1 00:00:08,160 --> 00:00:13,730 I've been multiplying matrices already, 2 00:00:13,730 --> 00:00:17,640 but certainly time for me to discuss the rules 3 00:00:17,640 --> 00:00:19,390 for matrix multiplication. 4 00:00:19,390 --> 00:00:24,980 And the interesting part is the many ways you can do it, 5 00:00:24,980 --> 00:00:27,970 and they all give the same answer. 6 00:00:27,970 --> 00:00:29,780 And they're all important. 7 00:00:29,780 --> 00:00:33,840 So matrix multiplication, and then, come inverses. 8 00:00:33,840 --> 00:00:38,520 So we mentioned the inverse of a matrix. 9 00:00:38,520 --> 00:00:39,660 That's a big deal. 10 00:00:39,660 --> 00:00:43,400 Lots to do about inverses and how to find them. 11 00:00:43,400 --> 00:00:48,420 Okay, so I'll begin with how to multiply two matrices. 12 00:00:48,420 --> 00:00:57,330 First way, okay, so suppose I have a matrix A multiplying 13 00:00:57,330 --> 00:01:03,500 a matrix B and -- giving me a result -- 14 00:01:03,500 --> 00:01:05,990 well, I could call it C. 15 00:01:05,990 --> 00:01:07,870 A times B. 16 00:01:07,870 --> 00:01:10,340 Okay. 17 00:01:10,340 --> 00:01:18,360 So, let me just review the rule for this entry. 18 00:01:18,360 --> 00:01:22,000 That's the entry in row i and column j. 19 00:01:22,000 --> 00:01:24,340 So that's the i j entry. 20 00:01:24,340 --> 00:01:27,350 Right there is C i j. 21 00:01:27,350 --> 00:01:30,900 We always write the row number and then the column number. 22 00:01:30,900 --> 00:01:31,670 So I might -- 23 00:01:31,670 --> 00:01:37,370 I might -- maybe I take it C 3 4, just to make it specific. 24 00:01:37,370 --> 00:01:39,700 So instead of i j, let me use numbers. 25 00:01:39,700 --> 00:01:41,290 C 3 4. 26 00:01:41,290 --> 00:01:45,210 So where does that come from, the three four entry? 27 00:01:45,210 --> 00:01:53,750 It comes from row three, here, row three and column four, 28 00:01:53,750 --> 00:01:55,620 as you know. 29 00:01:55,620 --> 00:01:56,710 Column four. 30 00:01:56,710 --> 00:01:58,750 And can I just write down, or can we 31 00:01:58,750 --> 00:02:05,430 write down the formula for it? 32 00:02:05,430 --> 00:02:08,539 If we look at the whole row and the whole column, 33 00:02:08,539 --> 00:02:15,240 the quick way for me to say it is row three of A -- 34 00:02:15,240 --> 00:02:17,470 I could use a dot for dot product. 35 00:02:17,470 --> 00:02:19,740 I won't often use that, actually. 36 00:02:19,740 --> 00:02:25,790 Dot column four of B. 37 00:02:25,790 --> 00:02:28,370 But this gives us a chance to just, like, 38 00:02:28,370 --> 00:02:31,460 use a little matrix notation. 39 00:02:31,460 --> 00:02:32,500 What are the entries? 40 00:02:32,500 --> 00:02:38,440 What's this first entry in row three? 41 00:02:38,440 --> 00:02:42,160 That number that's sitting right there is... 42 00:02:42,160 --> 00:02:46,690 A, so it's got two indices and what are they? 43 00:02:46,690 --> 00:02:47,550 3 1. 44 00:02:47,550 --> 00:02:51,540 So there's an a 3 1 there. 45 00:02:51,540 --> 00:02:56,120 Now what's the first guy at the top of column four? 46 00:02:56,120 --> 00:02:58,430 So what's sitting up there? 47 00:02:58,430 --> 00:03:01,970 B 1 4, right. 48 00:03:01,970 --> 00:03:09,670 So that this dot product starts with A 3 1 times B 1 4. 49 00:03:09,670 --> 00:03:13,870 And then what's the next -- so this is like I'm accumulating 50 00:03:13,870 --> 00:03:20,710 this sum, then comes the next guy, A 3 2, second column, 51 00:03:20,710 --> 00:03:24,270 times B 2 4, second row. 52 00:03:24,270 --> 00:03:31,120 So it's b A 3 2, B 2 4 and so on. 53 00:03:31,120 --> 00:03:33,060 Just practice with indices. 54 00:03:33,060 --> 00:03:39,100 Oh, let me even practice with a summation formula. 55 00:03:39,100 --> 00:03:41,890 So this is -- 56 00:03:41,890 --> 00:03:44,910 most of the course, I use whole vectors. 57 00:03:44,910 --> 00:03:49,680 I very seldom, get down to the details 58 00:03:49,680 --> 00:03:53,070 of these particular entries, but here we'd better do it. 59 00:03:53,070 --> 00:03:56,710 So it's some kind of a sum, right? 60 00:03:56,710 --> 00:04:03,930 Of things in row three, column K shall I say? 61 00:04:03,930 --> 00:04:09,670 Times things in row K, column four. 62 00:04:09,670 --> 00:04:12,680 Do you see that that's what we're seeing here? 63 00:04:12,680 --> 00:04:17,470 This is K is one, here K is two, on along -- 64 00:04:17,470 --> 00:04:22,070 so the sum goes all the way along the row and down 65 00:04:22,070 --> 00:04:25,640 the column, say, one to N. 66 00:04:25,640 --> 00:04:29,990 So that's what the C three four entry looks like. 67 00:04:29,990 --> 00:04:34,280 A sum of a three K b K four. 68 00:04:34,280 --> 00:04:38,500 Just takes a little practice to do that. 69 00:04:38,500 --> 00:04:39,220 Okay. 70 00:04:39,220 --> 00:04:42,540 And -- well, maybe I should say -- 71 00:04:42,540 --> 00:04:45,480 when are we allowed to multiply these matrices? 72 00:04:45,480 --> 00:04:48,160 What are the shapes of these things? 73 00:04:48,160 --> 00:04:50,100 The shapes are -- 74 00:04:50,100 --> 00:04:54,895 if we allow them to be not necessarily square matrices. 75 00:04:54,895 --> 00:04:56,770 If they're square, they've got to be the same 76 00:04:56,770 --> 00:04:57,850 size. 77 00:04:57,850 --> 00:05:01,580 If they're rectangular, they're not the same size. 78 00:05:01,580 --> 00:05:04,560 If they're rectangular, this might be -- well, 79 00:05:04,560 --> 00:05:08,490 I always think of A as m by n. 80 00:05:08,490 --> 00:05:11,460 m rows, n columns. 81 00:05:11,460 --> 00:05:13,500 So that sum goes to n. 82 00:05:13,500 --> 00:05:19,220 Now what's the point -- how many rows does B have to have? 83 00:05:19,220 --> 00:05:19,780 n. 84 00:05:19,780 --> 00:05:22,930 The number of rows in B, the number of guys 85 00:05:22,930 --> 00:05:25,710 that we meet coming down has to match the number of ones 86 00:05:25,710 --> 00:05:26,610 across. 87 00:05:26,610 --> 00:05:30,750 So B will have to be n by something. 88 00:05:30,750 --> 00:05:31,360 Whatever. 89 00:05:31,360 --> 00:05:39,760 P. So the number of columns here has to match the number of rows 90 00:05:39,760 --> 00:05:41,610 there, and then what's the result? 91 00:05:41,610 --> 00:05:43,580 What's the shape of the result? 92 00:05:43,580 --> 00:05:45,770 What's the shape of C, the output? 93 00:05:45,770 --> 00:05:51,410 Well, it's got these same m rows -- it's got m rows. 94 00:05:51,410 --> 00:05:54,000 And how many columns? 95 00:05:54,000 --> 00:05:58,560 P. m by P. Okay. 96 00:05:58,560 --> 00:06:02,610 So there are m times P little numbers in there, entries, 97 00:06:02,610 --> 00:06:05,750 and each one, looks like that. 98 00:06:05,750 --> 00:06:06,850 Okay. 99 00:06:06,850 --> 00:06:08,990 So that's the standard rule. 100 00:06:08,990 --> 00:06:11,765 That's the way people think of multiplying matrices. 101 00:06:15,010 --> 00:06:17,810 I do it too. 102 00:06:17,810 --> 00:06:21,990 But I want to talk about other ways 103 00:06:21,990 --> 00:06:24,540 to look at that same calculation, 104 00:06:24,540 --> 00:06:30,170 looking at whole columns and whole rows. 105 00:06:30,170 --> 00:06:30,890 Okay. 106 00:06:30,890 --> 00:06:34,030 So can I do A B C again? 107 00:06:34,030 --> 00:06:35,670 A B equaling C again? 108 00:06:35,670 --> 00:06:38,490 But now, tell me about... 109 00:06:42,190 --> 00:06:43,040 I'll put it up here. 110 00:06:45,770 --> 00:06:58,560 So here goes A, again, times B producing C. 111 00:06:58,560 --> 00:07:01,560 And again, this is m by n. 112 00:07:01,560 --> 00:07:06,520 This is n by P and this is m by P. Okay. 113 00:07:06,520 --> 00:07:09,290 Now I want to look at whole columns. 114 00:07:09,290 --> 00:07:12,590 I want to look at the columns of -- 115 00:07:12,590 --> 00:07:16,480 here's the second way to multiply matrices. 116 00:07:16,480 --> 00:07:19,450 Because I'm going to build on what I know already. 117 00:07:19,450 --> 00:07:24,150 How do I multiply a matrix by a column? 118 00:07:24,150 --> 00:07:27,500 I know how to multiply this matrix by that column. 119 00:07:30,270 --> 00:07:33,250 Shall I call that column one? 120 00:07:33,250 --> 00:07:35,430 That tells me column one of the answer. 121 00:07:38,460 --> 00:07:42,890 The matrix times the first column is that first column. 122 00:07:42,890 --> 00:07:45,140 Because none of this stuff entered 123 00:07:45,140 --> 00:07:46,910 that part of the answer. 124 00:07:46,910 --> 00:07:49,550 The matrix times the second column 125 00:07:49,550 --> 00:07:51,770 is the second column of the answer. 126 00:07:51,770 --> 00:07:53,780 Do you see what I'm saying? 127 00:07:53,780 --> 00:07:57,270 That I could think of multiplying a matrix 128 00:07:57,270 --> 00:08:00,470 by a vector, which I already knew how to do, 129 00:08:00,470 --> 00:08:08,860 and I can think of just P columns sitting side by side, 130 00:08:08,860 --> 00:08:12,000 just like resting next to each other. 131 00:08:12,000 --> 00:08:15,800 And I multiply A times each one of those. 132 00:08:15,800 --> 00:08:19,650 And I get the P columns of the answer. 133 00:08:19,650 --> 00:08:21,970 Do you see this as -- this is quite nice, 134 00:08:21,970 --> 00:08:27,980 to be able to think, okay, matrix multiplication works 135 00:08:27,980 --> 00:08:31,710 so that I can just think of having several columns, 136 00:08:31,710 --> 00:08:35,090 multiplying by A and getting the columns of the answer. 137 00:08:35,090 --> 00:08:43,169 So, like, here's column one shall I call that column one? 138 00:08:43,169 --> 00:08:47,540 And what's going in there is A times column one. 139 00:08:53,310 --> 00:08:54,680 Okay. 140 00:08:54,680 --> 00:08:57,220 So that's the picture a column at a time. 141 00:08:57,220 --> 00:08:58,700 So what does that tell me? 142 00:08:58,700 --> 00:09:01,400 What does that tell me about these columns? 143 00:09:01,400 --> 00:09:09,580 These columns of C are combinations, 144 00:09:09,580 --> 00:09:23,410 because we've seen that before, of columns of A. 145 00:09:23,410 --> 00:09:27,890 Every one of these comes from A times this, 146 00:09:27,890 --> 00:09:30,720 and A times a vector is a combination 147 00:09:30,720 --> 00:09:34,390 of the columns of A. 148 00:09:34,390 --> 00:09:40,640 And it makes sense, because the columns of A have length m 149 00:09:40,640 --> 00:09:43,380 and the columns of C have length m. 150 00:09:43,380 --> 00:09:48,260 And every column of C is some combination 151 00:09:48,260 --> 00:09:49,440 of the columns of A. 152 00:09:49,440 --> 00:09:51,410 And it's these numbers in here that 153 00:09:51,410 --> 00:09:53,670 tell me what combination it is. 154 00:09:53,670 --> 00:09:57,760 Do you see that? 155 00:09:57,760 --> 00:10:01,970 That in that answer, C, I'm seeing stuff that's 156 00:10:01,970 --> 00:10:04,740 combinations of these columns. 157 00:10:04,740 --> 00:10:09,390 Now, suppose I look at it -- that's two ways now. 158 00:10:09,390 --> 00:10:12,950 The third way is look at it by rows. 159 00:10:12,950 --> 00:10:17,420 So now let me change to rows. 160 00:10:17,420 --> 00:10:18,590 Okay. 161 00:10:18,590 --> 00:10:23,320 So now I can think of a row of A -- 162 00:10:23,320 --> 00:10:29,210 a row of A multiplying all these rows here and producing a row 163 00:10:29,210 --> 00:10:32,210 of the product. 164 00:10:32,210 --> 00:10:36,160 So this row takes a combination of these rows 165 00:10:36,160 --> 00:10:37,840 and that's the answer. 166 00:10:37,840 --> 00:10:50,680 So these rows of C are combinations of what? 167 00:10:50,680 --> 00:10:54,870 Tell me how to finish that. 168 00:10:54,870 --> 00:11:01,000 The rows of C, when I have a matrix B, it's got its rows 169 00:11:01,000 --> 00:11:04,840 and I multiply by A, and what does that do? 170 00:11:04,840 --> 00:11:06,750 It mixes the rows up. 171 00:11:06,750 --> 00:11:13,660 It creates combinations of the rows of B, thanks. 172 00:11:13,660 --> 00:11:17,730 Rows of B. 173 00:11:17,730 --> 00:11:24,990 That's what I wanted to see, that this answer -- 174 00:11:24,990 --> 00:11:27,820 I can see where the pieces are coming from. 175 00:11:27,820 --> 00:11:31,330 The rows in the answer are coming as combinations of these 176 00:11:31,330 --> 00:11:32,090 rows. 177 00:11:32,090 --> 00:11:37,020 The columns in the answer are coming as combinations of those 178 00:11:37,020 --> 00:11:37,750 columns. 179 00:11:37,750 --> 00:11:39,920 And so that's three ways. 180 00:11:39,920 --> 00:11:44,620 Now you can say, okay, what's the fourth way? 181 00:11:44,620 --> 00:11:48,450 The fourth way -- 182 00:11:48,450 --> 00:11:52,440 so that's -- now we've got, like, the regular way, 183 00:11:52,440 --> 00:11:58,360 the column way, the row way and -- 184 00:11:58,360 --> 00:11:59,680 what's left? 185 00:11:59,680 --> 00:12:06,450 The one that I can -- 186 00:12:06,450 --> 00:12:12,110 well, one way is columns times rows. 187 00:12:12,110 --> 00:12:15,940 What happens if I multiply -- 188 00:12:15,940 --> 00:12:21,480 So this was row times column, it gave a number. 189 00:12:21,480 --> 00:12:21,980 Okay. 190 00:12:21,980 --> 00:12:25,670 Now I want to ask you about column times row. 191 00:12:25,670 --> 00:12:37,610 If I multiply a column of A times a row of B, 192 00:12:37,610 --> 00:12:40,460 what shape I ending up with? 193 00:12:40,460 --> 00:12:43,380 So if I take a column times a row, 194 00:12:43,380 --> 00:12:47,710 that's definitely different from taking a row times a column. 195 00:12:47,710 --> 00:12:52,730 So a column of A was -- what's the shape of a column of A? 196 00:12:52,730 --> 00:12:56,640 m by one. 197 00:12:56,640 --> 00:12:59,430 A column of A is a column. 198 00:12:59,430 --> 00:13:03,190 It's got m entries and one column. 199 00:13:03,190 --> 00:13:04,780 And what's a row of B? 200 00:13:04,780 --> 00:13:10,000 It's got one row and P columns. 201 00:13:10,000 --> 00:13:12,870 So what's the shape -- what do I get if I multiply a column 202 00:13:12,870 --> 00:13:13,590 by a row? 203 00:13:16,350 --> 00:13:18,170 I get a big matrix. 204 00:13:18,170 --> 00:13:20,540 I get a full-sized matrix. 205 00:13:20,540 --> 00:13:25,790 If I multiply a column by a row -- 206 00:13:25,790 --> 00:13:28,090 should we just do one? 207 00:13:28,090 --> 00:13:34,470 Let me take the column two three four times the row one six. 208 00:13:38,510 --> 00:13:40,825 That product there -- 209 00:13:40,825 --> 00:13:42,950 I mean, when I'm just following the rules of matrix 210 00:13:42,950 --> 00:13:46,490 multiplication, those rules are just looking like -- 211 00:13:46,490 --> 00:13:51,570 kind of petite, kind of small, because the rows here 212 00:13:51,570 --> 00:13:53,830 are so short and the columns there are so short, 213 00:13:53,830 --> 00:13:56,610 but they're the same length, one entry. 214 00:13:56,610 --> 00:13:57,530 So what's the answer? 215 00:14:00,270 --> 00:14:03,770 What's the answer if I do two three four times one six, just 216 00:14:03,770 --> 00:14:05,320 for practice? 217 00:14:05,320 --> 00:14:09,180 Well, what's the first row of the answer? 218 00:14:09,180 --> 00:14:11,030 Two twelve. 219 00:14:11,030 --> 00:14:16,420 And the second row of the answer is three eighteen. 220 00:14:16,420 --> 00:14:24,220 And the third row of the answer is four twenty four. 221 00:14:24,220 --> 00:14:29,400 That's a very special matrix, there. 222 00:14:29,400 --> 00:14:30,840 Very special matrix. 223 00:14:30,840 --> 00:14:32,830 What can you tell me about its columns, 224 00:14:32,830 --> 00:14:34,220 the columns of that matrix? 225 00:14:37,640 --> 00:14:41,410 They're multiples of this guy, right? 226 00:14:41,410 --> 00:14:42,690 They're multiples of that one. 227 00:14:42,690 --> 00:14:44,340 Which follows our rule. 228 00:14:44,340 --> 00:14:47,540 We said that the columns of the answer were combinations, 229 00:14:47,540 --> 00:14:50,300 but there's only -- to take a combination of one guy, 230 00:14:50,300 --> 00:14:52,300 it's just a multiple. 231 00:14:52,300 --> 00:14:54,000 The rows of the answer, what can you 232 00:14:54,000 --> 00:14:55,530 tell me about those three rows? 233 00:14:58,060 --> 00:15:01,480 They're all multiples of this row. 234 00:15:01,480 --> 00:15:04,660 They're all multiples of one six, as we expected. 235 00:15:04,660 --> 00:15:06,970 But I'm getting a full-sized matrix. 236 00:15:06,970 --> 00:15:15,800 And now, just to complete this thought, if I have -- 237 00:15:15,800 --> 00:15:17,280 let me write down the fourth way. 238 00:15:21,110 --> 00:15:35,630 A B is a sum of columns of A times rows of B. 239 00:15:35,630 --> 00:15:40,680 So that, for example, if my matrix was two three four 240 00:15:40,680 --> 00:15:47,450 and then had another column, say, seven eight nine, 241 00:15:47,450 --> 00:15:53,010 and my matrix here has -- say, started with one six and then 242 00:15:53,010 --> 00:16:00,890 had another column like zero zero, then -- 243 00:16:00,890 --> 00:16:04,630 here's the fourth way, okay? 244 00:16:04,630 --> 00:16:07,690 I've got two columns there, I've got two rows there. 245 00:16:07,690 --> 00:16:10,920 So the beautiful rule is -- 246 00:16:10,920 --> 00:16:13,080 see, the whole thing by columns and rows 247 00:16:13,080 --> 00:16:19,730 is that I can take the first column times the first row 248 00:16:19,730 --> 00:16:25,100 and add the second column times the second row. 249 00:16:32,690 --> 00:16:34,790 So that's the fourth way -- 250 00:16:34,790 --> 00:16:38,760 that I can take columns times rows, 251 00:16:38,760 --> 00:16:41,800 first column times first row, second column times second 252 00:16:41,800 --> 00:16:42,960 row and add. 253 00:16:42,960 --> 00:16:44,110 Actually, what will I get? 254 00:16:44,110 --> 00:16:46,765 What will the answer be for that matrix multiplication? 255 00:16:49,730 --> 00:16:52,130 Well, this one it's just going to give us zero, 256 00:16:52,130 --> 00:16:56,070 so in fact I'm back to this -- that's the answer, 257 00:16:56,070 --> 00:16:59,310 for that matrix multiplication. 258 00:16:59,310 --> 00:17:05,569 I'm happy to put up here these facts about matrix 259 00:17:05,569 --> 00:17:10,040 multiplication, because it gives me a chance to write down 260 00:17:10,040 --> 00:17:12,099 special matrices like this. 261 00:17:12,099 --> 00:17:15,050 This is a special matrix. 262 00:17:15,050 --> 00:17:18,569 All those rows lie on the same line. 263 00:17:18,569 --> 00:17:21,900 All those rows lie on the line through one six. 264 00:17:21,900 --> 00:17:25,240 If I draw a picture of all these row vectors, 265 00:17:25,240 --> 00:17:28,119 they're all the same direction. 266 00:17:28,119 --> 00:17:30,730 If I draw a picture of these two column vectors, 267 00:17:30,730 --> 00:17:33,900 they're in the same direction. 268 00:17:33,900 --> 00:17:37,430 Later, I would use this language. 269 00:17:37,430 --> 00:17:39,310 Not too much later, either. 270 00:17:39,310 --> 00:17:42,400 I would say the row space, which is 271 00:17:42,400 --> 00:17:44,380 like all the combinations of the rows, 272 00:17:44,380 --> 00:17:47,060 is just a line for this matrix. 273 00:17:47,060 --> 00:17:51,240 The row space is the line through the vector one six. 274 00:17:51,240 --> 00:17:54,390 All the rows lie on that line. 275 00:17:54,390 --> 00:17:57,710 And the column space is also a line. 276 00:17:57,710 --> 00:18:01,510 All the columns lie on the line through the vector two 277 00:18:01,510 --> 00:18:02,950 three four. 278 00:18:02,950 --> 00:18:07,340 So this is like a really minimal matrix. 279 00:18:07,340 --> 00:18:10,940 And it's because of these ones. 280 00:18:10,940 --> 00:18:11,440 Okay. 281 00:18:11,440 --> 00:18:18,580 So that's a third way. 282 00:18:18,580 --> 00:18:23,160 Now I want to say one more thing about matrix multiplication 283 00:18:23,160 --> 00:18:26,100 while we're on the subject. 284 00:18:26,100 --> 00:18:28,060 And it's this. 285 00:18:28,060 --> 00:18:29,740 You could also multiply -- 286 00:18:29,740 --> 00:18:34,390 You could also cut the matrix into blocks 287 00:18:34,390 --> 00:18:37,160 and do the multiplication by blocks. 288 00:18:37,160 --> 00:18:46,690 Yet that's actually so, useful that I want to mention it. 289 00:18:46,690 --> 00:18:47,745 Block multiplication. 290 00:18:50,540 --> 00:18:54,500 So I could take my matrix A and I could chop it up, like, 291 00:18:54,500 --> 00:18:58,460 maybe just for simplicity, let me chop it into two -- 292 00:18:58,460 --> 00:18:59,770 into four square blocks. 293 00:18:59,770 --> 00:19:00,980 Suppose it's square. 294 00:19:00,980 --> 00:19:03,990 Let's just take a nice case. 295 00:19:03,990 --> 00:19:07,800 And B, suppose it's square also, same size. 296 00:19:11,340 --> 00:19:13,460 So these sizes don't have to be the same. 297 00:19:13,460 --> 00:19:16,540 What they have to do is match properly. 298 00:19:16,540 --> 00:19:18,590 Here they certainly will match. 299 00:19:18,590 --> 00:19:22,340 So here's the rule for block multiplication, 300 00:19:22,340 --> 00:19:28,880 that if this has blocks like, A -- 301 00:19:28,880 --> 00:19:34,220 so maybe A1, A2, A3, A4 are the blocks here, 302 00:19:34,220 --> 00:19:38,210 and these blocks are B1, B2,3 and B4? 303 00:19:38,210 --> 00:19:44,770 Then the answer I can find block. 304 00:19:44,770 --> 00:19:46,500 And if you tell me what's in that block, 305 00:19:46,500 --> 00:19:49,560 then I'm going to be quiet about matrix multiplication 306 00:19:49,560 --> 00:19:51,870 for the rest of the day. 307 00:19:51,870 --> 00:19:54,920 What goes into that block? 308 00:19:54,920 --> 00:19:57,870 You see, these might be -- this matrix might be -- 309 00:19:57,870 --> 00:20:02,400 these matrices might be, like, twenty by twenty with blocks 310 00:20:02,400 --> 00:20:06,030 that are ten by ten, to take the easy case where all the blocks 311 00:20:06,030 --> 00:20:08,940 are the same shape. 312 00:20:08,940 --> 00:20:13,110 And the point is that I could multiply those by blocks. 313 00:20:13,110 --> 00:20:15,940 And what goes in here? 314 00:20:15,940 --> 00:20:20,920 What's that block in the answer? 315 00:20:20,920 --> 00:20:25,710 A1 B1, that's a matrix times a matrix, 316 00:20:25,710 --> 00:20:28,270 it's the right size, ten by ten. 317 00:20:28,270 --> 00:20:30,610 Any more? 318 00:20:30,610 --> 00:20:36,760 Plus, what else goes in there? 319 00:20:36,760 --> 00:20:38,160 A2 B3, right? 320 00:20:38,160 --> 00:20:41,490 It's just like block rows times block columns. 321 00:20:45,610 --> 00:20:49,070 Nobody, I think, not even Gauss could see instantly 322 00:20:49,070 --> 00:20:51,040 that it works. 323 00:20:51,040 --> 00:20:55,130 But somehow, if we check it through, all five ways 324 00:20:55,130 --> 00:20:58,020 we're doing the same multiplications. 325 00:20:58,020 --> 00:21:02,430 So this familiar multiplication is 326 00:21:02,430 --> 00:21:04,270 what we're really doing when we do it 327 00:21:04,270 --> 00:21:10,240 by columns, by rows by columns times rows and by blocks. 328 00:21:10,240 --> 00:21:10,740 Okay. 329 00:21:10,740 --> 00:21:13,680 I just have to, like, get the rules straight 330 00:21:13,680 --> 00:21:16,991 for matrix multiplication. 331 00:21:16,991 --> 00:21:17,490 Okay. 332 00:21:22,740 --> 00:21:24,490 All right, I'm ready for the second topic, 333 00:21:24,490 --> 00:21:27,590 which is inverses. 334 00:21:27,590 --> 00:21:28,479 Okay. 335 00:21:28,479 --> 00:21:29,270 Ready for inverses. 336 00:21:34,140 --> 00:21:39,040 And let me do it for square matrices first. 337 00:21:44,270 --> 00:21:44,770 Okay. 338 00:21:44,770 --> 00:21:52,270 So I've got a square matrix A. 339 00:21:52,270 --> 00:21:55,370 And it may or may not have an inverse, right? 340 00:21:55,370 --> 00:21:57,260 Not all matrices have inverses. 341 00:21:57,260 --> 00:22:01,770 In fact, that's the most important question you can ask 342 00:22:01,770 --> 00:22:06,260 about the matrix, is if it's -- if you know it's square, 343 00:22:06,260 --> 00:22:08,540 is it invertible or not? 344 00:22:08,540 --> 00:22:12,500 If it is invertible, then there is some other matrix, 345 00:22:12,500 --> 00:22:15,680 shall I call it A inverse? 346 00:22:15,680 --> 00:22:21,600 And what's the -- if A inverse exists -- 347 00:22:21,600 --> 00:22:24,270 there's a big "if" here. 348 00:22:24,270 --> 00:22:34,120 If this matrix exists, and it'll be really central to figure out 349 00:22:34,120 --> 00:22:35,940 when does it exist? 350 00:22:35,940 --> 00:22:40,350 And then if it does exist, how would you find it? 351 00:22:40,350 --> 00:22:45,710 But what's the equation here that I haven't -- 352 00:22:45,710 --> 00:22:47,900 that I have to finish now? 353 00:22:47,900 --> 00:22:53,280 This matrix, if it exists multiplies A and produces, 354 00:22:53,280 --> 00:22:54,755 I think, the identity. 355 00:23:10,840 --> 00:23:12,680 But a real -- 356 00:23:12,680 --> 00:23:17,600 an inverse for a square matrix could be on the right as well 357 00:23:17,600 --> 00:23:19,240 -- 358 00:23:19,240 --> 00:23:25,760 this is true, too, that it's -- 359 00:23:25,760 --> 00:23:28,280 if I have a -- yeah in fact, this is not -- 360 00:23:28,280 --> 00:23:31,570 this is probably the -- 361 00:23:31,570 --> 00:23:38,400 this is something that's not easy to prove, but it works. 362 00:23:38,400 --> 00:23:40,240 That a left -- 363 00:23:40,240 --> 00:23:43,670 square matrices, a left inverse is also a right 364 00:23:43,670 --> 00:23:44,570 inverse. 365 00:23:44,570 --> 00:23:49,090 If I can find a matrix on the left that gets the identity, 366 00:23:49,090 --> 00:23:50,990 then also that matrix on the right 367 00:23:50,990 --> 00:23:53,630 will produce that identity. 368 00:23:53,630 --> 00:23:57,310 For rectangular matrices, we'll see a left inverse 369 00:23:57,310 --> 00:23:58,950 that isn't a right inverse. 370 00:23:58,950 --> 00:24:01,660 In fact, the shapes wouldn't allow it. 371 00:24:01,660 --> 00:24:03,700 But for square matrices, the shapes 372 00:24:03,700 --> 00:24:09,820 allow it and it happens, if A has an inverse. 373 00:24:09,820 --> 00:24:14,011 Okay, so give me some cases -- 374 00:24:14,011 --> 00:24:14,510 let's see. 375 00:24:14,510 --> 00:24:17,090 I hate to be negative here, but let's talk 376 00:24:17,090 --> 00:24:20,070 about the case with no inverse. 377 00:24:20,070 --> 00:24:31,210 So -- these matrices are called invertible or non-singular -- 378 00:24:36,640 --> 00:24:38,960 those are the good ones. 379 00:24:38,960 --> 00:24:41,510 And we want to be able to identify how -- 380 00:24:41,510 --> 00:24:44,180 if we're given a matrix, has it got an inverse? 381 00:24:44,180 --> 00:24:46,950 Can I talk about the singular case? 382 00:24:52,210 --> 00:24:52,940 No inverse. 383 00:24:57,420 --> 00:24:58,950 All right. 384 00:24:58,950 --> 00:25:02,720 Best to start with an example. 385 00:25:02,720 --> 00:25:06,880 Tell me an example -- let's get an example up here. 386 00:25:06,880 --> 00:25:09,030 Let's make it two by two -- 387 00:25:09,030 --> 00:25:13,540 of a matrix that has not got an inverse. 388 00:25:13,540 --> 00:25:16,160 And let's see why. 389 00:25:16,160 --> 00:25:19,230 Let me write one up. 390 00:25:19,230 --> 00:25:20,950 No inverse. 391 00:25:20,950 --> 00:25:22,620 Let's see why. 392 00:25:22,620 --> 00:25:30,780 Let me write up -- one three two six. 393 00:25:35,180 --> 00:25:37,820 Why does that matrix have no inverse? 394 00:25:40,950 --> 00:25:43,245 You could answer that various ways. 395 00:25:45,760 --> 00:25:48,390 Give me one reason. 396 00:25:48,390 --> 00:25:51,620 Well, you could -- if you know about determinants, 397 00:25:51,620 --> 00:25:57,060 which you're not supposed to, you could take its determinant 398 00:25:57,060 --> 00:25:58,740 and you would get -- 399 00:25:58,740 --> 00:26:00,330 Zero. 400 00:26:00,330 --> 00:26:03,090 Okay. 401 00:26:03,090 --> 00:26:04,120 Now -- all right. 402 00:26:07,640 --> 00:26:10,330 Let me ask you other reasons. 403 00:26:10,330 --> 00:26:13,120 I mean, as for other reasons that that matrix 404 00:26:13,120 --> 00:26:15,810 isn't invertible. 405 00:26:15,810 --> 00:26:18,765 Here, I could use what I'm saying here. 406 00:26:24,550 --> 00:26:27,800 Suppose A times other matrix gave the identity. 407 00:26:32,060 --> 00:26:34,750 Why is that not possible? 408 00:26:34,750 --> 00:26:39,010 Because -- oh, yeah -- 409 00:26:39,010 --> 00:26:41,220 I'm thinking about columns here. 410 00:26:41,220 --> 00:26:46,200 If I multiply this matrix A by some other matrix, then the -- 411 00:26:46,200 --> 00:26:50,240 the result -- what can you tell me about the columns? 412 00:26:50,240 --> 00:26:55,730 They're all multiples of those columns, right? 413 00:26:55,730 --> 00:26:59,340 If I multiply A by another matrix that -- 414 00:26:59,340 --> 00:27:04,130 the product has columns that come from those columns. 415 00:27:04,130 --> 00:27:06,480 So can I get the identity matrix? 416 00:27:06,480 --> 00:27:08,120 No way. 417 00:27:08,120 --> 00:27:11,670 The columns of the identity matrix, like one zero -- 418 00:27:11,670 --> 00:27:14,740 it's not a combination of those columns, 419 00:27:14,740 --> 00:27:16,520 because those two columns lie on the -- 420 00:27:16,520 --> 00:27:19,140 both lie on the same line. 421 00:27:19,140 --> 00:27:22,140 Every combination is just going to be on that line 422 00:27:22,140 --> 00:27:24,480 and I can't get one zero. 423 00:27:24,480 --> 00:27:33,530 So, do you see that sort of column picture of the matrix 424 00:27:33,530 --> 00:27:34,750 not being invertible. 425 00:27:34,750 --> 00:27:37,580 In fact, here's another reason. 426 00:27:37,580 --> 00:27:40,720 This is even a more important reason. 427 00:27:40,720 --> 00:27:42,260 Well, how can I say more important? 428 00:27:42,260 --> 00:27:44,810 All those are important. 429 00:27:44,810 --> 00:27:47,910 This is another way to see it. 430 00:27:47,910 --> 00:27:51,410 A matrix has no inverse -- 431 00:27:51,410 --> 00:27:55,710 yeah -- here -- now this is important. 432 00:27:55,710 --> 00:27:59,180 A matrix has no -- a square matrix won't have an inverse 433 00:27:59,180 --> 00:28:07,980 if there's no inverse because I can solve -- 434 00:28:07,980 --> 00:28:20,810 I can find an X of -- a vector X with A times -- 435 00:28:20,810 --> 00:28:23,140 this A times X giving zero. 436 00:28:27,010 --> 00:28:31,310 This is the reason I like best. 437 00:28:31,310 --> 00:28:33,170 That matrix won't have an inverse. 438 00:28:33,170 --> 00:28:40,490 Can you -- well, let me change I to U. 439 00:28:40,490 --> 00:28:46,780 So tell me a vector X that, solves A X equals zero. 440 00:28:46,780 --> 00:28:49,610 I mean, this is, like, the key equation. 441 00:28:49,610 --> 00:28:51,310 In mathematics, all the key equations 442 00:28:51,310 --> 00:28:53,490 have zero on the right-hand side. 443 00:28:53,490 --> 00:28:55,490 So what's the X? 444 00:28:55,490 --> 00:28:58,290 Tell me an X here -- 445 00:28:58,290 --> 00:29:01,360 so now I'm going to put -- slip in the X that you tell me 446 00:29:01,360 --> 00:29:05,500 and I'm going to get zero. 447 00:29:05,500 --> 00:29:09,000 What X would do that job? 448 00:29:09,000 --> 00:29:11,370 Three and negative one? 449 00:29:11,370 --> 00:29:13,610 Is that the one you picked, or -- yeah. 450 00:29:13,610 --> 00:29:18,720 Or another -- well, if you picked zero with zero, 451 00:29:18,720 --> 00:29:21,060 I'm not so excited, right? 452 00:29:21,060 --> 00:29:23,490 Because that would always work. 453 00:29:23,490 --> 00:29:26,550 So it's really the fact that this vector 454 00:29:26,550 --> 00:29:28,840 isn't zero that's important. 455 00:29:28,840 --> 00:29:33,750 It's a non-zero vector and three negative one would do it. 456 00:29:33,750 --> 00:29:37,130 That just says three of this column minus one of that column 457 00:29:37,130 --> 00:29:38,861 is the zero column. 458 00:29:38,861 --> 00:29:39,360 Okay. 459 00:29:42,240 --> 00:29:46,860 So now I know that A couldn't be invertible. 460 00:29:46,860 --> 00:29:49,890 But what's the reasoning? 461 00:29:49,890 --> 00:29:54,720 If A X is zero, suppose I multiplied by A inverse. 462 00:29:54,720 --> 00:29:56,660 Yeah, well here's the reason. 463 00:29:56,660 --> 00:30:02,180 Here -- this is why this spells disaster for an inverse. 464 00:30:02,180 --> 00:30:06,580 The matrix can't have an inverse if some combination 465 00:30:06,580 --> 00:30:09,140 of the columns gives z- it gives nothing. 466 00:30:09,140 --> 00:30:12,630 Because, I could take A X equals zero, 467 00:30:12,630 --> 00:30:21,200 I could multiply by A inverse and what would I discover? 468 00:30:21,200 --> 00:30:23,840 Suppose I take that equation and I multiply by -- 469 00:30:23,840 --> 00:30:26,860 if A inverse existed, which of course I'm going to come 470 00:30:26,860 --> 00:30:30,560 to the conclusion it can't because if it existed, 471 00:30:30,560 --> 00:30:33,570 if there was an A inverse to this dopey matrix, 472 00:30:33,570 --> 00:30:36,990 I would multiply that equation by that inverse and I would 473 00:30:36,990 --> 00:30:42,270 discover X is zero. 474 00:30:42,270 --> 00:30:45,320 If I multiply A by A inverse on the left, I get X. 475 00:30:45,320 --> 00:30:48,830 If I multiply by A inverse on the right, I get zero. 476 00:30:48,830 --> 00:30:50,790 So I would discover X was zero. 477 00:30:50,790 --> 00:30:53,050 But it -- X is not zero. 478 00:30:53,050 --> 00:30:54,800 X -- this guy wasn't zero. 479 00:30:54,800 --> 00:30:55,460 There it is. 480 00:30:55,460 --> 00:30:58,170 It's three minus one. 481 00:30:58,170 --> 00:31:06,250 So, conclusion -- only, it takes us some time to really work 482 00:31:06,250 --> 00:31:08,040 with that conclusion -- 483 00:31:08,040 --> 00:31:14,300 our conclusion will be that non-invertible matrices, 484 00:31:14,300 --> 00:31:19,490 singular matrices, some combinations of their columns 485 00:31:19,490 --> 00:31:22,290 gives the zero column. 486 00:31:22,290 --> 00:31:26,090 They they take some vector X into zero. 487 00:31:26,090 --> 00:31:30,650 And there's no way A inverse can recover, right? 488 00:31:30,650 --> 00:31:32,520 That's what this equation says. 489 00:31:32,520 --> 00:31:36,980 This equation says I take this vector X and multiplying 490 00:31:36,980 --> 00:31:39,470 by A gives zero. 491 00:31:39,470 --> 00:31:42,220 But then when I multiply by A inverse, 492 00:31:42,220 --> 00:31:44,690 I can never escape from zero. 493 00:31:44,690 --> 00:31:47,980 So there couldn't be an A inverse. 494 00:31:47,980 --> 00:31:51,701 Where here -- okay, now fix -- 495 00:31:51,701 --> 00:31:52,200 all right. 496 00:31:52,200 --> 00:31:57,200 Now let me take -- all right, back to the positive side. 497 00:31:57,200 --> 00:32:01,530 Let's take a matrix that does have an inverse. 498 00:32:01,530 --> 00:32:03,840 And why not invert it? 499 00:32:03,840 --> 00:32:04,500 Okay. 500 00:32:04,500 --> 00:32:08,782 Can I -- so let me take on this third board a matrix -- 501 00:32:08,782 --> 00:32:09,990 shall I fix that up a little? 502 00:32:12,810 --> 00:32:15,660 Tell me a matrix that has got an inverse. 503 00:32:18,680 --> 00:32:22,930 Well, let me say one three two -- what shall I put there? 504 00:32:25,620 --> 00:32:28,665 Well, don't put six, I guess is -- right? 505 00:32:31,570 --> 00:32:35,000 Do I any favorites here? 506 00:32:35,000 --> 00:32:36,910 One? 507 00:32:36,910 --> 00:32:38,280 Or eight? 508 00:32:41,470 --> 00:32:42,140 I don't care. 509 00:32:42,140 --> 00:32:43,450 What, seven? 510 00:32:43,450 --> 00:32:43,950 Seven. 511 00:32:43,950 --> 00:32:44,540 Okay. 512 00:32:44,540 --> 00:32:46,060 Seven is a lucky number. 513 00:32:46,060 --> 00:32:48,750 All right, seven, okay. 514 00:32:48,750 --> 00:32:49,380 Okay. 515 00:32:49,380 --> 00:32:51,080 So -- now what's our idea? 516 00:32:51,080 --> 00:32:53,740 We believe that this matrix is invertible. 517 00:32:53,740 --> 00:32:57,060 Those who like determinants have quickly taken its determinant 518 00:32:57,060 --> 00:32:59,320 and found it wasn't zero. 519 00:32:59,320 --> 00:33:04,610 Those who like columns, and probably that -- 520 00:33:04,610 --> 00:33:08,450 that department is not totally popular yet -- 521 00:33:08,450 --> 00:33:11,080 but those who like columns will look at those two columns 522 00:33:11,080 --> 00:33:15,090 and say, hey, they point in different directions. 523 00:33:15,090 --> 00:33:18,080 So I can get anything. 524 00:33:18,080 --> 00:33:19,860 Now, let me see, what do I mean? 525 00:33:19,860 --> 00:33:21,880 How I going to computer A inverse? 526 00:33:21,880 --> 00:33:24,060 So A inverse -- 527 00:33:24,060 --> 00:33:28,360 here's A inverse, now, and I have to find it. 528 00:33:28,360 --> 00:33:33,219 And what do I get when I do this multiplication? 529 00:33:33,219 --> 00:33:33,760 The identity. 530 00:33:40,810 --> 00:33:43,940 You know, forgive me for taking two by two-s, but -- 531 00:33:43,940 --> 00:33:49,240 lt's good to keep the computations manageable and let 532 00:33:49,240 --> 00:33:51,090 the ideas come out. 533 00:33:51,090 --> 00:33:55,230 Okay, now what's the idea I want? 534 00:33:55,230 --> 00:33:57,520 I'm looking for this matrix A inverse, how 535 00:33:57,520 --> 00:33:58,750 I going to find it? 536 00:33:58,750 --> 00:34:04,235 Right now, I've got four numbers to find. 537 00:34:08,290 --> 00:34:12,159 I'm going to look at the first column. 538 00:34:12,159 --> 00:34:16,860 Let me take this first column, A B. 539 00:34:16,860 --> 00:34:18,699 What's up there? 540 00:34:18,699 --> 00:34:20,670 What -- tell me this. 541 00:34:20,670 --> 00:34:25,260 What equation does the first column satisfy? 542 00:34:25,260 --> 00:34:31,489 The first column satisfies A times that column is one zero. 543 00:34:31,489 --> 00:34:34,179 The first column of the answer. 544 00:34:34,179 --> 00:34:38,820 And the second column, C D, satisfies A times 545 00:34:38,820 --> 00:34:41,139 that second column is zero one. 546 00:34:41,139 --> 00:34:48,909 You see that finding the inverse is like solving two systems. 547 00:34:48,909 --> 00:34:52,210 One system, when the right-hand side is one zero -- 548 00:34:52,210 --> 00:34:54,139 I'm just going to split it into two pieces. 549 00:34:57,427 --> 00:34:58,760 I don't even need to rewrite it. 550 00:34:58,760 --> 00:35:04,480 I can take A times -- so let me put it here. 551 00:35:04,480 --> 00:35:18,100 A times column j of A inverse is column j of the identity. 552 00:35:21,350 --> 00:35:23,010 I've got n equations. 553 00:35:23,010 --> 00:35:26,500 I've got, well, two in this case. 554 00:35:26,500 --> 00:35:29,100 And they have the same matrix, A, 555 00:35:29,100 --> 00:35:30,860 but they have different right-hand sides. 556 00:35:30,860 --> 00:35:32,840 The right-hand sides are just the columns 557 00:35:32,840 --> 00:35:35,800 of the identity, this guy and this guy. 558 00:35:35,800 --> 00:35:37,450 And these are the two solutions. 559 00:35:37,450 --> 00:35:39,610 Do you see what I'm going -- 560 00:35:39,610 --> 00:35:45,120 I'm looking at that equation by columns. 561 00:35:45,120 --> 00:35:47,050 I'm looking at A times this column, 562 00:35:47,050 --> 00:35:49,910 giving that guy, and A times that column giving that guy. 563 00:35:49,910 --> 00:35:55,500 So -- Essentially -- so this is like the Gauss -- 564 00:35:55,500 --> 00:35:56,410 we're back to Gauss. 565 00:35:56,410 --> 00:36:00,820 We're back to solving systems of equations, but we're solving -- 566 00:36:00,820 --> 00:36:05,160 we've got two right-hand sides instead of one. 567 00:36:05,160 --> 00:36:07,010 That's where Jordan comes in. 568 00:36:07,010 --> 00:36:09,980 So at the very beginning of the lecture, 569 00:36:09,980 --> 00:36:13,230 I mentioned Gauss-Jordan, let me write it up again. 570 00:36:13,230 --> 00:36:13,730 Okay. 571 00:36:13,730 --> 00:36:16,630 Here's the Gauss-Jordan idea. 572 00:36:21,600 --> 00:36:35,000 Gauss-Jordan solve two equations at once. 573 00:36:39,460 --> 00:36:39,960 Okay. 574 00:36:39,960 --> 00:36:43,630 Let me show you how the mechanics go. 575 00:36:43,630 --> 00:36:48,970 How do I solve a single equation? 576 00:36:48,970 --> 00:36:55,760 So the two equations are one three two seven, 577 00:36:55,760 --> 00:37:01,630 multiplying A B gives one zero. 578 00:37:01,630 --> 00:37:03,560 And the other equation is the same 579 00:37:03,560 --> 00:37:12,130 one three two seven multiplying C D gives zero one. 580 00:37:12,130 --> 00:37:15,010 Okay. 581 00:37:15,010 --> 00:37:17,410 That'll tell me the two columns of the inverse. 582 00:37:17,410 --> 00:37:19,080 I'll have inverse. 583 00:37:19,080 --> 00:37:22,940 In other words, if I can solve with this matrix A, 584 00:37:22,940 --> 00:37:24,850 if I can solve with that right-hand side 585 00:37:24,850 --> 00:37:28,340 and that right-hand side, I'm invertible. 586 00:37:28,340 --> 00:37:29,460 I've got it. 587 00:37:29,460 --> 00:37:30,690 Okay. 588 00:37:30,690 --> 00:37:36,410 And Jordan sort of said to Gauss, solve them together, 589 00:37:36,410 --> 00:37:39,650 look at the matrix -- if we just solve this one, 590 00:37:39,650 --> 00:37:43,140 I would look at one three two seven, 591 00:37:43,140 --> 00:37:46,030 and how do I deal with the right-hand side? 592 00:37:46,030 --> 00:37:48,960 I stick it on as an extra column, right? 593 00:37:56,130 --> 00:37:58,460 That's this augmented matrix. 594 00:37:58,460 --> 00:38:01,750 That's the matrix when I'm watching the right-hand side 595 00:38:01,750 --> 00:38:04,470 at the same time, doing the same thing to the right side 596 00:38:04,470 --> 00:38:06,520 that I do to the left? 597 00:38:06,520 --> 00:38:09,610 So I just carry it along as an extra column. 598 00:38:09,610 --> 00:38:12,100 Now I'm going to carry along two extra columns. 599 00:38:17,200 --> 00:38:20,920 And I'm going to do whatever Gauss wants, right? 600 00:38:20,920 --> 00:38:24,150 I'm going to do elimination. 601 00:38:24,150 --> 00:38:27,010 I'm going to get this to be simple 602 00:38:27,010 --> 00:38:30,070 and this thing will turn into the inverse. 603 00:38:30,070 --> 00:38:32,460 This is what's coming. 604 00:38:32,460 --> 00:38:35,800 I'm going to do elimination steps to make this 605 00:38:35,800 --> 00:38:39,240 into the identity, and lo and behold, 606 00:38:39,240 --> 00:38:41,550 the inverse will show up here. 607 00:38:41,550 --> 00:38:43,330 K--- let's do it. 608 00:38:43,330 --> 00:38:43,830 Okay. 609 00:38:43,830 --> 00:38:46,880 So what are the elimination steps? 610 00:38:46,880 --> 00:38:51,680 So you see -- here's my matrix A and here's the identity, like, 611 00:38:51,680 --> 00:38:52,795 stuck on, augmented on. 612 00:38:52,795 --> 00:38:53,670 STUDENT: I'm sorry... 613 00:38:53,670 --> 00:38:54,211 STRANG: Yeah? 614 00:38:54,211 --> 00:38:58,800 STUDENT: -- is the two and the three supposed to be switched? 615 00:38:58,800 --> 00:39:01,070 STRANG: Did I -- oh, no, they weren't supposed to be 616 00:39:01,070 --> 00:39:01,570 switched. 617 00:39:01,570 --> 00:39:02,070 Sorry. 618 00:39:02,070 --> 00:39:04,680 Thanks. 619 00:39:04,680 --> 00:39:05,930 Okay. 620 00:39:05,930 --> 00:39:08,425 Thank you very much. 621 00:39:08,425 --> 00:39:09,800 And there -- I've got them right. 622 00:39:09,800 --> 00:39:11,215 Okay, thanks. 623 00:39:14,520 --> 00:39:15,510 Okay. 624 00:39:15,510 --> 00:39:17,780 So let's do elimination. 625 00:39:17,780 --> 00:39:19,890 All right, it's going to be simple, right? 626 00:39:19,890 --> 00:39:23,690 So I take two of this row away from this row. 627 00:39:23,690 --> 00:39:27,180 So this row stays the same and two 628 00:39:27,180 --> 00:39:28,680 of those come away from this. 629 00:39:28,680 --> 00:39:32,560 That leaves me with a zero and a one and two of these away from 630 00:39:32,560 --> 00:39:36,970 this is that what you're getting -- 631 00:39:36,970 --> 00:39:39,930 after one elimination step -- 632 00:39:39,930 --> 00:39:42,790 Let me sort of separate the -- 633 00:39:42,790 --> 00:39:45,040 the left half from the right half. 634 00:39:45,040 --> 00:39:48,690 So two of that first row got subtracted from the second row. 635 00:39:48,690 --> 00:39:53,240 Now this is an upper triangular form. 636 00:39:53,240 --> 00:39:57,120 Gauss would quit, but Jordan says keeps going. 637 00:39:57,120 --> 00:39:59,250 Use elimination upwards. 638 00:39:59,250 --> 00:40:03,710 Subtract a multiple of equation two from equation one 639 00:40:03,710 --> 00:40:05,970 to get rid of the three. 640 00:40:05,970 --> 00:40:08,970 So let's go the whole way. 641 00:40:08,970 --> 00:40:14,420 So now I'm going to -- this guy is fine, but I'm going to -- 642 00:40:14,420 --> 00:40:15,770 what do I do now? 643 00:40:15,770 --> 00:40:19,850 What's my final step that produces the inverse? 644 00:40:19,850 --> 00:40:22,200 I multiply this by the right number 645 00:40:22,200 --> 00:40:26,060 to get up to ther to remove that three. 646 00:40:26,060 --> 00:40:28,350 So I guess, I -- since this is a one, 647 00:40:28,350 --> 00:40:30,830 there's the pivot sitting there. 648 00:40:30,830 --> 00:40:33,750 I multiply it by three and subtract from that, 649 00:40:33,750 --> 00:40:35,080 so what do I get? 650 00:40:35,080 --> 00:40:38,460 I'll have one zero -- oh, yeah that was my whole point. 651 00:40:38,460 --> 00:40:41,640 I'll multiply this by three and subtract from that, 652 00:40:41,640 --> 00:40:46,450 which will give me seven. 653 00:40:46,450 --> 00:40:49,340 And I multiply this by three and subtract from that, 654 00:40:49,340 --> 00:40:50,920 which gives me a minus three. 655 00:41:00,060 --> 00:41:06,500 And what's my hope, belief? 656 00:41:06,500 --> 00:41:10,780 Here I started with A and the identity, 657 00:41:10,780 --> 00:41:16,410 and I ended up with the identity and who? 658 00:41:16,410 --> 00:41:18,640 That better be A inverse. 659 00:41:24,090 --> 00:41:26,980 That's the Gauss Jordan idea. 660 00:41:26,980 --> 00:41:33,690 Start with this long matrix, double-length A I, eliminate, 661 00:41:33,690 --> 00:41:37,450 eliminate until this part is down to I, 662 00:41:37,450 --> 00:41:40,530 then this one will -- must be for some reason, 663 00:41:40,530 --> 00:41:45,230 and we've got to find the reason -- must be A inverse. 664 00:41:45,230 --> 00:41:46,690 Shall I just check that it works? 665 00:41:50,160 --> 00:41:55,080 Let me just check that -- can I multiply this matrix this part 666 00:41:55,080 --> 00:42:00,300 times A, I'll carry A over here and just do that 667 00:42:00,300 --> 00:42:01,190 multiplication. 668 00:42:01,190 --> 00:42:03,310 You'll see I'll do it the old fashioned way. 669 00:42:03,310 --> 00:42:06,000 Seven minus six is a one. 670 00:42:06,000 --> 00:42:08,670 Twenty one minus twenty one is a zero, 671 00:42:08,670 --> 00:42:13,280 minus two plus two is a zero, minus six plus seven is a one. 672 00:42:13,280 --> 00:42:15,780 Check. 673 00:42:15,780 --> 00:42:18,290 So that is the inverse. 674 00:42:18,290 --> 00:42:20,950 That's the Gauss-Jordan idea. 675 00:42:20,950 --> 00:42:24,190 So, you'll -- one of the homework problems or more than 676 00:42:24,190 --> 00:42:30,710 one for Wednesday will ask you to go through those steps. 677 00:42:30,710 --> 00:42:33,560 I think you just got to go through Gauss-Jordan a couple 678 00:42:33,560 --> 00:42:38,600 of times, but I -- 679 00:42:38,600 --> 00:42:43,690 yeah -- just to see the mechanics. 680 00:42:43,690 --> 00:42:48,000 But the, important thing is, why -- 681 00:42:48,000 --> 00:42:50,190 is, like, what happened? 682 00:42:50,190 --> 00:42:53,570 Why did we -- why did we get A inverse there? 683 00:42:53,570 --> 00:42:54,690 Let me ask you that. 684 00:42:57,860 --> 00:43:03,190 We got -- so we take -- 685 00:43:03,190 --> 00:43:08,020 We do row reduction, we do elimination on this long matrix 686 00:43:08,020 --> 00:43:12,840 A I until the first half 687 00:43:12,840 --> 00:43:14,990 Then a second half is A inverse. is up. 688 00:43:14,990 --> 00:43:20,050 Well, how do I see that? 689 00:43:20,050 --> 00:43:22,310 Let me put up here how I see that. 690 00:43:22,310 --> 00:43:30,100 So here's my Gauss-Jordan thing, and I'm doing stuff to it. 691 00:43:30,100 --> 00:43:34,320 So I'm -- well, whole lot of E's. 692 00:43:37,530 --> 00:43:40,150 Remember those are those elimination matrices. 693 00:43:40,150 --> 00:43:42,861 Those are the -- those are the things that we figured out last 694 00:43:42,861 --> 00:43:43,360 time. 695 00:43:43,360 --> 00:43:49,370 Yes, that's what an elimination step is it's in matrix form, 696 00:43:49,370 --> 00:43:51,430 I'm multiplying by some Es. 697 00:43:51,430 --> 00:43:55,120 And the result -- well, so I'm multiplying by a whole bunch 698 00:43:55,120 --> 00:43:55,650 of Es. 699 00:43:55,650 --> 00:43:57,530 So, I get a -- 700 00:43:57,530 --> 00:44:02,070 can I call the overall matrix E? 701 00:44:02,070 --> 00:44:06,420 That's the elimination matrix, the product of all those little 702 00:44:06,420 --> 00:44:06,920 pieces. 703 00:44:06,920 --> 00:44:09,050 What do I mean by little pieces? 704 00:44:09,050 --> 00:44:11,160 Well, there was an elimination matrix 705 00:44:11,160 --> 00:44:14,400 that subtracted two of that away from that. 706 00:44:14,400 --> 00:44:16,180 Then there was an elimination matrix 707 00:44:16,180 --> 00:44:19,160 that subtracted three of that away from that. 708 00:44:19,160 --> 00:44:21,370 I guess in this case, that was all. 709 00:44:21,370 --> 00:44:24,280 So there were just two Es in this case, one 710 00:44:24,280 --> 00:44:26,630 that did this step and one that did this step 711 00:44:26,630 --> 00:44:31,690 and together they gave me an E that does both steps. 712 00:44:31,690 --> 00:44:38,560 And the net result was to get an I here. 713 00:44:38,560 --> 00:44:45,490 And you can tell me what that has to be. 714 00:44:45,490 --> 00:44:50,240 This is, like, the picture of what happened. 715 00:44:50,240 --> 00:44:53,350 If E multiplied A, whatever that E is -- 716 00:44:53,350 --> 00:44:59,780 we never figured it out in this way. 717 00:44:59,780 --> 00:45:06,692 But whatever that E times that E is, E times A is -- 718 00:45:09,410 --> 00:45:12,120 What's E times A? 719 00:45:12,120 --> 00:45:13,480 It's I. 720 00:45:13,480 --> 00:45:20,510 That E, whatever the heck it was, multiplied A and produced 721 00:45:20,510 --> 00:45:22,710 So E must be -- 722 00:45:22,710 --> 00:45:30,640 E A equaling I tells us what E is, I. namely it is -- 723 00:45:30,640 --> 00:45:32,110 STUDENT: It's the inverse of A. 724 00:45:32,110 --> 00:45:33,720 STRANG: It's the inverse of A. 725 00:45:33,720 --> 00:45:35,660 Great. 726 00:45:35,660 --> 00:45:39,780 And therefore, when the second half, when E multiplies I, 727 00:45:39,780 --> 00:45:41,470 it's E -- 728 00:45:41,470 --> 00:45:44,760 Put this A inverse. 729 00:45:44,760 --> 00:45:48,450 You see the picture looking that way? 730 00:45:48,450 --> 00:45:49,910 E times A is the identity. 731 00:45:49,910 --> 00:45:53,480 It tells us what E has to be. 732 00:45:53,480 --> 00:45:55,600 It has to be the inverse, and therefore, 733 00:45:55,600 --> 00:45:57,870 on the right-hand side, where E -- 734 00:45:57,870 --> 00:46:00,580 where we just smartly tucked on the identity, 735 00:46:00,580 --> 00:46:03,060 it's turning in, step by step -- 736 00:46:03,060 --> 00:46:05,870 It's turning into A inverse. 737 00:46:05,870 --> 00:46:12,410 There is the statement of Gauss-Jordan elimination. 738 00:46:12,410 --> 00:46:14,590 That's how you find the inverse. 739 00:46:14,590 --> 00:46:19,210 Where we can look at it as elimination, 740 00:46:19,210 --> 00:46:24,260 as solving n equations at the same time -- -- 741 00:46:24,260 --> 00:46:28,170 and tacking on n columns, solving those equations and up 742 00:46:28,170 --> 00:46:32,740 goes the n columns of A inverse Okay, thanks. 743 00:46:32,740 --> 00:46:34,640 See you on Wednesday.