1 00:00:00,000 --> 00:00:07,610 2 00:00:07,610 --> 00:00:08,942 CHRISTINE BREINER: Welcome back to recitation. 3 00:00:08,942 --> 00:00:12,700 In this video, what I want to work on is using what we know 4 00:00:12,700 --> 00:00:16,360 about matrix multiplication and finding inverses of 5 00:00:16,360 --> 00:00:19,420 matrices to solve a system of equations. 6 00:00:19,420 --> 00:00:22,860 So we've set up the system already as if it's already in 7 00:00:22,860 --> 00:00:23,690 matrix form. 8 00:00:23,690 --> 00:00:27,550 And what I'd like us to do is, for this particular A-- this 3 9 00:00:27,550 --> 00:00:29,820 by 3 matrix A-- 10 00:00:29,820 --> 00:00:33,250 find a vector x, so that Ax equals b. 11 00:00:33,250 --> 00:00:35,750 Where b is equal to these two things. 12 00:00:35,750 --> 00:00:36,815 So you're going to do two problems. 13 00:00:36,815 --> 00:00:39,480 You're going to do when b equals 1, 2, negative 3. 14 00:00:39,480 --> 00:00:42,220 And you're going to do when b is equal to 0, 0, 0. 15 00:00:42,220 --> 00:00:45,220 So you want to find vector x so that Ax 16 00:00:45,220 --> 00:00:47,050 equals this value here. 17 00:00:47,050 --> 00:00:49,350 And what I'd like you to do is I'd like you to use the 18 00:00:49,350 --> 00:00:53,160 strategy that you saw in the lecture, which is find A 19 00:00:53,160 --> 00:00:56,110 inverse, and then take A inverse b. 20 00:00:56,110 --> 00:00:59,920 So we really want to practice understanding how to find the 21 00:00:59,920 --> 00:01:01,350 inverse of a matrix. 22 00:01:01,350 --> 00:01:04,340 So why don't you work on this, pause the video, when you feel 23 00:01:04,340 --> 00:01:06,690 comfortable, confident, that you have the right answer, 24 00:01:06,690 --> 00:01:09,200 then bring the video back up, and you can compare 25 00:01:09,200 --> 00:01:10,450 your work with mine. 26 00:01:10,450 --> 00:01:18,030 27 00:01:18,030 --> 00:01:19,080 OK, welcome back. 28 00:01:19,080 --> 00:01:21,330 Well, hopefully you were able to make some headway and you 29 00:01:21,330 --> 00:01:24,740 feel confident in your answers for 1 and 2. 30 00:01:24,740 --> 00:01:28,970 I am going to find the inverse of the matrix A first, and 31 00:01:28,970 --> 00:01:30,610 then solve the problem. 32 00:01:30,610 --> 00:01:32,850 And because there's a lot of computation, 33 00:01:32,850 --> 00:01:33,980 I may make a mistake. 34 00:01:33,980 --> 00:01:35,660 So I'm going to have to check every once in awhile 35 00:01:35,660 --> 00:01:38,830 that I'm doing OK. 36 00:01:38,830 --> 00:01:40,310 so hopefully-- it's too bad you can't tell me if I've made 37 00:01:40,310 --> 00:01:42,020 a mistake, but hopefully my studio audience 38 00:01:42,020 --> 00:01:43,260 will help me out. 39 00:01:43,260 --> 00:01:46,360 So the first thing I need to do is I need to find the 40 00:01:46,360 --> 00:01:49,170 determinant of A. So I'm going to do that first, and then I'm 41 00:01:49,170 --> 00:01:53,540 going to find the cofactor matrix and go from there. 42 00:01:53,540 --> 00:02:00,880 So if I want to find the determinant of A. I guess I'll 43 00:02:00,880 --> 00:02:04,860 just use the first row here, because it's pretty easy. 44 00:02:04,860 --> 00:02:07,020 So the determinant of A is going to be 3 times the 45 00:02:07,020 --> 00:02:09,170 determinant of this matrix-- 46 00:02:09,170 --> 00:02:10,920 this 2 by 2 matrix. 47 00:02:10,920 --> 00:02:13,280 So it's going to be 3 times-- 48 00:02:13,280 --> 00:02:15,830 and then I get a 2 times negative 1, which is negative 49 00:02:15,830 --> 00:02:20,750 2, and then minus 0-- so I get a 3 times negative 2. 50 00:02:20,750 --> 00:02:21,050 Oops. 51 00:02:21,050 --> 00:02:24,010 And I was about to write plus, but I should write minus. 52 00:02:24,010 --> 00:02:25,970 I take minus 1 times-- 53 00:02:25,970 --> 00:02:28,880 because this is my minus, I take negative of this thing 54 00:02:28,880 --> 00:02:32,920 times the matrix that is these two components in the first 55 00:02:32,920 --> 00:02:33,990 column and these two components 56 00:02:33,990 --> 00:02:35,290 in the second column. 57 00:02:35,290 --> 00:02:35,580 Right? 58 00:02:35,580 --> 00:02:39,360 We take away the column and the row that the 1 is 59 00:02:39,360 --> 00:02:41,490 contained in and we look at what remains-- the 2 by 2 60 00:02:41,490 --> 00:02:42,460 matrix that remains. 61 00:02:42,460 --> 00:02:44,240 And we find the determinant of that. 62 00:02:44,240 --> 00:02:46,060 So we get negative 1 times negative 1, 63 00:02:46,060 --> 00:02:47,450 which gives me a 1. 64 00:02:47,450 --> 00:02:50,960 And then negative 1 times 0 gives me a 0. 65 00:02:50,960 --> 00:02:56,470 So I just have the negative 1 from the row 1, column 2 spot, 66 00:02:56,470 --> 00:02:59,080 and then the determinant of the matrix that 67 00:02:59,080 --> 00:03:02,490 remains is 1, OK-- 68 00:03:02,490 --> 00:03:04,180 of the minor matrix that remains. 69 00:03:04,180 --> 00:03:07,110 And then the last one I should put a plus, but notice that it 70 00:03:07,110 --> 00:03:10,590 is a minus already, so I'm going to put just minus 1 71 00:03:10,590 --> 00:03:13,170 times what remains. 72 00:03:13,170 --> 00:03:13,990 What's this minor? 73 00:03:13,990 --> 00:03:17,740 This one is this 2 by 2 matrix I'm looking at, right? 74 00:03:17,740 --> 00:03:19,950 So I need to take the determinant of this 2 by 2 75 00:03:19,950 --> 00:03:23,620 matrix and multiply it by that negative 1 to get the third 76 00:03:23,620 --> 00:03:25,670 component here I have to add in. 77 00:03:25,670 --> 00:03:28,130 Negative 1 times negative 1 is 1. 78 00:03:28,130 --> 00:03:30,490 And then I subtract negative 1 times 2. 79 00:03:30,490 --> 00:03:31,660 So this is where I have to be careful. 80 00:03:31,660 --> 00:03:33,990 It's 1 minus negative 2. 81 00:03:33,990 --> 00:03:35,400 So I'm going to get a 3. 82 00:03:35,400 --> 00:03:36,670 OK. 83 00:03:36,670 --> 00:03:39,910 1 here minus a negative 2-- so 1 plus 2-- 84 00:03:39,910 --> 00:03:41,710 I'm going to get a 3. 85 00:03:41,710 --> 00:03:42,480 OK. 86 00:03:42,480 --> 00:03:46,010 And so negative 6 minus 1 minus 3-- 87 00:03:46,010 --> 00:03:48,720 looks like I get a negative 10. 88 00:03:48,720 --> 00:03:50,040 That's good, because I think that's what 89 00:03:50,040 --> 00:03:51,210 I'm supposed to get. 90 00:03:51,210 --> 00:03:52,340 OK. 91 00:03:52,340 --> 00:03:54,460 Now what I want to do is I want to find the matrix of 92 00:03:54,460 --> 00:03:58,090 minors for A. And then I'm going to find-- 93 00:03:58,090 --> 00:04:02,090 so I'm going to find the matrix of minors first, and 94 00:04:02,090 --> 00:04:05,180 then I'm going to switch the signs appropriately so I get 95 00:04:05,180 --> 00:04:07,010 the cofactors correct. 96 00:04:07,010 --> 00:04:07,880 OK? 97 00:04:07,880 --> 00:04:11,710 So some of them I already have. But, the whole matrix of 98 00:04:11,710 --> 00:04:14,370 minors, I'm going to just go through and do it again to be 99 00:04:14,370 --> 00:04:14,970 very careful. 100 00:04:14,970 --> 00:04:16,960 So the first one I delete. 101 00:04:16,960 --> 00:04:19,910 For the first row and column spot, I delete row 1 and 102 00:04:19,910 --> 00:04:23,030 column 1, and I look at the determinant of that matrix. 103 00:04:23,030 --> 00:04:27,520 That's 2 times negative 1 is negative 2 minus 0, so I get a 104 00:04:27,520 --> 00:04:29,510 negative 2 there. 105 00:04:29,510 --> 00:04:33,650 For the first row, second column I come back-- and I'm 106 00:04:33,650 --> 00:04:36,650 now again looking, I'm deleting this column and row-- 107 00:04:36,650 --> 00:04:39,230 and so I'm looking at the determinant of this matrix. 108 00:04:39,230 --> 00:04:42,890 So I get negative 1 times negative 1 is 1, minus 109 00:04:42,890 --> 00:04:45,150 0, so I get a 1. 110 00:04:45,150 --> 00:04:46,900 Again, I'm going to change all the signs later. 111 00:04:46,900 --> 00:04:49,600 So I'm going to do that in the second step. 112 00:04:49,600 --> 00:04:52,020 Now I'm in row 1, column 3. 113 00:04:52,020 --> 00:04:55,110 So I'm going to delete row 1, column 3 and look at the 114 00:04:55,110 --> 00:04:56,970 determinant of that matrix. 115 00:04:56,970 --> 00:05:00,010 I get negative 1 times negative 1 is 1, minus the 116 00:05:00,010 --> 00:05:01,820 negative 2, so there's my 3. 117 00:05:01,820 --> 00:05:06,310 Those I already knew, but I didn't want to just plop them 118 00:05:06,310 --> 00:05:06,750 in from here. 119 00:05:06,750 --> 00:05:08,000 But notice that is what you get here. 120 00:05:08,000 --> 00:05:09,410 Negative 2, 1, and 3. 121 00:05:09,410 --> 00:05:11,570 That's exactly where they come from, right? 122 00:05:11,570 --> 00:05:13,920 We got them by the same method. 123 00:05:13,920 --> 00:05:17,750 OK, and so now I want to find the minors for the rest of it. 124 00:05:17,750 --> 00:05:22,170 So let's look at, when I delete row 2, column 1, I'm 125 00:05:22,170 --> 00:05:24,200 left with 1, negative 1 here. 126 00:05:24,200 --> 00:05:26,390 Negative 1, negative 1 here. 127 00:05:26,390 --> 00:05:26,700 Right? 128 00:05:26,700 --> 00:05:29,980 So 1 times negative 1 is negative 1. 129 00:05:29,980 --> 00:05:31,830 And then negative and negative is positive. 130 00:05:31,830 --> 00:05:35,935 So it's negative 1 minus negative 1, so I 131 00:05:35,935 --> 00:05:38,200 get negative 2. 132 00:05:38,200 --> 00:05:40,660 That one was a lot of signs, so you might want to check. 133 00:05:40,660 --> 00:05:42,300 Maybe I should check. 134 00:05:42,300 --> 00:05:44,430 OK, maybe I should check. 135 00:05:44,430 --> 00:05:48,180 I'm deleting this column and this row, so I get 1 times 136 00:05:48,180 --> 00:05:49,290 negative 1. 137 00:05:49,290 --> 00:05:51,290 That's a negative 1, right? 138 00:05:51,290 --> 00:05:52,720 Negative 1 minus-- 139 00:05:52,720 --> 00:05:55,030 negative 1 times negative 1 is 1-- and so there's 140 00:05:55,030 --> 00:05:56,080 negative 1 minus 1. 141 00:05:56,080 --> 00:05:56,810 That looks good. 142 00:05:56,810 --> 00:05:58,700 Negative 2. 143 00:05:58,700 --> 00:05:59,060 Right? 144 00:05:59,060 --> 00:05:59,800 Negative, negative, negative. 145 00:05:59,800 --> 00:06:00,960 Yeah. 146 00:06:00,960 --> 00:06:02,150 OK. 147 00:06:02,150 --> 00:06:07,270 And then I'm looking at row 2, column 2. 148 00:06:07,270 --> 00:06:11,130 So now I'm deleting this row and this column. 149 00:06:11,130 --> 00:06:12,440 All right. 150 00:06:12,440 --> 00:06:14,190 And so I have these sort of diagonals here. 151 00:06:14,190 --> 00:06:16,150 That's what I'm interested in, right? 152 00:06:16,150 --> 00:06:17,570 So I get 3 times negative 1. 153 00:06:17,570 --> 00:06:20,100 That's negative 3. 154 00:06:20,100 --> 00:06:23,290 And then minus 1, because I have negative 1 times negative 155 00:06:23,290 --> 00:06:24,260 1 is positive 1. 156 00:06:24,260 --> 00:06:26,210 So negative 3 minus 1. 157 00:06:26,210 --> 00:06:29,020 So I should get negative 4. 158 00:06:29,020 --> 00:06:29,480 Right? 159 00:06:29,480 --> 00:06:30,760 And then I'm over here. 160 00:06:30,760 --> 00:06:35,800 So I need to delete this column and this row. 161 00:06:35,800 --> 00:06:39,010 So I get 3 times negative 1 is negative 3. 162 00:06:39,010 --> 00:06:40,720 Minus the negative 1, that's plus 1. 163 00:06:40,720 --> 00:06:46,340 So negative 3 plus 1 is negative 2. 164 00:06:46,340 --> 00:06:48,450 And before I go on, I'm going to check those first 2 rows. 165 00:06:48,450 --> 00:06:49,950 Because if I made a mistake now, it's only 166 00:06:49,950 --> 00:06:50,990 going to get worse. 167 00:06:50,990 --> 00:06:52,730 What did I have? 168 00:06:52,730 --> 00:06:53,420 Yes. 169 00:06:53,420 --> 00:06:53,690 OK. 170 00:06:53,690 --> 00:06:54,930 So far so good. 171 00:06:54,930 --> 00:06:55,750 Whew. 172 00:06:55,750 --> 00:06:56,520 All right. 173 00:06:56,520 --> 00:06:58,210 Next, final row. 174 00:06:58,210 --> 00:07:04,800 OK, final row is, I'm going to delete this column and row 175 00:07:04,800 --> 00:07:06,470 here, and I'm looking at this matrix. 176 00:07:06,470 --> 00:07:08,720 1 times 0 is 0. 177 00:07:08,720 --> 00:07:11,160 2 times negative 1 is negative 1, but I subtract that. 178 00:07:11,160 --> 00:07:15,730 So it's 0 minus negative 2, so it's 2. 179 00:07:15,730 --> 00:07:19,790 And then row 3, column 2. 180 00:07:19,790 --> 00:07:23,300 So row 3, I delete row 3 and column 2. 181 00:07:23,300 --> 00:07:25,090 3 times 0 is 0. 182 00:07:25,090 --> 00:07:26,740 0 minus-- 183 00:07:26,740 --> 00:07:29,340 negative 1 times negative 1 is 1-- so 0 minus 1, that's 184 00:07:29,340 --> 00:07:31,730 negative 1. 185 00:07:31,730 --> 00:07:35,690 And then the last spot, I'm deleting this 186 00:07:35,690 --> 00:07:37,310 row and this column. 187 00:07:37,310 --> 00:07:40,740 So 3 times 2 is 6 minus negative 1. 188 00:07:40,740 --> 00:07:43,280 I get 7. 189 00:07:43,280 --> 00:07:44,960 All right, let's check that row. 190 00:07:44,960 --> 00:07:46,080 2, negative 1, 7. 191 00:07:46,080 --> 00:07:46,490 OK. 192 00:07:46,490 --> 00:07:49,530 I have not done the cofactor matrix yet, because now I need 193 00:07:49,530 --> 00:07:51,530 to change the appropriate signs. 194 00:07:51,530 --> 00:07:54,330 OK, so if this is the matrix of minors, then if I want to 195 00:07:54,330 --> 00:07:57,060 change it to the cofactor matrix, what do I have to do? 196 00:07:57,060 --> 00:08:00,190 I'm going to scratch this out and write the cofactor matrix 197 00:08:00,190 --> 00:08:03,230 so that we can just change the signs appropriately. 198 00:08:03,230 --> 00:08:04,560 I'm going to do it all right here. 199 00:08:04,560 --> 00:08:05,570 And how does it work? 200 00:08:05,570 --> 00:08:08,450 Well, remember I'm going to go plus, minus, plus. 201 00:08:08,450 --> 00:08:09,940 Minus, plus, minus. 202 00:08:09,940 --> 00:08:10,830 Plus, minus, plus. 203 00:08:10,830 --> 00:08:12,960 I have to do this grid that starts with plus and 204 00:08:12,960 --> 00:08:13,960 alternates minus. 205 00:08:13,960 --> 00:08:18,190 So this sign stays the same, this sign switches, this sign 206 00:08:18,190 --> 00:08:19,090 stays the same. 207 00:08:19,090 --> 00:08:20,630 That's the plus, minus, plus. 208 00:08:20,630 --> 00:08:23,200 This one is going to be minus, plus, minus. 209 00:08:23,200 --> 00:08:24,690 So the minus switches that. 210 00:08:24,690 --> 00:08:26,490 Plus keeps that the same. 211 00:08:26,490 --> 00:08:28,430 Minus switches that. 212 00:08:28,430 --> 00:08:30,010 And then I was at minus, plus, minus. 213 00:08:30,010 --> 00:08:32,300 So I'm going to have plus, minus, plus. 214 00:08:32,300 --> 00:08:34,690 And so these two stay the same, and this one switches. 215 00:08:34,690 --> 00:08:36,404 So a lot of things that were negative became positive. 216 00:08:36,404 --> 00:08:38,650 And I had to change-- 217 00:08:38,650 --> 00:08:41,090 maybe I threw in one negative, maybe not. 218 00:08:41,090 --> 00:08:43,720 But, so out all the signs I kept, this one stayed the 219 00:08:43,720 --> 00:08:45,550 same, this one stayed the same, this one stayed the 220 00:08:45,550 --> 00:08:49,140 same, these two stayed the same, and then these 4 221 00:08:49,140 --> 00:08:53,010 switched, because it's the plus, minus, plus sort of grid 222 00:08:53,010 --> 00:08:54,980 that I have to put on top of this. 223 00:08:54,980 --> 00:08:56,870 OK, so that's the cofactor matrix. 224 00:08:56,870 --> 00:08:58,620 We're getting closer. 225 00:08:58,620 --> 00:09:02,830 OK, now we need the transpose of this, right? 226 00:09:02,830 --> 00:09:08,220 227 00:09:08,220 --> 00:09:09,025 So if I look at the transpose, actually, know what I'm going 228 00:09:09,025 --> 00:09:12,070 to do-- because I'm also just going to have to take the 229 00:09:12,070 --> 00:09:14,330 transpose and then multiply it by 1 over the determinant-- 230 00:09:14,330 --> 00:09:16,190 I'm going to do that all at once. 231 00:09:16,190 --> 00:09:16,980 OK. 232 00:09:16,980 --> 00:09:19,380 Because we can do that all at once, and then we don't have 233 00:09:19,380 --> 00:09:21,750 to worry about it. 234 00:09:21,750 --> 00:09:25,560 So A inverse I know is going to be negative 1/10, because 235 00:09:25,560 --> 00:09:27,720 the determinant was minus 10. 236 00:09:27,720 --> 00:09:31,130 So it's 1 over the determinant times the 237 00:09:31,130 --> 00:09:32,790 transpose of this matrix. 238 00:09:32,790 --> 00:09:34,544 So the transpose of this matrix-- remember what I'm 239 00:09:34,544 --> 00:09:37,650 going to do is essentially you fix the diagonal and you're 240 00:09:37,650 --> 00:09:39,000 going to flip. 241 00:09:39,000 --> 00:09:40,220 That's really what, in the square matrix, that's how you 242 00:09:40,220 --> 00:09:41,230 can think about it. 243 00:09:41,230 --> 00:09:43,310 But every column is going to become a row. 244 00:09:43,310 --> 00:09:45,620 So I'm going to write this as my first row. 245 00:09:45,620 --> 00:09:47,500 This first column is going to become my first row. 246 00:09:47,500 --> 00:09:52,810 So it's going to be negative 2, 2, 2 as my first row. 247 00:09:52,810 --> 00:09:54,720 And then the next column is going to be negative 1, 248 00:09:54,720 --> 00:09:57,670 negative 4, 1. 249 00:09:57,670 --> 00:09:59,010 I mean next row. 250 00:09:59,010 --> 00:10:00,930 I will take a column and change it to a row. 251 00:10:00,930 --> 00:10:05,220 The next row is going to be negative 1, negative 4, 1. 252 00:10:05,220 --> 00:10:06,360 And then the last one. 253 00:10:06,360 --> 00:10:08,620 I take this column and I change it to a row. 254 00:10:08,620 --> 00:10:09,970 It's going to be 3, 2, 7. 255 00:10:09,970 --> 00:10:14,290 256 00:10:14,290 --> 00:10:14,780 OK. 257 00:10:14,780 --> 00:10:16,530 And because again, I want to make sure-- this 258 00:10:16,530 --> 00:10:18,220 one is really messy-- 259 00:10:18,220 --> 00:10:21,550 I want to make sure I have something similar for that, or 260 00:10:21,550 --> 00:10:22,800 exactly that. 261 00:10:22,800 --> 00:10:23,530 OK. 262 00:10:23,530 --> 00:10:25,850 I think I'm still doing all right. 263 00:10:25,850 --> 00:10:27,670 Now, let's get to solving the problem. 264 00:10:27,670 --> 00:10:29,920 Because so far, we just were finding the inverse matrix. 265 00:10:29,920 --> 00:10:31,860 So I'm going to leave it in this form, instead of dividing 266 00:10:31,860 --> 00:10:35,290 by 10 in every spot, because that will be annoying. 267 00:10:35,290 --> 00:10:39,870 So let's think about how do I want to solve the system. 268 00:10:39,870 --> 00:10:43,990 That I had, I had Ax equals b. 269 00:10:43,990 --> 00:10:47,470 And actually, I mean, my strategy is to find the 270 00:10:47,470 --> 00:10:48,770 inverse matrix. 271 00:10:48,770 --> 00:10:51,300 I didn't talk to you about why we know the inverse matrix 272 00:10:51,300 --> 00:10:54,150 actually exists. 273 00:10:54,150 --> 00:10:55,540 But ultimately, you haven't even seen this yet in the 274 00:10:55,540 --> 00:10:56,450 lecture videos, really. 275 00:10:56,450 --> 00:10:59,690 Except that you know that the determinant of A being 276 00:10:59,690 --> 00:11:01,790 non-zero gives you an inverse matrix. 277 00:11:01,790 --> 00:11:03,750 That's all you know, I think, at this point. 278 00:11:03,750 --> 00:11:06,360 That you have the determinant of A. It's non-zero, so you 279 00:11:06,360 --> 00:11:08,100 can find an inverse matrix. 280 00:11:08,100 --> 00:11:11,290 Makes sense based on the formulation you have, because 281 00:11:11,290 --> 00:11:14,300 if the determinant is 0, then this quantity 1 over the 282 00:11:14,300 --> 00:11:17,590 determinant of A, you've run into quite a bit of trouble. 283 00:11:17,590 --> 00:11:20,920 So that's just as a little sidebar, we know the inverse 284 00:11:20,920 --> 00:11:22,960 matrix exists for A. 285 00:11:22,960 --> 00:11:24,340 So what we do-- 286 00:11:24,340 --> 00:11:25,470 this is again the strategy-- 287 00:11:25,470 --> 00:11:29,370 you multiply A inverse A times x on the left side. 288 00:11:29,370 --> 00:11:31,480 Ooh. 289 00:11:31,480 --> 00:11:34,000 Is equal to-- sorry-- that should be the lowercase b. 290 00:11:34,000 --> 00:11:35,370 Should be a vector there. 291 00:11:35,370 --> 00:11:39,800 It is equal to A inverse b on the right hand side. 292 00:11:39,800 --> 00:11:42,880 And you notice, it's very important, in the matrix 293 00:11:42,880 --> 00:11:46,780 multiplication video we saw that it's very important the 294 00:11:46,780 --> 00:11:48,700 order in which you multiply matrices. 295 00:11:48,700 --> 00:11:52,120 And since I'm putting A inverse on the far left of 296 00:11:52,120 --> 00:11:55,940 this side of the equality, I have to put it on the far left 297 00:11:55,940 --> 00:11:58,620 of the right hand side of the equality. 298 00:11:58,620 --> 00:11:58,960 Right? 299 00:11:58,960 --> 00:12:01,190 And in fact, you would run into trouble if you tried to 300 00:12:01,190 --> 00:12:02,430 switch the order of these. 301 00:12:02,430 --> 00:12:02,760 OK? 302 00:12:02,760 --> 00:12:05,250 We wouldn't be able to multiply them. 303 00:12:05,250 --> 00:12:06,000 All right? 304 00:12:06,000 --> 00:12:10,000 So A inverse A, we know is just the identity matrix. 305 00:12:10,000 --> 00:12:13,620 So you get the identity matrix times x is equal 306 00:12:13,620 --> 00:12:16,430 to A inverse b. 307 00:12:16,430 --> 00:12:22,640 So you can find x by finding A inverse times b. 308 00:12:22,640 --> 00:12:23,470 Right? 309 00:12:23,470 --> 00:12:25,140 And so now we have A inverse. 310 00:12:25,140 --> 00:12:26,860 Let's see if we can solve the problem. 311 00:12:26,860 --> 00:12:28,830 One point I want to make is that now 312 00:12:28,830 --> 00:12:30,080 that you have A inverse-- 313 00:12:30,080 --> 00:12:32,440 I've tried to ask you to solve the problem for 314 00:12:32,440 --> 00:12:33,160 two different b's-- 315 00:12:33,160 --> 00:12:35,320 you don't have to go and find A inverse again, right? 316 00:12:35,320 --> 00:12:36,320 You're done finding A inverse. 317 00:12:36,320 --> 00:12:39,130 You just now have to do the multiplication. 318 00:12:39,130 --> 00:12:42,660 So now for number 1, we had b was equal to-- 319 00:12:42,660 --> 00:12:44,030 I'm going to write it here, so I don't have to 320 00:12:44,030 --> 00:12:45,070 keep looking over-- 321 00:12:45,070 --> 00:12:49,120 1, 2, negative 3. 322 00:12:49,120 --> 00:12:56,040 So A inverse b is going to be equal to-- well I should get 323 00:12:56,040 --> 00:12:57,860 another vector, so I should just have 324 00:12:57,860 --> 00:12:59,610 three components here. 325 00:12:59,610 --> 00:13:02,200 And I'm probably going to have to write out what I get, 326 00:13:02,200 --> 00:13:04,370 because it might be long. 327 00:13:04,370 --> 00:13:05,570 But let's see-- actually, you know what I'm going to do to 328 00:13:05,570 --> 00:13:07,050 make it easier? 329 00:13:07,050 --> 00:13:08,620 Because there's a lot of junk going on here. 330 00:13:08,620 --> 00:13:10,510 So what I'm going to do to make it easier is put the 331 00:13:10,510 --> 00:13:13,340 negative 1/10 in front to start. 332 00:13:13,340 --> 00:13:15,740 Because that negative 1/10 is going to come along with every 333 00:13:15,740 --> 00:13:18,800 term, so I'm just going to put the negative 1/10 in front and 334 00:13:18,800 --> 00:13:20,290 deal with it at the end. 335 00:13:20,290 --> 00:13:21,540 OK? 336 00:13:21,540 --> 00:13:24,020 So now I'm just going to multiply b-- which is this 1, 337 00:13:24,020 --> 00:13:25,260 2, negative 3-- 338 00:13:25,260 --> 00:13:27,860 by this big matrix here without the 339 00:13:27,860 --> 00:13:29,480 negative 1/10 in front. 340 00:13:29,480 --> 00:13:30,510 OK? 341 00:13:30,510 --> 00:13:31,320 So let's look at that. 342 00:13:31,320 --> 00:13:35,190 We're just going to have first row times the column, and 343 00:13:35,190 --> 00:13:37,020 that's going to give me the first position. 344 00:13:37,020 --> 00:13:40,115 So negative 2 times 1 is negative 2. 345 00:13:40,115 --> 00:13:42,030 I'm going to write them all down. 346 00:13:42,030 --> 00:13:45,350 Plus 2 times 2 is 4. 347 00:13:45,350 --> 00:13:49,300 Plus 2 times negative 3 is negative 6. 348 00:13:49,300 --> 00:13:50,730 So that's the first position. 349 00:13:50,730 --> 00:13:52,920 We'll simplify in a moment. 350 00:13:52,920 --> 00:13:55,180 So the next one, I get negative 1 times 1. 351 00:13:55,180 --> 00:13:56,490 That's negative 1. 352 00:13:56,490 --> 00:13:58,450 Then I get negative 4 times 2. 353 00:13:58,450 --> 00:13:59,550 That's negative 8. 354 00:13:59,550 --> 00:14:01,190 So minus 8. 355 00:14:01,190 --> 00:14:05,080 And then I get 1 times negative 3, so minus 3. 356 00:14:05,080 --> 00:14:07,750 So we've got two of the rows done. 357 00:14:07,750 --> 00:14:09,940 We just have to simplify them in a moment. 358 00:14:09,940 --> 00:14:12,360 And now we just do this third component. 359 00:14:12,360 --> 00:14:17,080 So it's the third row of A inverse without that scalar in 360 00:14:17,080 --> 00:14:20,010 front, times the only column of b to 361 00:14:20,010 --> 00:14:21,810 give me the last position. 362 00:14:21,810 --> 00:14:22,450 Right? 363 00:14:22,450 --> 00:14:26,460 So 3 times 1 is 3, plus 2 times 2 is 4, so I get 3 plus 364 00:14:26,460 --> 00:14:31,250 4, and then 7 times negative 3 is minus 21. 365 00:14:31,250 --> 00:14:32,090 OK. 366 00:14:32,090 --> 00:14:33,940 So what do I get when I write it all out? 367 00:14:33,940 --> 00:14:35,500 I get negative 1/10. 368 00:14:35,500 --> 00:14:37,380 And then-- 369 00:14:37,380 --> 00:14:40,750 so negative 8 plus 4-- that looks like a minus 4. 370 00:14:40,750 --> 00:14:41,660 Right? 371 00:14:41,660 --> 00:14:44,210 8, 9, 10, 11, 12. 372 00:14:44,210 --> 00:14:46,170 That looks like a negative 12. 373 00:14:46,170 --> 00:14:47,310 It's a lot of adding for me. 374 00:14:47,310 --> 00:14:48,820 I make a lot of adding mistakes, so 375 00:14:48,820 --> 00:14:49,950 we should be careful. 376 00:14:49,950 --> 00:14:52,170 This looks like negative 14. 377 00:14:52,170 --> 00:14:53,030 OK. 378 00:14:53,030 --> 00:14:58,600 So this is a matrix that, it's just a vector, right? 379 00:14:58,600 --> 00:15:00,040 All the negative signs will drop out. 380 00:15:00,040 --> 00:15:01,750 I'll get some fractions. 381 00:15:01,750 --> 00:15:05,370 But if it is the correct answer-- which I'm really 382 00:15:05,370 --> 00:15:07,510 hoping it is, because I just did this whole problem and I 383 00:15:07,510 --> 00:15:09,340 hope it's the correct answer-- if it's the correct answer, 384 00:15:09,340 --> 00:15:10,640 then what should it do? 385 00:15:10,640 --> 00:15:13,760 When I take the original A that I had and I multiply it 386 00:15:13,760 --> 00:15:16,020 by this, I should get b. 387 00:15:16,020 --> 00:15:17,870 I should get 1, 2, negative 3. 388 00:15:17,870 --> 00:15:19,670 So you can check your work very easily 389 00:15:19,670 --> 00:15:20,910 to see if it works. 390 00:15:20,910 --> 00:15:25,350 You can take A times this, and see if you get b. 391 00:15:25,350 --> 00:15:26,340 Right? 392 00:15:26,340 --> 00:15:30,740 And then you'll know if this is the x we were looking for. 393 00:15:30,740 --> 00:15:32,090 OK? 394 00:15:32,090 --> 00:15:34,550 And then let's look at number two. 395 00:15:34,550 --> 00:15:38,260 I just said that b equals 0, 0, 0. 396 00:15:38,260 --> 00:15:42,499 And the point I want to make there is that since this has 397 00:15:42,499 --> 00:15:45,230 an inverse, A inverse, since A has an inverse, A inverse b is 398 00:15:45,230 --> 00:15:46,370 going to be-- 399 00:15:46,370 --> 00:15:46,960 in this case-- 400 00:15:46,960 --> 00:15:53,860 A inverse times 0, 0, 0, which is going to give you 0, 0, 0. 401 00:15:53,860 --> 00:15:56,280 So the only solution we have in this case-- 402 00:15:56,280 --> 00:15:59,080 because A inverse, if I look and I try and multiply every 403 00:15:59,080 --> 00:16:02,220 row by this column, right, I'm going to get 0 in the first 404 00:16:02,220 --> 00:16:05,060 spot, 0 in the second spot, and 0 in the third spot-- 405 00:16:05,060 --> 00:16:07,910 so the solution I get-- 406 00:16:07,910 --> 00:16:11,940 the x I'm looking for, so that Ax equals 0, 0, 0-- 407 00:16:11,940 --> 00:16:13,760 is 0, 0, 0. 408 00:16:13,760 --> 00:16:17,810 And what I just want to mention to you, is that that 409 00:16:17,810 --> 00:16:20,390 is true because A is invertible. 410 00:16:20,390 --> 00:16:23,260 If A were not invertible, you could get other solutions. 411 00:16:23,260 --> 00:16:24,230 Other things might work. 412 00:16:24,230 --> 00:16:26,630 And that's also true, actually, in this case as 413 00:16:26,630 --> 00:16:29,860 well, but it's a little harder to see that it could be 414 00:16:29,860 --> 00:16:32,480 potentially a weird thing. 415 00:16:32,480 --> 00:16:39,550 To solve Ax equals 0, 0, 0, it's sort of like, naturally 416 00:16:39,550 --> 00:16:41,380 we see 0, 0, 0 as a solution. 417 00:16:41,380 --> 00:16:44,090 Right away you can see that, and that's one that we get. 418 00:16:44,090 --> 00:16:46,050 The point I want to make is because A is invertible, 419 00:16:46,050 --> 00:16:47,620 that's the only solution. 420 00:16:47,620 --> 00:16:49,890 And if A were not invertible, you could get other 421 00:16:49,890 --> 00:16:51,260 solutions to that. 422 00:16:51,260 --> 00:16:53,920 So that's something that we haven't seen yet-- we haven't 423 00:16:53,920 --> 00:16:54,780 dealt with yet-- 424 00:16:54,780 --> 00:16:57,140 but that is something that can happen. 425 00:16:57,140 --> 00:16:59,940 So I just want to point out that there could be an oddity 426 00:16:59,940 --> 00:17:01,490 if A were not invertible. 427 00:17:01,490 --> 00:17:04,150 But since A is invertible, we get just one solution for both 428 00:17:04,150 --> 00:17:05,470 of these things. 429 00:17:05,470 --> 00:17:06,740 OK. 430 00:17:06,740 --> 00:17:09,120 So I'm going to go back and just remind you of a few 431 00:17:09,120 --> 00:17:11,470 things of how we found the inverse matrix, 432 00:17:11,470 --> 00:17:13,210 and then I will stop. 433 00:17:13,210 --> 00:17:16,340 So we were given a matrix A. And to go through the steps of 434 00:17:16,340 --> 00:17:18,570 finding the inverse matrix, what did we do? 435 00:17:18,570 --> 00:17:21,900 The first thing we did was we found the determinant. 436 00:17:21,900 --> 00:17:23,690 Then we found the matrix of minors. 437 00:17:23,690 --> 00:17:26,810 And then I just took that matrix of minors, put the plus 438 00:17:26,810 --> 00:17:28,980 minus grid on top of it so that I got 439 00:17:28,980 --> 00:17:31,410 the cofactor matrix. 440 00:17:31,410 --> 00:17:31,650 Right? 441 00:17:31,650 --> 00:17:33,850 And then once I had the cofactor matrix, you just have 442 00:17:33,850 --> 00:17:35,580 to transpose it. 443 00:17:35,580 --> 00:17:36,350 So I came over here. 444 00:17:36,350 --> 00:17:38,220 I transposed that, and I put 1 over the 445 00:17:38,220 --> 00:17:40,260 determinant of A in front. 446 00:17:40,260 --> 00:17:43,160 So the scalar is 1 over the determinant of A times the 447 00:17:43,160 --> 00:17:45,190 transpose of the cofactor matrix. 448 00:17:45,190 --> 00:17:47,180 And that's what gives me A inverse. 449 00:17:47,180 --> 00:17:49,710 So there are a fair number of steps, but you can do them 450 00:17:49,710 --> 00:17:52,160 very systematically, and then you have the inverse matrix 451 00:17:52,160 --> 00:17:53,300 that you're looking for. 452 00:17:53,300 --> 00:17:58,220 And then you can solve for x when you're looking for Ax 453 00:17:58,220 --> 00:18:01,030 equals b, and you know b and you know A. And you do this 454 00:18:01,030 --> 00:18:04,010 same process we just outlined here again, and 455 00:18:04,010 --> 00:18:05,590 that gives it to you. 456 00:18:05,590 --> 00:18:07,620 OK, I think I'll stop there. 457 00:18:07,620 --> 00:18:07,955