> 00:12:12.110 well, one way is columns times rows. 00:12:12.110 --> 00:12:15.940 What happens if I multiply -- 00:12:15.940 --> 00:12:21.480 So this was row times column, it gave a number. 00:12:21.480 --> 00:12:21.980 Okay. 00:12:21.980 --> 00:12:25.670 Now I want to ask you about column times row. 00:12:25.670 --> 00:12:37.610 If I multiply a column of A times a row of B, 00:12:37.610 --> 00:12:40.460 what shape I ending up with? 00:12:40.460 --> 00:12:43.380 So if I take a column times a row, 00:12:43.380 --> 00:12:47.710 that's definitely different from taking a row times a column. 00:12:47.710 --> 00:12:52.730 So a column of A was -- what's the shape of a column of A? 00:12:52.730 --> 00:12:56.640 m by one. 00:12:56.640 --> 00:12:59.430 A column of A is a column. 00:12:59.430 --> 00:13:03.190 It's got m entries and one column. 00:13:03.190 --> 00:13:04.780 And what's a row of B? 00:13:04.780 --> 00:13:10.000 It's got one row and P columns. 00:13:10.000 --> 00:13:12.870 So what's the shape -- what do I get if I multiply a column 00:13:12.870 --> 00:13:13.590 by a row? 00:13:16.350 --> 00:13:18.170 I get a big matrix. 00:13:18.170 --> 00:13:20.540 I get a full-sized matrix. 00:13:20.540 --> 00:13:25.790 If I multiply a column by a row -- 00:13:25.790 --> 00:13:28.090 should we just do one? 00:13:28.090 --> 00:13:34.470 Let me take the column two three four times the row one six. 00:13:38.510 --> 00:13:40.825 That product there -- 00:13:40.825 --> 00:13:42.950 I mean, when I'm just following the rules of matrix 00:13:42.950 --> 00:13:46.490 multiplication, those rules are just looking like -- 00:13:46.490 --> 00:13:51.570 kind of petite, kind of small, because the rows here 00:13:51.570 --> 00:13:53.830 are so short and the columns there are so short, 00:13:53.830 --> 00:13:56.610 but they're the same length, one entry. 00:13:56.610 --> 00:13:57.530 So what's the answer? 00:14:00.270 --> 00:14:03.770 What's the answer if I do two three four times one six, just 00:14:03.770 --> 00:14:05.320 for practice? 00:14:05.320 --> 00:14:09.180 Well, what's the first row of the answer? 00:14:09.180 --> 00:14:11.030 Two twelve. 00:14:11.030 --> 00:14:16.420 And the second row of the answer is three eighteen. 00:14:16.420 --> 00:14:24.220 And the third row of the answer is four twenty four. 00:14:24.220 --> 00:14:29.400 That's a very special matrix, there. 00:14:29.400 --> 00:14:30.840 Very special matrix. 00:14:30.840 --> 00:14:32.830 What can you tell me about its columns, 00:14:32.830 --> 00:14:34.220 the columns of that matrix? 00:14:37.640 --> 00:14:41.410 They're multiples of this guy, right? 00:14:41.410 --> 00:14:42.690 They're multiples of that one. 00:14:42.690 --> 00:14:44.340 Which follows our rule. 00:14:44.340 --> 00:14:47.540 We said that the columns of the answer were combinations, 00:14:47.540 --> 00:14:50.300 but there's only -- to take a combination of one guy, 00:14:50.300 --> 00:14:52.300 it's just a multiple. 00:14:52.300 --> 00:14:54.000 The rows of the answer, what can you 00:14:54.000 --> 00:14:55.530 tell me about those three rows? 00:14:58.060 --> 00:15:01.480 They're all multiples of this row. 00:15:01.480 --> 00:15:04.660 They're all multiples of one six, as we expected. 00:15:04.660 --> 00:15:06.970 But I'm getting a full-sized matrix. 00:15:06.970 --> 00:15:15.800 And now, just to complete this thought, if I have -- 00:15:15.800 --> 00:15:17.280 let me write down the fourth way. 00:15:21.110 --> 00:15:35.630 A B is a sum of columns of A times rows of B. 00:15:35.630 --> 00:15:40.680 So that, for example, if my matrix was two three four 00:15:40.680 --> 00:15:47.450 and then had another column, say, seven eight nine, 00:15:47.450 --> 00:15:53.010 and my matrix here has -- say, started with one six and then 00:15:53.010 --> 00:16:00.890 had another column like zero zero, then -- 00:16:00.890 --> 00:16:04.630 here's the fourth way, okay? 00:16:04.630 --> 00:16:07.690 I've got two columns there, I've got two rows there. 00:16:07.690 --> 00:16:10.920 So the beautiful rule is -- 00:16:10.920 --> 00:16:13.080 see, the whole thing by columns and rows 00:16:13.080 --> 00:16:19.730 is that I can take the first column times the first row 00:16:19.730 --> 00:16:25.100 and add the second column times the second row. 00:16:32.690 --> 00:16:34.790 So that's the fourth way -- 00:16:34.790 --> 00:16:38.760 that I can take columns times rows, 00:16:38.760 --> 00:16:41.800 first column times first row, second column times second 00:16:41.800 --> 00:16:42.960 row and add. 00:16:42.960 --> 00:16:44.110 Actually, what will I get? 00:16:44.110 --> 00:16:46.765 What will the answer be for that matrix multiplication? 00:16:49.730 --> 00:16:52.130 Well, this one it's just going to give us zero, 00:16:52.130 --> 00:16:56.070 so in fact I'm back to this -- that's the answer, 00:16:56.070 --> 00:16:59.310 for that matrix multiplication. 00:16:59.310 --> 00:17:05.569 I'm happy to put up here these facts about matrix 00:17:05.569 --> 00:17:10.040 multiplication, because it gives me a chance to write down 00:17:10.040 --> 00:17:12.099 special matrices like this. 00:17:12.099 --> 00:17:15.050 This is a special matrix. 00:17:15.050 --> 00:17:18.569 All those rows lie on the same line. 00:17:18.569 --> 00:17:21.900 All those rows lie on the line through one six. 00:17:21.900 --> 00:17:25.240 If I draw a picture of all these row vectors, 00:17:25.240 --> 00:17:28.119 they're all the same direction. 00:17:28.119 --> 00:17:30.730 If I draw a picture of these two column vectors, 00:17:30.730 --> 00:17:33.900 they're in the same direction. 00:17:33.900 --> 00:17:37.430 Later, I would use this language. 00:17:37.430 --> 00:17:39.310 Not too much later, either. 00:17:39.310 --> 00:17:42.400 I would say the row space, which is 00:17:42.400 --> 00:17:44.380 like all the combinations of the rows, 00:17:44.380 --> 00:17:47.060 is just a line for this matrix. 00:17:47.060 --> 00:17:51.240 The row space is the line through the vector one six. 00:17:51.240 --> 00:17:54.390 All the rows lie on that line. 00:17:54.390 --> 00:17:57.710 And the column space is also a line. 00:17:57.710 --> 00:18:01.510 All the columns lie on the line through the vector two 00:18:01.510 --> 00:18:02.950 three four. 00:18:02.950 --> 00:18:07.340 So this is like a really minimal matrix. 00:18:07.340 --> 00:18:10.940 And it's because of these ones. 00:18:10.940 --> 00:18:11.440 Okay. 00:18:11.440 --> 00:18:18.580 So that's a third way. 00:18:18.580 --> 00:18:23.160 Now I want to say one more thing about matrix multiplication 00:18:23.160 --> 00:18:26.100 while we're on the subject. 00:18:26.100 --> 00:18:28.060 And it's this. 00:18:28.060 --> 00:18:29.740 You could also multiply -- 00:18:29.740 --> 00:18:34.390 You could also cut the matrix into blocks 00:18:34.390 --> 00:18:37.160 and do the multiplication by blocks. 00:18:37.160 --> 00:18:46.690 Yet that's actually so, useful that I want to mention it. 00:18:46.690 --> 00:18:47.745 Block multiplication. 00:18:50.540 --> 00:18:54.500 So I could take my matrix A and I could chop it up, like, 00:18:54.500 --> 00:18:58.460 maybe just for simplicity, let me chop it into two -- 00:18:58.460 --> 00:18:59.770 into four square blocks. 00:18:59.770 --> 00:19:00.980 Suppose it's square. 00:19:00.980 --> 00:19:03.990 Let's just take a nice case. 00:19:03.990 --> 00:19:07.800 And B, suppose it's square also, same size. 00:19:11.340 --> 00:19:13.460 So these sizes don't have to be the same. 00:19:13.460 --> 00:19:16.540 What they have to do is match properly. 00:19:16.540 --> 00:19:18.590 Here they certainly will match. 00:19:18.590 --> 00:19:22.340 So here's the rule for block multiplication, 00:19:22.340 --> 00:19:28.880 that if this has blocks like, A -- 00:19:28.880 --> 00:19:34.220 so maybe A1, A2, A3, A4 are the blocks here, 00:19:34.220 --> 00:19:38.210 and these blocks are B1, B2,3 and B4? 00:19:38.210 --> 00:19:44.770 Then the answer I can find block. 00:19:44.770 --> 00:19:46.500 And if you tell me what's in that block, 00:19:46.500 --> 00:19:49.560 then I'm going to be quiet about matrix multiplication 00:19:49.560 --> 00:19:51.870 for the rest of the day. 00:19:51.870 --> 00:19:54.920 What goes into that block? 00:19:54.920 --> 00:19:57.870 You see, these might be -- this matrix might be -- 00:19:57.870 --> 00:20:02.400 these matrices might be, like, twenty by twenty with blocks 00:20:02.400 --> 00:20:06.030 that are ten by ten, to take the easy case where all the blocks 00:20:06.030 --> 00:20:08.940 are the same shape. 00:20:08.940 --> 00:20:13.110 And the point is that I could multiply those by blocks. 00:20:13.110 --> 00:20:15.940 And what goes in here? 00:20:15.940 --> 00:20:20.920 What's that block in the answer? 00:20:20.920 --> 00:20:25.710 A1 B1, that's a matrix times a matrix, 00:20:25.710 --> 00:20:28.270 it's the right size, ten by ten. 00:20:28.270 --> 00:20:30.610 Any more? 00:20:30.610 --> 00:20:36.760 Plus, what else goes in there? 00:20:36.760 --> 00:20:38.160 A2 B3, right? 00:20:38.160 --> 00:20:41.490 It's just like block rows times block columns. 00:20:45.610 --> 00:20:49.070 Nobody, I think, not even Gauss could see instantly 00:20:49.070 --> 00:20:51.040 that it works. 00:20:51.040 --> 00:20:55.130 But somehow, if we check it through, all five ways 00:20:55.130 --> 00:20:58.020 we're doing the same multiplications. 00:20:58.020 --> 00:21:02.430 So this familiar multiplication is 00:21:02.430 --> 00:21:04.270 what we're really doing when we do it 00:21:04.270 --> 00:21:10.240 by columns, by rows by columns times rows and by blocks. 00:21:10.240 --> 00:21:10.740 Okay. 00:21:10.740 --> 00:21:13.680 I just have to, like, get the rules straight 00:21:13.680 --> 00:21:16.991 for matrix multiplication. 00:21:16.991 --> 00:21:17.490 Okay. 00:21:22.740 --> 00:21:24.490 All right, I'm ready for the second topic, 00:21:24.490 --> 00:21:27.590 which is inverses. 00:21:27.590 --> 00:21:28.479 Okay. 00:21:28.479 --> 00:21:29.270 Ready for inverses. 00:21:34.140 --> 00:21:39.040 And let me do it for square matrices first. 00:21:44.270 --> 00:21:44.770 Okay. 00:21:44.770 --> 00:21:52.270 So I've got a square matrix A. 00:21:52.270 --> 00:21:55.370 And it may or may not have an inverse, right? 00:21:55.370 --> 00:21:57.260 Not all matrices have inverses. 00:21:57.260 --> 00:22:01.770 In fact, that's the most important question you can ask 00:22:01.770 --> 00:22:06.260 about the matrix, is if it's -- if you know it's square, 00:22:06.260 --> 00:22:08.540 is it invertible or not? 00:22:08.540 --> 00:22:12.500 If it is invertible, then there is some other matrix, 00:22:12.500 --> 00:22:15.680 shall I call it A inverse? 00:22:15.680 --> 00:22:21.600 And what's the -- if A inverse exists -- 00:22:21.600 --> 00:22:24.270 there's a big "if" here. 00:22:24.270 --> 00:22:34.120 If this matrix exists, and it'll be really central to figure out 00:22:34.120 --> 00:22:35.940 when does it exist? 00:22:35.940 --> 00:22:40.350 And then if it does exist, how would you find it? 00:22:40.350 --> 00:22:45.710 But what's the equation here that I haven't -- 00:22:45.710 --> 00:22:47.900 that I have to finish now? 00:22:47.900 --> 00:22:53.280 This matrix, if it exists multiplies A and produces, 00:22:53.280 --> 00:22:54.755 I think, the identity. 00:23:10.840 --> 00:23:12.680 But a real -- 00:23:12.680 --> 00:23:17.600 an inverse for a square matrix could be on the right as well 00:23:17.600 --> 00:23:19.240 -- 00:23:19.240 --> 00:23:25.760 this is true, too, that it's -- 00:23:25.760 --> 00:23:28.280 if I have a -- yeah in fact, this is not -- 00:23:28.280 --> 00:23:31.570 this is probably the -- 00:23:31.570 --> 00:23:38.400 this is something that's not easy to prove, but it works. 00:23:38.400 --> 00:23:40.240 That a left -- 00:23:40.240 --> 00:23:43.670 square matrices, a left inverse is also a right 00:23:43.670 --> 00:23:44.570 inverse. 00:23:44.570 --> 00:23:49.090 If I can find a matrix on the left that gets the identity, 00:23:49.090 --> 00:23:50.990 then also that matrix on the right 00:23:50.990 --> 00:23:53.630 will produce that identity. 00:23:53.630 --> 00:23:57.310 For rectangular matrices, we'll see a left inverse 00:23:57.310 --> 00:23:58.950 that isn't a right inverse. 00:23:58.950 --> 00:24:01.660 In fact, the shapes wouldn't allow it. 00:24:01.660 --> 00:24:03.700 But for square matrices, the shapes 00:24:03.700 --> 00:24:09.820 allow it and it happens, if A has an inverse. 00:24:09.820 --> 00:24:14.011 Okay, so give me some cases -- 00:24:14.011 --> 00:24:14.510 let's see. 00:24:14.510 --> 00:24:17.090 I hate to be negative here, but let's talk 00:24:17.090 --> 00:24:20.070 about the case with no inverse. 00:24:20.070 --> 00:24:31.210 So -- these matrices are called invertible or non-singular -- 00:24:36.640 --> 00:24:38.960 those are the good ones. 00:24:38.960 --> 00:24:41.510 And we want to be able to identify how -- 00:24:41.510 --> 00:24:44.180 if we're given a matrix, has it got an inverse? 00:24:44.180 --> 00:24:46.950 Can I talk about the singular case? 00:24:52.210 --> 00:24:52.940 No inverse. 00:24:57.420 --> 00:24:58.950 All right. 00:24:58.950 --> 00:25:02.720 Best to start with an example. 00:25:02.720 --> 00:25:06.880 Tell me an example -- let's get an example up here. 00:25:06.880 --> 00:25:09.030 Let's make it two by two -- 00:25:09.030 --> 00:25:13.540 of a matrix that has not got an inverse. 00:25:13.540 --> 00:25:16.160 And let's see why. 00:25:16.160 --> 00:25:19.230 Let me write one up. 00:25:19.230 --> 00:25:20.950 No inverse. 00:25:20.950 --> 00:25:22.620 Let's see why. 00:25:22.620 --> 00:25:30.780 Let me write up -- one three two six. 00:25:35.180 --> 00:25:37.820 Why does that matrix have no inverse? 00:25:40.950 --> 00:25:43.245 You could answer that various ways. 00:25:45.760 --> 00:25:48.390 Give me one reason. 00:25:48.390 --> 00:25:51.620 Well, you could -- if you know about determinants, 00:25:51.620 --> 00:25:57.060 which you're not supposed to, you could take its determinant 00:25:57.060 --> 00:25:58.740 and you would get -- 00:25:58.740 --> 00:26:00.330 Zero. 00:26:00.330 --> 00:26:03.090 Okay. 00:26:03.090 --> 00:26:04.120 Now -- all right. 00:26:07.640 --> 00:26:10.330 Let me ask you other reasons. 00:26:10.330 --> 00:26:13.120 I mean, as for other reasons that that matrix 00:26:13.120 --> 00:26:15.810 isn't invertible. 00:26:15.810 --> 00:26:18.765 Here, I could use what I'm saying here. 00:26:24.550 --> 00:26:27.800 Suppose A times other matrix gave the identity. 00:26:32.060 --> 00:26:34.750 Why is that not possible? 00:26:34.750 --> 00:26:39.010 Because -- oh, yeah -- 00:26:39.010 --> 00:26:41.220 I'm thinking about columns here. 00:26:41.220 --> 00:26:46.200 If I multiply this matrix A by some other matrix, then the -- 00:26:46.200 --> 00:26:50.240 the result -- what can you tell me about the columns? 00:26:50.240 --> 00:26:55.730 They're all multiples of those columns, right? 00:26:55.730 --> 00:26:59.340 If I multiply A by another matrix that -- 00:26:59.340 --> 00:27:04.130 the product has columns that come from those columns. 00:27:04.130 --> 00:27:06.480 So can I get the identity matrix? 00:27:06.480 --> 00:27:08.120 No way. 00:27:08.120 --> 00:27:11.670 The columns of the identity matrix, like one zero -- 00:27:11.670 --> 00:27:14.740 it's not a combination of those columns, 00:27:14.740 --> 00:27:16.520 because those two columns lie on the -- 00:27:16.520 --> 00:27:19.140 both lie on the same line. 00:27:19.140 --> 00:27:22.140 Every combination is just going to be on that line 00:27:22.140 --> 00:27:24.480 and I can't get one zero. 00:27:24.480 --> 00:27:33.530 So, do you see that sort of column picture of the matrix 00:27:33.530 --> 00:27:34.750 not being invertible. 00:27:34.750 --> 00:27:37.580 In fact, here's another reason. 00:27:37.580 --> 00:27:40.720 This is even a more important reason. 00:27:40.720 --> 00:27:42.260 Well, how can I say more important? 00:27:42.260 --> 00:27:44.810 All those are important. 00:27:44.810 --> 00:27:47.910 This is another way to see it. 00:27:47.910 --> 00:27:51.410 A matrix has no inverse -- 00:27:51.410 --> 00:27:55.710 yeah -- here -- now this is important. 00:27:55.710 --> 00:27:59.180 A matrix has no -- a square matrix won't have an inverse 00:27:59.180 --> 00:28:07.980 if there's no inverse because I can solve -- 00:28:07.980 --> 00:28:20.810 I can find an X of -- a vector X with A times -- 00:28:20.810 --> 00:28:23.140 this A times X giving zero. 00:28:27.010 --> 00:28:31.310 This is the reason I like best. 00:28:31.310 --> 00:28:33.170 That matrix won't have an inverse. 00:28:33.170 --> 00:28:40.490 Can you -- well, let me change I to U. 00:28:40.490 --> 00:28:46.780 So tell me a vector X that, solves A X equals zero. 00:28:46.780 --> 00:28:49.610 I mean, this is, like, the key equation. 00:28:49.610 --> 00:28:51.310 In mathematics, all the key equations 00:28:51.310 --> 00:28:53.490 have zero on the right-hand side. 00:28:53.490 --> 00:28:55.490 So what's the X? 00:28:55.490 --> 00:28:58.290 Tell me an X here -- 00:28:58.290 --> 00:29:01.360 so now I'm going to put -- slip in the X that you tell me 00:29:01.360 --> 00:29:05.500 and I'm going to get zero. 00:29:05.500 --> 00:29:09.000 What X would do that job? 00:29:09.000 --> 00:29:11.370 Three and negative one? 00:29:11.370 --> 00:29:13.610 Is that the one you picked, or -- yeah. 00:29:13.610 --> 00:29:18.720 Or another -- well, if you picked zero with zero, 00:29:18.720 --> 00:29:21.060 I'm not so excited, right? 00:29:21.060 --> 00:29:23.490 Because that would always work. 00:29:23.490 --> 00:29:26.550 So it's really the fact that this vector 00:29:26.550 --> 00:29:28.840 isn't zero that's important. 00:29:28.840 --> 00:29:33.750 It's a non-zero vector and three negative one would do it. 00:29:33.750 --> 00:29:37.130 That just says three of this column minus one of that column 00:29:37.130 --> 00:29:38.861 is the zero column. 00:29:38.861 --> 00:29:39.360 Okay. 00:29:42.240 --> 00:29:46.860 So now I know that A couldn't be invertible. 00:29:46.860 --> 00:29:49.890 But what's the reasoning? 00:29:49.890 --> 00:29:54.720 If A X is zero, suppose I multiplied by A inverse. 00:29:54.720 --> 00:29:56.660 Yeah, well here's the reason. 00:29:56.660 --> 00:30:02.180 Here -- this is why this spells disaster for an inverse. 00:30:02.180 --> 00:30:06.580 The matrix can't have an inverse if some combination 00:30:06.580 --> 00:30:09.140 of the columns gives z- it gives nothing. 00:30:09.140 --> 00:30:12.630 Because, I could take A X equals zero, 00:30:12.630 --> 00:30:21.200 I could multiply by A inverse and what would I discover? 00:30:21.200 --> 00:30:23.840 Suppose I take that equation and I multiply by -- 00:30:23.840 --> 00:30:26.860 if A inverse existed, which of course I'm going to come 00:30:26.860 --> 00:30:30.560 to the conclusion it can't because if it existed, 00:30:30.560 --> 00:30:33.570 if there was an A inverse to this dopey matrix, 00:30:33.570 --> 00:30:36.990 I would multiply that equation by that inverse and I would 00:30:36.990 --> 00:30:42.270 discover X is zero. 00:30:42.270 --> 00:30:45.320 If I multiply A by A inverse on the left, I get X. 00:30:45.320 --> 00:30:48.830 If I multiply by A inverse on the right, I get zero. 00:30:48.830 --> 00:30:50.790 So I would discover X was zero. 00:30:50.790 --> 00:30:53.050 But it -- X is not zero. 00:30:53.050 --> 00:30:54.800 X -- this guy wasn't zero. 00:30:54.800 --> 00:30:55.460 There it is. 00:30:55.460 --> 00:30:58.170 It's three minus one. 00:30:58.170 --> 00:31:06.250 So, conclusion -- only, it takes us some time to really work 00:31:06.250 --> 00:31:08.040 with that conclusion -- 00:31:08.040 --> 00:31:14.300 our conclusion will be that non-invertible matrices, 00:31:14.300 --> 00:31:19.490 singular matrices, some combinations of their columns 00:31:19.490 --> 00:31:22.290 gives the zero column. 00:31:22.290 --> 00:31:26.090 They they take some vector X into zero. 00:31:26.090 --> 00:31:30.650 And there's no way A inverse can recover, right? 00:31:30.650 --> 00:31:32.520 That's what this equation says. 00:31:32.520 --> 00:31:36.980 This equation says I take this vector X and multiplying 00:31:36.980 --> 00:31:39.470 by A gives zero. 00:31:39.470 --> 00:31:42.220 But then when I multiply by A inverse, 00:31:42.220 --> 00:31:44.690 I can never escape from zero. 00:31:44.690 --> 00:31:47.980 So there couldn't be an A inverse. 00:31:47.980 --> 00:31:51.701 Where here -- okay, now fix -- 00:31:51.701 --> 00:31:52.200 all right. 00:31:52.200 --> 00:31:57.200 Now let me take -- all right, back to the positive side. 00:31:57.200 --> 00:32:01.530 Let's take a matrix that does have an inverse. 00:32:01.530 --> 00:32:03.840 And why not invert it? 00:32:03.840 --> 00:32:04.500 Okay. 00:32:04.500 --> 00:32:08.782 Can I -- so let me take on this third board a matrix -- 00:32:08.782 --> 00:32:09.990 shall I fix that up a little? 00:32:12.810 --> 00:32:15.660 Tell me a matrix that has got an inverse. 00:32:18.680 --> 00:32:22.930 Well, let me say one three two -- what shall I put there? 00:32:25.620 --> 00:32:28.665 Well, don't put six, I guess is -- right? 00:32:31.570 --> 00:32:35.000 Do I any favorites here? 00:32:35.000 --> 00:32:36.910 One? 00:32:36.910 --> 00:32:38.280 Or eight? 00:32:41.470 --> 00:32:42.140 I don't care. 00:32:42.140 --> 00:32:43.450 What, seven? 00:32:43.450 --> 00:32:43.950 Seven. 00:32:43.950 --> 00:32:44.540 Okay. 00:32:44.540 --> 00:32:46.060 Seven is a lucky number. 00:32:46.060 --> 00:32:48.750 All right, seven, okay. 00:32:48.750 --> 00:32:49.380 Okay. 00:32:49.380 --> 00:32:51.080 So -- now what's our idea? 00:32:51.080 --> 00:32:53.740 We believe that this matrix is invertible. 00:32:53.740 --> 00:32:57.060 Those who like determinants have quickly taken its determinant 00:32:57.060 --> 00:32:59.320 and found it wasn't zero. 00:32:59.320 --> 00:33:04.610 Those who like columns, and probably that -- 00:33:04.610 --> 00:33:08.450 that department is not totally popular yet -- 00:33:08.450 --> 00:33:11.080 but those who like columns will look at those two columns 00:33:11.080 --> 00:33:15.090 and say, hey, they point in different directions. 00:33:15.090 --> 00:33:18.080 So I can get anything. 00:33:18.080 --> 00:33:19.860 Now, let me see, what do I mean? 00:33:19.860 --> 00:33:21.880 How I going to computer A inverse? 00:33:21.880 --> 00:33:24.060 So A inverse -- 00:33:24.060 --> 00:33:28.360 here's A inverse, now, and I have to find it. 00:33:28.360 --> 00:33:33.219 And what do I get when I do this multiplication? 00:33:33.219 --> 00:33:33.760 The identity. 00:33:40.810 --> 00:33:43.940 You know, forgive me for taking two by two-s, but -- 00:33:43.940 --> 00:33:49.240 lt's good to keep the computations manageable and let 00:33:49.240 --> 00:33:51.090 the ideas come out. 00:33:51.090 --> 00:33:55.230 Okay, now what's the idea I want? 00:33:55.230 --> 00:33:57.520 I'm looking for this matrix A inverse, how 00:33:57.520 --> 00:33:58.750 I going to find it? 00:33:58.750 --> 00:34:04.235 Right now, I've got four numbers to find. 00:34:08.290 --> 00:34:12.159 I'm going to look at the first column. 00:34:12.159 --> 00:34:16.860 Let me take this first column, A B. 00:34:16.860 --> 00:34:18.699 What's up there? 00:34:18.699 --> 00:34:20.670 What -- tell me this. 00:34:20.670 --> 00:34:25.260 What equation does the first column satisfy? 00:34:25.260 --> 00:34:31.489 The first column satisfies A times that column is one zero. 00:34:31.489 --> 00:34:34.179 The first column of the answer. 00:34:34.179 --> 00:34:38.820 And the second column, C D, satisfies A times 00:34:38.820 --> 00:34:41.139 that second column is zero one. 00:34:41.139 --> 00:34:48.909 You see that finding the inverse is like solving two systems. 00:34:48.909 --> 00:34:52.210 One system, when the right-hand side is one zero -- 00:34:52.210 --> 00:34:54.139 I'm just going to split it into two pieces. 00:34:57.427 --> 00:34:58.760 I don't even need to rewrite it. 00:34:58.760 --> 00:35:04.480 I can take A times -- so let me put it here. 00:35:04.480 --> 00:35:18.100 A times column j of A inverse is column j of the identity. 00:35:21.350 --> 00:35:23.010 I've got n equations. 00:35:23.010 --> 00:35:26.500 I've got, well, two in this case. 00:35:26.500 --> 00:35:29.100 And they have the same matrix, A, 00:35:29.100 --> 00:35:30.860 but they have different right-hand sides. 00:35:30.860 --> 00:35:32.840 The right-hand sides are just the columns 00:35:32.840 --> 00:35:35.800 of the identity, this guy and this guy. 00:35:35.800 --> 00:35:37.450 And these are the two solutions. 00:35:37.450 --> 00:35:39.610 Do you see what I'm going -- 00:35:39.610 --> 00:35:45.120 I'm looking at that equation by columns. 00:35:45.120 --> 00:35:47.050 I'm looking at A times this column, 00:35:47.050 --> 00:35:49.910 giving that guy, and A times that column giving that guy. 00:35:49.910 --> 00:35:55.500 So -- Essentially -- so this is like the Gauss -- 00:35:55.500 --> 00:35:56.410 we're back to Gauss. 00:35:56.410 --> 00:36:00.820 We're back to solving systems of equations, but we're solving -- 00:36:00.820 --> 00:36:05.160 we've got two right-hand sides instead of one. 00:36:05.160 --> 00:36:07.010 That's where Jordan comes in. 00:36:07.010 --> 00:36:09.980 So at the very beginning of the lecture, 00:36:09.980 --> 00:36:13.230 I mentioned Gauss-Jordan, let me write it up again. 00:36:13.230 --> 00:36:13.730 Okay. 00:36:13.730 --> 00:36:16.630 Here's the Gauss-Jordan idea. 00:36:21.600 --> 00:36:35.000 Gauss-Jordan solve two equations at once. 00:36:39.460 --> 00:36:39.960 Okay. 00:36:39.960 --> 00:36:43.630 Let me show you how the mechanics go. 00:36:43.630 --> 00:36:48.970 How do I solve a single equation? 00:36:48.970 --> 00:36:55.760 So the two equations are one three two seven, 00:36:55.760 --> 00:37:01.630 multiplying A B gives one zero. 00:37:01.630 --> 00:37:03.560 And the other equation is the same 00:37:03.560 --> 00:37:12.130 one three two seven multiplying C D gives zero one. 00:37:12.130 --> 00:37:15.010 Okay. 00:37:15.010 --> 00:37:17.410 That'll tell me the two columns of the inverse. 00:37:17.410 --> 00:37:19.080 I'll have inverse. 00:37:19.080 --> 00:37:22.940 In other words, if I can solve with this matrix A, 00:37:22.940 --> 00:37:24.850 if I can solve with that right-hand side 00:37:24.850 --> 00:37:28.340 and that right-hand side, I'm invertible. 00:37:28.340 --> 00:37:29.460 I've got it. 00:37:29.460 --> 00:37:30.690 Okay. 00:37:30.690 --> 00:37:36.410 And Jordan sort of said to Gauss, solve them together, 00:37:36.410 --> 00:37:39.650 look at the matrix -- if we just solve this one, 00:37:39.650 --> 00:37:43.140 I would look at one three two seven, 00:37:43.140 --> 00:37:46.030 and how do I deal with the right-hand side? 00:37:46.030 --> 00:37:48.960 I stick it on as an extra column, right? 00:37:56.130 --> 00:37:58.460 That's this augmented matrix. 00:37:58.460 --> 00:38:01.750 That's the matrix when I'm watching the right-hand side 00:38:01.750 --> 00:38:04.470 at the same time, doing the same thing to the right side 00:38:04.470 --> 00:38:06.520 that I do to the left? 00:38:06.520 --> 00:38:09.610 So I just carry it along as an extra column. 00:38:09.610 --> 00:38:12.100 Now I'm going to carry along two extra columns. 00:38:17.200 --> 00:38:20.920 And I'm going to do whatever Gauss wants, right? 00:38:20.920 --> 00:38:24.150 I'm going to do elimination. 00:38:24.150 --> 00:38:27.010 I'm going to get this to be simple 00:38:27.010 --> 00:38:30.070 and this thing will turn into the inverse. 00:38:30.070 --> 00:38:32.460 This is what's coming. 00:38:32.460 --> 00:38:35.800 I'm going to do elimination steps to make this 00:38:35.800 --> 00:38:39.240 into the identity, and lo and behold, 00:38:39.240 --> 00:38:41.550 the inverse will show up here. 00:38:41.550 --> 00:38:43.330 K--- let's do it. 00:38:43.330 --> 00:38:43.830 Okay. 00:38:43.830 --> 00:38:46.880 So what are the elimination steps? 00:38:46.880 --> 00:38:51.680 So you see -- here's my matrix A and here's the identity, like, 00:38:51.680 --> 00:38:52.795 stuck on, augmented on. 00:38:52.795 --> 00:38:53.670 STUDENT: I'm sorry... 00:38:53.670 --> 00:38:54.211 STRANG: Yeah? 00:38:54.211 --> 00:38:58.800 STUDENT: -- is the two and the three supposed to be switched? 00:38:58.800 --> 00:39:01.070 STRANG: Did I -- oh, no, they weren't supposed to be 00:39:01.070 --> 00:39:01.570 switched. 00:39:01.570 --> 00:39:02.070 Sorry. 00:39:02.070 --> 00:39:04.680 Thanks. 00:39:04.680 --> 00:39:05.930 Okay. 00:39:05.930 --> 00:39:08.425 Thank you very much. 00:39:08.425 --> 00:39:09.800 And there -- I've got them right. 00:39:09.800 --> 00:39:11.215 Okay, thanks. 00:39:14.520 --> 00:39:15.510 Okay. 00:39:15.510 --> 00:39:17.780 So let's do elimination. 00:39:17.780 --> 00:39:19.890 All right, it's going to be simple, right? 00:39:19.890 --> 00:39:23.690 So I take two of this row away from this row. 00:39:23.690 --> 00:39:27.180 So this row stays the same and two 00:39:27.180 --> 00:39:28.680 of those come away from this. 00:39:28.680 --> 00:39:32.560 That leaves me with a zero and a one and two of these away from 00:39:32.560 --> 00:39:36.970 this is that what you're getting -- 00:39:36.970 --> 00:39:39.930 after one elimination step -- 00:39:39.930 --> 00:39:42.790 Let me sort of separate the -- 00:39:42.790 --> 00:39:45.040 the left half from the right half. 00:39:45.040 --> 00:39:48.690 So two of that first row got subtracted from the second row. 00:39:48.690 --> 00:39:53.240 Now this is an upper triangular form. 00:39:53.240 --> 00:39:57.120 Gauss would quit, but Jordan says keeps going. 00:39:57.120 --> 00:39:59.250 Use elimination upwards. 00:39:59.250 --> 00:40:03.710 Subtract a multiple of equation two from equation one 00:40:03.710 --> 00:40:05.970 to get rid of the three. 00:40:05.970 --> 00:40:08.970 So let's go the whole way. 00:40:08.970 --> 00:40:14.420 So now I'm going to -- this guy is fine, but I'm going to -- 00:40:14.420 --> 00:40:15.770 what do I do now? 00:40:15.770 --> 00:40:19.850 What's my final step that produces the inverse? 00:40:19.850 --> 00:40:22.200 I multiply this by the right number 00:40:22.200 --> 00:40:26.060 to get up to ther to remove that three. 00:40:26.060 --> 00:40:28.350 So I guess, I -- since this is a one, 00:40:28.350 --> 00:40:30.830 there's the pivot sitting there. 00:40:30.830 --> 00:40:33.750 I multiply it by three and subtract from that, 00:40:33.750 --> 00:40:35.080 so what do I get? 00:40:35.080 --> 00:40:38.460 I'll have one zero -- oh, yeah that was my whole point. 00:40:38.460 --> 00:40:41.640 I'll multiply this by three and subtract from that, 00:40:41.640 --> 00:40:46.450 which will give me seven. 00:40:46.450 --> 00:40:49.340 And I multiply this by three and subtract from that, 00:40:49.340 --> 00:40:50.920 which gives me a minus three. 00:41:00.060 --> 00:41:06.500 And what's my hope, belief? 00:41:06.500 --> 00:41:10.780 Here I started with A and the identity, 00:41:10.780 --> 00:41:16.410 and I ended up with the identity and who? 00:41:16.410 --> 00:41:18.640 That better be A inverse. 00:41:24.090 --> 00:41:26.980 That's the Gauss Jordan idea. 00:41:26.980 --> 00:41:33.690 Start with this long matrix, double-length A I, eliminate, 00:41:33.690 --> 00:41:37.450 eliminate until this part is down to I, 00:41:37.450 --> 00:41:40.530 then this one will -- must be for some reason, 00:41:40.530 --> 00:41:45.230 and we've got to find the reason -- must be A inverse. 00:41:45.230 --> 00:41:46.690 Shall I just check that it works? 00:41:50.160 --> 00:41:55.080 Let me just check that -- can I multiply this matrix this part 00:41:55.080 --> 00:42:00.300 times A, I'll carry A over here and just do that 00:42:00.300 --> 00:42:01.190 multiplication. 00:42:01.190 --> 00:42:03.310 You'll see I'll do it the old fashioned way. 00:42:03.310 --> 00:42:06.000 Seven minus six is a one. 00:42:06.000 --> 00:42:08.670 Twenty one minus twenty one is a zero, 00:42:08.670 --> 00:42:13.280 minus two plus two is a zero, minus six plus seven is a one. 00:42:13.280 --> 00:42:15.780 Check. 00:42:15.780 --> 00:42:18.290 So that is the inverse. 00:42:18.290 --> 00:42:20.950 That's the Gauss-Jordan idea. 00:42:20.950 --> 00:42:24.190 So, you'll -- one of the homework problems or more than 00:42:24.190 --> 00:42:30.710 one for Wednesday will ask you to go through those steps. 00:42:30.710 --> 00:42:33.560 I think you just got to go through Gauss-Jordan a couple 00:42:33.560 --> 00:42:38.600 of times, but I -- 00:42:38.600 --> 00:42:43.690 yeah -- just to see the mechanics. 00:42:43.690 --> 00:42:48.000 But the, important thing is, why -- 00:42:48.000 --> 00:42:50.190 is, like, what happened? 00:42:50.190 --> 00:42:53.570 Why did we -- why did we get A inverse there? 00:42:53.570 --> 00:42:54.690 Let me ask you that. 00:42:57.860 --> 00:43:03.190 We got -- so we take -- 00:43:03.190 --> 00:43:08.020 We do row reduction, we do elimination on this long matrix 00:43:08.020 --> 00:43:12.840 A I until the first half 00:43:12.840 --> 00:43:14.990 Then a second half is A inverse. is up. 00:43:14.990 --> 00:43:20.050 Well, how do I see that? 00:43:20.050 --> 00:43:22.310 Let me put up here how I see that. 00:43:22.310 --> 00:43:30.100 So here's my Gauss-Jordan thing, and I'm doing stuff to it. 00:43:30.100 --> 00:43:34.320 So I'm -- well, whole lot of E's. 00:43:37.530 --> 00:43:40.150 Remember those are those elimination matrices. 00:43:40.150 --> 00:43:42.861 Those are the -- those are the things that we figured out last 00:43:42.861 --> 00:43:43.360 time. 00:43:43.360 --> 00:43:49.370 Yes, that's what an elimination step is it's in matrix form, 00:43:49.370 --> 00:43:51.430 I'm multiplying by some Es. 00:43:51.430 --> 00:43:55.120 And the result -- well, so I'm multiplying by a whole bunch 00:43:55.120 --> 00:43:55.650 of Es. 00:43:55.650 --> 00:43:57.530 So, I get a -- 00:43:57.530 --> 00:44:02.070 can I call the overall matrix E? 00:44:02.070 --> 00:44:06.420 That's the elimination matrix, the product of all those little 00:44:06.420 --> 00:44:06.920 pieces. 00:44:06.920 --> 00:44:09.050 What do I mean by little pieces? 00:44:09.050 --> 00:44:11.160 Well, there was an elimination matrix 00:44:11.160 --> 00:44:14.400 that subtracted two of that away from that. 00:44:14.400 --> 00:44:16.180 Then there was an elimination matrix 00:44:16.180 --> 00:44:19.160 that subtracted three of that away from that. 00:44:19.160 --> 00:44:21.370 I guess in this case, that was all. 00:44:21.370 --> 00:44:24.280 So there were just two Es in this case, one 00:44:24.280 --> 00:44:26.630 that did this step and one that did this step 00:44:26.630 --> 00:44:31.690 and together they gave me an E that does both steps. 00:44:31.690 --> 00:44:38.560 And the net result was to get an I here. 00:44:38.560 --> 00:44:45.490 And you can tell me what that has to be. 00:44:45.490 --> 00:44:50.240 This is, like, the picture of what happened. 00:44:50.240 --> 00:44:53.350 If E multiplied A, whatever that E is -- 00:44:53.350 --> 00:44:59.780 we never figured it out in this way. 00:44:59.780 --> 00:45:06.692 But whatever that E times that E is, E times A is -- 00:45:09.410 --> 00:45:12.120 What's E times A? 00:45:12.120 --> 00:45:13.480 It's I. 00:45:13.480 --> 00:45:20.510 That E, whatever the heck it was, multiplied A and produced 00:45:20.510 --> 00:45:22.710 So E must be -- 00:45:22.710 --> 00:45:30.640 E A equaling I tells us what E is, I. namely it is -- 00:45:30.640 --> 00:45:32.110 STUDENT: It's the inverse of A. 00:45:32.110 --> 00:45:33.720 STRANG: It's the inverse of A. 00:45:33.720 --> 00:45:35.660 Great. 00:45:35.660 --> 00:45:39.780 And therefore, when the second half, when E multiplies I, 00:45:39.780 --> 00:45:41.470 it's E -- 00:45:41.470 --> 00:45:44.760 Put this A inverse. 00:45:44.760 --> 00:45:48.450 You see the picture looking that way? 00:45:48.450 --> 00:45:49.910 E times A is the identity. 00:45:49.910 --> 00:45:53.480 It tells us what E has to be. 00:45:53.480 --> 00:45:55.600 It has to be the inverse, and therefore, 00:45:55.600 --> 00:45:57.870 on the right-hand side, where E -- 00:45:57.870 --> 00:46:00.580 where we just smartly tucked on the identity, 00:46:00.580 --> 00:46:03.060 it's turning in, step by step -- 00:46:03.060 --> 00:46:05.870 It's turning into A inverse. 00:46:05.870 --> 00:46:12.410 There is the statement of Gauss-Jordan elimination. 00:46:12.410 --> 00:46:14.590 That's how you find the inverse. 00:46:14.590 --> 00:46:19.210 Where we can look at it as elimination, 00:46:19.210 --> 00:46:24.260 as solving n equations at the same time -- -- 00:46:24.260 --> 00:46:28.170 and tacking on n columns, solving those equations and up 00:46:28.170 --> 00:46:32.740 goes the n columns of A inverse Okay, thanks. 00:46:32.740 --> 00:46:34.640 See you on Wednesday.