1 00:00:00,000 --> 00:00:10,250 OK, this is lecture twenty. 2 00:00:10,250 --> 00:00:15,680 And this is the final lecture on determinants. 3 00:00:15,680 --> 00:00:18,390 And it's about the applications. 4 00:00:18,390 --> 00:00:22,730 So we worked hard in the last two lectures 5 00:00:22,730 --> 00:00:25,900 to get a formula for the determinant 6 00:00:25,900 --> 00:00:27,980 and the properties of the determinant. 7 00:00:27,980 --> 00:00:35,250 Now to use the determinant and, and always this determinant 8 00:00:35,250 --> 00:00:40,710 packs all this information into a single number. 9 00:00:40,710 --> 00:00:45,800 And that number can give us formulas 10 00:00:45,800 --> 00:00:50,510 for all sorts of, things that we've been calculating already 11 00:00:50,510 --> 00:00:51,550 without formulas. 12 00:00:51,550 --> 00:00:54,110 Now what was A inverse? 13 00:00:54,110 --> 00:00:57,690 So, so I'm beginning with the formula for A inverse. 14 00:00:57,690 --> 00:01:00,400 15 00:01:00,400 --> 00:01:02,450 Two, two by two formula we know, 16 00:01:02,450 --> 00:01:03,070 right? 17 00:01:03,070 --> 00:01:09,700 The two by two formula for A inverse, the inverse of a b c d 18 00:01:09,700 --> 00:01:22,160 inverse is one over the determinant times d a -b -c. 19 00:01:22,160 --> 00:01:25,630 20 00:01:25,630 --> 00:01:31,140 Somehow I want to see what's going on for three by three 21 00:01:31,140 --> 00:01:33,140 and n by n. 22 00:01:33,140 --> 00:01:37,200 And actually maybe you can see what's going on from this two 23 00:01:37,200 --> 00:01:38,870 by two case. 24 00:01:38,870 --> 00:01:41,060 So there's a formula for the inverse, 25 00:01:41,060 --> 00:01:44,390 and what did I divide by? 26 00:01:44,390 --> 00:01:45,880 The determinant. 27 00:01:45,880 --> 00:01:50,220 So what I'm looking for is a formula 28 00:01:50,220 --> 00:01:53,590 where it has one over the determinant 29 00:01:53,590 --> 00:01:56,690 and, and you remember why that makes good sense, 30 00:01:56,690 --> 00:02:02,150 because then that's perfect as long as the determinant isn't 31 00:02:02,150 --> 00:02:06,330 zero, and that's exactly when there is an inverse. 32 00:02:06,330 --> 00:02:11,640 But now I have to ask can we recognize any of this stuff? 33 00:02:11,640 --> 00:02:16,410 Do you recognize what that number d is from the past? 34 00:02:16,410 --> 00:02:19,730 From last, from the last lecture? 35 00:02:19,730 --> 00:02:24,200 My hint is think cofactors. 36 00:02:24,200 --> 00:02:28,480 Because my formula is going to be, my formula for the inverse 37 00:02:28,480 --> 00:02:30,760 is going to be one over the determinant 38 00:02:30,760 --> 00:02:33,660 times a matrix of cofactors. 39 00:02:33,660 --> 00:02:35,290 So you remember that D? 40 00:02:35,290 --> 00:02:37,700 What's that the cofactor of? 41 00:02:37,700 --> 00:02:39,230 Remember cofactors? 42 00:02:39,230 --> 00:02:41,530 If -- that's the one one cofactor, 43 00:02:41,530 --> 00:02:47,570 because if I strike out row and column one, I'm left with d. 44 00:02:47,570 --> 00:02:51,080 And what's minus b? 45 00:02:51,080 --> 00:02:51,930 OK. 46 00:02:51,930 --> 00:02:53,780 Which cofactor is that one? 47 00:02:53,780 --> 00:03:00,900 Oh, minus b is the cofactor of c, right? 48 00:03:00,900 --> 00:03:04,670 If I strike out the c, I'm left with a b there. 49 00:03:04,670 --> 00:03:07,400 And why the minus sign? 50 00:03:07,400 --> 00:03:12,310 Because this c was in a two one position, and two plus one is 51 00:03:12,310 --> 00:03:12,910 odd. 52 00:03:12,910 --> 00:03:17,000 So there was a minus went into the cofactor, and that's it. 53 00:03:17,000 --> 00:03:17,780 OK. 54 00:03:17,780 --> 00:03:21,710 I'll write down next what my formula is. 55 00:03:21,710 --> 00:03:26,900 Here's the big formula for the A -- for A inverse. 56 00:03:26,900 --> 00:03:35,800 It's one over the determinant of A and then some matrix. 57 00:03:35,800 --> 00:03:40,280 And that matrix is the matrix of cofactors, 58 00:03:40,280 --> 00:03:41,370 c. 59 00:03:41,370 --> 00:03:45,330 Only one thing, it turns -- you'll see, 60 00:03:45,330 --> 00:03:47,920 I have to, I transpose. 61 00:03:47,920 --> 00:03:50,760 So this is the matrix of cofactors, the -- 62 00:03:50,760 --> 00:03:52,090 what I'll just -- 63 00:03:52,090 --> 00:03:55,710 but why don't we just call it the cofactor matrix. 64 00:03:55,710 --> 00:04:02,400 So the one one entry of, of c is the cof- is the one one 65 00:04:02,400 --> 00:04:05,840 cofactor, the thing that we get by throwing away row and column 66 00:04:05,840 --> 00:04:06,340 one. 67 00:04:06,340 --> 00:04:09,770 It's the d. 68 00:04:09,770 --> 00:04:15,420 And, because of the transpose, what I see up here 69 00:04:15,420 --> 00:04:19,140 is the cofactor of this guy down here, right? 70 00:04:19,140 --> 00:04:22,450 That's where the transpose came in. 71 00:04:22,450 --> 00:04:27,420 What I see here, this is the cofactor not of this one, 72 00:04:27,420 --> 00:04:28,690 because I've transposed. 73 00:04:28,690 --> 00:04:30,920 This is the cofactor of the b. 74 00:04:30,920 --> 00:04:35,350 When I throw away the b, the b row and the b column, 75 00:04:35,350 --> 00:04:38,240 I'm left with c, and then I have that minus sign 76 00:04:38,240 --> 00:04:38,940 again. 77 00:04:38,940 --> 00:04:44,580 And of course the two two entry is the cofactor 78 00:04:44,580 --> 00:04:48,660 of d, and that's this a. 79 00:04:48,660 --> 00:04:49,930 So there's the formula. 80 00:04:49,930 --> 00:04:50,790 OK. 81 00:04:50,790 --> 00:04:53,080 But we got to think why. 82 00:04:53,080 --> 00:04:56,680 I mean, it worked in this two by two case, 83 00:04:56,680 --> 00:04:59,590 but a lot of other formulas would have worked just as well. 84 00:04:59,590 --> 00:05:02,200 We, we have to see why that's true. 85 00:05:02,200 --> 00:05:05,790 In other words, why is it -- 86 00:05:05,790 --> 00:05:09,530 so this is what I aim to find. 87 00:05:09,530 --> 00:05:12,570 And, and let's just sort of look to see what is that telling 88 00:05:12,570 --> 00:05:13,210 us. 89 00:05:13,210 --> 00:05:17,400 That tells us that the -- the expression for A inverse -- 90 00:05:17,400 --> 00:05:20,670 let's look at a three by three. 91 00:05:20,670 --> 00:05:27,240 Can I just write down a a b c d e f g h i? 92 00:05:27,240 --> 00:05:30,780 And I'm looking for its inverse. 93 00:05:30,780 --> 00:05:33,340 And what kind of a formula -- do I see there? 94 00:05:33,340 --> 00:05:34,720 I mean, what -- 95 00:05:34,720 --> 00:05:38,630 the determinant is a bunch of products of three factors, 96 00:05:38,630 --> 00:05:39,370 right? 97 00:05:39,370 --> 00:05:43,190 The determinant of this matrix'll involve a e i, 98 00:05:43,190 --> 00:05:49,160 and b f times g, and c times d times h, and minus c 99 00:05:49,160 --> 00:05:50,900 e g, and so on. 100 00:05:50,900 --> 00:05:54,810 So things with three factors go in here. 101 00:05:54,810 --> 00:06:03,140 Things with how many factors do things in the cofactor matrix 102 00:06:03,140 --> 00:06:03,640 have? 103 00:06:03,640 --> 00:06:06,030 What's a typical cofactor? 104 00:06:06,030 --> 00:06:09,000 What's the cofactor of a? 105 00:06:09,000 --> 00:06:12,800 The cofactor of a, the one one entry up here in the inverse 106 00:06:12,800 --> 00:06:14,700 is? 107 00:06:14,700 --> 00:06:17,690 I throw away the row and column containing a 108 00:06:17,690 --> 00:06:19,740 and I take the determinant of what's left, 109 00:06:19,740 --> 00:06:21,230 that's the cofactor. 110 00:06:21,230 --> 00:06:24,390 And that's e i minus f h. 111 00:06:24,390 --> 00:06:25,890 Products of two things. 112 00:06:25,890 --> 00:06:29,560 Now, I'm just making the observation 113 00:06:29,560 --> 00:06:36,900 that the determinant of A involves products of n entries. 114 00:06:36,900 --> 00:06:41,200 115 00:06:41,200 --> 00:06:51,310 And the cofactor matrix involves products of n minus 1 entries. 116 00:06:51,310 --> 00:06:56,620 And, like, we never noticed any of this stuff 117 00:06:56,620 --> 00:06:58,780 when we were computing the inverse 118 00:06:58,780 --> 00:07:01,090 by the Gauss-Jordan method or whatever. 119 00:07:01,090 --> 00:07:02,380 You remember how we did it? 120 00:07:02,380 --> 00:07:06,620 We took the matrix A, we tucked the identity next to it, 121 00:07:06,620 --> 00:07:10,740 we did elimination till A became the identity. 122 00:07:10,740 --> 00:07:13,980 And then that, the identity suddenly was A inverse. 123 00:07:13,980 --> 00:07:16,870 Well, that was great numerically. 124 00:07:16,870 --> 00:07:21,300 But we never knew what was going on, basically. 125 00:07:21,300 --> 00:07:25,000 And now we see what the formula is 126 00:07:25,000 --> 00:07:27,830 in terms of letters, what's the algebra instead 127 00:07:27,830 --> 00:07:29,530 of the algorithm. 128 00:07:29,530 --> 00:07:30,140 OK. 129 00:07:30,140 --> 00:07:33,780 But I have to say why this is right, right? 130 00:07:33,780 --> 00:07:37,940 I still -- that's a pretty magic formula. 131 00:07:37,940 --> 00:07:40,150 Where does it come from? 132 00:07:40,150 --> 00:07:41,320 Well, I'll just check it. 133 00:07:41,320 --> 00:07:44,200 Having, having got it up there, let me -- 134 00:07:44,200 --> 00:07:48,510 I'll say, how can we check -- 135 00:07:48,510 --> 00:07:50,510 what do I want to check? 136 00:07:50,510 --> 00:07:55,150 I want to check that A times its inverse gives the identity. 137 00:07:55,150 --> 00:07:58,130 So I want, I want to check that A times this thing, 138 00:07:58,130 --> 00:08:02,000 A times this -- now I'm going to write in the inverse -- 139 00:08:02,000 --> 00:08:03,220 gives the identity. 140 00:08:03,220 --> 00:08:06,830 So I check that A times C transpose -- 141 00:08:06,830 --> 00:08:09,340 let me bring the determinant up here. 142 00:08:09,340 --> 00:08:14,490 Determinant of A times the identity. 143 00:08:14,490 --> 00:08:16,900 That's my job. 144 00:08:16,900 --> 00:08:24,340 That's it, that if this is true, and it is, then, then I've 145 00:08:24,340 --> 00:08:28,760 correctly identified A inverse as C transpose divided 146 00:08:28,760 --> 00:08:30,810 by the determinant. 147 00:08:30,810 --> 00:08:31,310 OK. 148 00:08:31,310 --> 00:08:34,260 But why is this true? 149 00:08:34,260 --> 00:08:35,220 Why is that true? 150 00:08:35,220 --> 00:08:38,059 Let me, let me put down what I'm doing here. 151 00:08:38,059 --> 00:08:40,220 I have A -- 152 00:08:40,220 --> 00:08:43,630 here, here's A, here's a11 -- 153 00:08:43,630 --> 00:08:48,150 I'm doing this multiplication -- along to a1n. 154 00:08:48,150 --> 00:08:53,210 And then down in this last row will be an an1 along to ann. 155 00:08:53,210 --> 00:08:59,660 And I'm multiplying by the cofactor matrix transposed. 156 00:08:59,660 --> 00:09:07,820 So when I transpose, it'll be c11 c12 down to c1n. 157 00:09:07,820 --> 00:09:11,340 Notice usually that one coming first 158 00:09:11,340 --> 00:09:15,000 would mean I'm in row one, but I've transposed, 159 00:09:15,000 --> 00:09:18,200 so that's, those are the cofactors. 160 00:09:18,200 --> 00:09:21,700 This first column are the cofactors from row one. 161 00:09:21,700 --> 00:09:28,670 And then the last column would be the cofactors from row n. 162 00:09:28,670 --> 00:09:33,960 And why should that come out to be anything good? 163 00:09:33,960 --> 00:09:37,790 In fact, why should it come out to be as good as this? 164 00:09:37,790 --> 00:09:43,701 Well, you can tell me what the one one entry in the product 165 00:09:43,701 --> 00:09:44,200 is. 166 00:09:44,200 --> 00:09:47,630 This is like you're seeing the main point if you just 167 00:09:47,630 --> 00:09:49,750 tell me one entry. 168 00:09:49,750 --> 00:09:54,470 What do I get up there in the one one entry 169 00:09:54,470 --> 00:10:00,600 when I do this row of this row from A times 170 00:10:00,600 --> 00:10:04,100 this column of cofactors? 171 00:10:04,100 --> 00:10:06,150 What, what will I get there? 172 00:10:06,150 --> 00:10:08,460 Because we have seen this. 173 00:10:08,460 --> 00:10:10,590 I mean, we're, right, building exactly 174 00:10:10,590 --> 00:10:15,290 on what the last lecture reached. 175 00:10:15,290 --> 00:10:21,130 So this is a11 times c11, a12 times c12, a1n times c1n. 176 00:10:21,130 --> 00:10:25,700 What does that what does that sum up to? 177 00:10:25,700 --> 00:10:32,900 That's the cofactor formula for the determinant. 178 00:10:32,900 --> 00:10:35,240 That's the, this cofactor formula, 179 00:10:35,240 --> 00:10:39,110 which I wrote, which we got last time. 180 00:10:39,110 --> 00:10:42,700 The determinant of A is, if I use row one, let, 181 00:10:42,700 --> 00:10:47,450 let I equal one, then I have a11 times 182 00:10:47,450 --> 00:10:51,600 its cofactor, a12 times its cofactor, and so on. 183 00:10:51,600 --> 00:10:54,650 And that gives me the determinant. 184 00:10:54,650 --> 00:10:59,470 And it worked in this, in this case. 185 00:10:59,470 --> 00:11:06,020 This row times this thing is, sure enough, ad minus bc. 186 00:11:06,020 --> 00:11:08,800 But this formula says it always works. 187 00:11:08,800 --> 00:11:12,830 So up here in this, in this position, 188 00:11:12,830 --> 00:11:15,470 I'm getting determinant of A. 189 00:11:15,470 --> 00:11:18,130 What about in the two two position? 190 00:11:18,130 --> 00:11:23,460 Row two times column two there, what, what is that? 191 00:11:23,460 --> 00:11:26,640 That's just the cofactors, that's just row two 192 00:11:26,640 --> 00:11:28,960 times its cofactors. 193 00:11:28,960 --> 00:11:32,980 So of course I get the determinant again. 194 00:11:32,980 --> 00:11:36,310 And in the last here, this is the last row 195 00:11:36,310 --> 00:11:37,780 times its cofactors. 196 00:11:37,780 --> 00:11:41,540 It's exactly -- you see, we're realizing that the cofactor 197 00:11:41,540 --> 00:11:44,780 formula is just this sum of products, 198 00:11:44,780 --> 00:11:47,160 so of course we think, hey, we've got a matrix 199 00:11:47,160 --> 00:11:48,510 multiplication there. 200 00:11:48,510 --> 00:11:57,870 And we get determinant of A. 201 00:11:57,870 --> 00:12:00,920 But there's one more idea here, right? 202 00:12:00,920 --> 00:12:01,990 Great. 203 00:12:01,990 --> 00:12:05,410 What else, what have I not -- so I haven't got that formula 204 00:12:05,410 --> 00:12:10,520 completely proved yet, because I've still got to do all 205 00:12:10,520 --> 00:12:14,130 the off-diagonal stuff, which I want to be zero, 206 00:12:14,130 --> 00:12:15,120 right? 207 00:12:15,120 --> 00:12:17,560 I just want this to be determinant 208 00:12:17,560 --> 00:12:22,850 of A times the identity, and then I'm, I'm a happy person. 209 00:12:22,850 --> 00:12:24,860 So why should that be? 210 00:12:24,860 --> 00:12:30,660 Why should it be that one row times the cofactors 211 00:12:30,660 --> 00:12:34,920 from a different row, which become a column because I 212 00:12:34,920 --> 00:12:39,360 transpose, give zero? 213 00:12:39,360 --> 00:12:44,210 In other words, the cofactor formula gives the determinant 214 00:12:44,210 --> 00:12:48,440 if the row and the, and the cofactors -- you know, 215 00:12:48,440 --> 00:12:54,050 if the entries of A and the cofactors are for the same row. 216 00:12:54,050 --> 00:12:58,470 But for some reason, if I take the cofactors from the -- 217 00:12:58,470 --> 00:13:01,920 entries from the first row and the cofactors from the second 218 00:13:01,920 --> 00:13:04,290 row, for some reason I automatically 219 00:13:04,290 --> 00:13:05,270 get zero. 220 00:13:05,270 --> 00:13:08,140 And it's sort of like interesting 221 00:13:08,140 --> 00:13:10,540 to say, why does that happen? 222 00:13:10,540 --> 00:13:14,620 And can I just check that -- of course, we know it happens, 223 00:13:14,620 --> 00:13:16,410 in this case. 224 00:13:16,410 --> 00:13:19,060 Here are the numbers from row one 225 00:13:19,060 --> 00:13:24,600 and here are the cofactors from row two, right? 226 00:13:24,600 --> 00:13:26,820 Those are the numbers in row one. 227 00:13:26,820 --> 00:13:29,510 And th- these are the cofactors from row two, 228 00:13:29,510 --> 00:13:32,110 because the cofactor of c is minus b 229 00:13:32,110 --> 00:13:33,530 and the cofactor of d is 230 00:13:33,530 --> 00:13:34,030 a. 231 00:13:34,030 --> 00:13:40,380 And sure enough, that row times this column gives -- 232 00:13:40,380 --> 00:13:41,840 please say it. 233 00:13:41,840 --> 00:13:43,370 Zero, right. 234 00:13:43,370 --> 00:13:43,870 OK. 235 00:13:43,870 --> 00:13:47,330 So now how come? 236 00:13:47,330 --> 00:13:48,250 How come? 237 00:13:48,250 --> 00:13:51,190 Can we even see it in this two by two case? 238 00:13:51,190 --> 00:13:54,460 Why did -- well, I mean, I guess we, you know, 239 00:13:54,460 --> 00:13:57,352 in one way we certainly do see it, because it's right here. 240 00:13:57,352 --> 00:13:59,310 I mean, do we just do it, and then we get zero. 241 00:13:59,310 --> 00:14:01,870 But we want to think of some reason 242 00:14:01,870 --> 00:14:07,280 why the answer's zero, some reason that we can use in the n 243 00:14:07,280 --> 00:14:08,820 by n case. 244 00:14:08,820 --> 00:14:10,590 So let -- here, here is my thinking. 245 00:14:10,590 --> 00:14:13,450 246 00:14:13,450 --> 00:14:16,310 We must be, if we're getting the answer's zero, 247 00:14:16,310 --> 00:14:20,330 we suspect that what we're doing somehow, 248 00:14:20,330 --> 00:14:24,020 we're taking the determinant of some matrix that 249 00:14:24,020 --> 00:14:26,660 has two equal rows. 250 00:14:26,660 --> 00:14:31,040 So I believe that if we multiply these by the cofactors from 251 00:14:31,040 --> 00:14:35,770 some other row, we're taking the determinant -- ye, 252 00:14:35,770 --> 00:14:38,660 what matrix are we taking the determinant of? 253 00:14:38,660 --> 00:14:40,380 Here it's, this is it. 254 00:14:40,380 --> 00:14:44,900 We're, when we do that times this, we're really taking -- 255 00:14:44,900 --> 00:14:48,510 can I put this in little letters down here? 256 00:14:48,510 --> 00:14:57,220 I'm taking -- let me look at the matrix a b a b. 257 00:14:57,220 --> 00:15:03,610 Let me call that matrix AS, A screwed up. 258 00:15:03,610 --> 00:15:04,290 OK. 259 00:15:04,290 --> 00:15:05,240 All right. 260 00:15:05,240 --> 00:15:09,260 So now that matrix is certainly singular. 261 00:15:09,260 --> 00:15:12,380 So if we find its determinant, we're going to get zero. 262 00:15:12,380 --> 00:15:16,810 But I claim that if we find its determinant by the cofactor 263 00:15:16,810 --> 00:15:19,960 rule, go along the first row, we would 264 00:15:19,960 --> 00:15:23,810 take a times the cofactor of a. 265 00:15:23,810 --> 00:15:27,180 And what is the -- 266 00:15:27,180 --> 00:15:32,440 see, how -- oh no -- let me go along the second row. 267 00:15:32,440 --> 00:15:33,810 OK. 268 00:15:33,810 --> 00:15:36,320 So let's see, which -- 269 00:15:36,320 --> 00:15:37,190 if I take -- 270 00:15:37,190 --> 00:15:40,510 I know I've got a singular matrix here. 271 00:15:40,510 --> 00:15:46,310 And I believe that when I do this multiplication, what 272 00:15:46,310 --> 00:15:51,110 I'm doing is using the cofactor formula for the determinant. 273 00:15:51,110 --> 00:15:52,700 And I know I'm going to get zero. 274 00:15:52,700 --> 00:15:54,530 Let me try this again. 275 00:15:54,530 --> 00:15:56,920 So the cofactor formula for the determinant 276 00:15:56,920 --> 00:16:04,170 says I should take a times its cofactor, which is this b, 277 00:16:04,170 --> 00:16:09,260 plus b times its cofactor, which is this minus a. 278 00:16:09,260 --> 00:16:10,040 OK. 279 00:16:10,040 --> 00:16:15,260 That's what we're doing, apart from a sign here. 280 00:16:15,260 --> 00:16:20,980 Oh yeah, so you know, there might be a minus multiplying 281 00:16:20,980 --> 00:16:21,950 everything. 282 00:16:21,950 --> 00:16:24,280 So if I take this determinant, it's A -- 283 00:16:24,280 --> 00:16:29,000 the determinant of this, the determinant of A screwed up is 284 00:16:29,000 --> 00:16:33,050 a times its cofactor, which is b, 285 00:16:33,050 --> 00:16:38,980 plus the second guy times its cofactor, which is minus a. 286 00:16:38,980 --> 00:16:41,790 And of course I get the answer zero, 287 00:16:41,790 --> 00:16:45,790 and this is exactly what's happening in that, in that, 288 00:16:45,790 --> 00:16:47,650 row times this wrong column. 289 00:16:47,650 --> 00:16:48,150 OK. 290 00:16:48,150 --> 00:16:52,740 291 00:16:52,740 --> 00:16:56,460 That's the two by two picture, and it's just the same here. 292 00:16:56,460 --> 00:17:01,870 That the reason I get a zero up in there is, the reason 293 00:17:01,870 --> 00:17:09,319 I get a zero is that when I multiply the first row of A 294 00:17:09,319 --> 00:17:12,880 and the last row of the cofactor matrix, 295 00:17:12,880 --> 00:17:16,450 it's as if I'm taking the determinant of this screwed up 296 00:17:16,450 --> 00:17:19,420 matrix that has first and last rows identical. 297 00:17:19,420 --> 00:17:22,270 298 00:17:22,270 --> 00:17:26,150 The book pins this down more specific -- 299 00:17:26,150 --> 00:17:29,410 and more carefully than I can do in the lecture. 300 00:17:29,410 --> 00:17:31,200 I hope you're seeing the point. 301 00:17:31,200 --> 00:17:34,130 That this is an identity. 302 00:17:34,130 --> 00:17:37,310 That it's a beautiful identity and it tells us what 303 00:17:37,310 --> 00:17:40,690 the inverse of the matrix is. 304 00:17:40,690 --> 00:17:43,820 So it gives us the inverse, the formula for the inverse. 305 00:17:43,820 --> 00:17:44,510 OK. 306 00:17:44,510 --> 00:17:49,180 So that's the first goal of my lecture, was to find this 307 00:17:49,180 --> 00:17:49,960 formula. 308 00:17:49,960 --> 00:17:51,490 It's done. 309 00:17:51,490 --> 00:17:53,800 OK. 310 00:17:53,800 --> 00:17:58,180 And of course I could invert, now, I can, 311 00:17:58,180 --> 00:18:00,900 I sort of like I can see what -- 312 00:18:00,900 --> 00:18:02,850 I can answer questions like this. 313 00:18:02,850 --> 00:18:08,510 Suppose I have a matrix, and let me move the one one 314 00:18:08,510 --> 00:18:09,740 entry. 315 00:18:09,740 --> 00:18:13,600 What happens to the inverse? 316 00:18:13,600 --> 00:18:15,780 Just, just think about that question. 317 00:18:15,780 --> 00:18:18,030 Suppose I have some matrix, I just write down 318 00:18:18,030 --> 00:18:21,300 some nice, non-singular matrix that's got an inverse, 319 00:18:21,300 --> 00:18:25,170 and then I move the one one entry a little bit. 320 00:18:25,170 --> 00:18:27,880 I add one to it, for example. 321 00:18:27,880 --> 00:18:30,730 What happens to the inverse matrix? 322 00:18:30,730 --> 00:18:33,160 Well, this formula should tell me. 323 00:18:33,160 --> 00:18:36,110 I have to look to see what happens to the determinant 324 00:18:36,110 --> 00:18:39,400 and what happens to all the cofactors. 325 00:18:39,400 --> 00:18:44,300 And, the picture, it's all there. 326 00:18:44,300 --> 00:18:44,960 It's all there. 327 00:18:44,960 --> 00:18:49,420 We can really understand how the inverse changes 328 00:18:49,420 --> 00:18:51,250 when the matrix changes. 329 00:18:51,250 --> 00:18:52,330 OK. 330 00:18:52,330 --> 00:18:58,910 Now my second application is to -- let me put that over here -- 331 00:18:58,910 --> 00:18:59,630 is to Ax=b. 332 00:18:59,630 --> 00:19:03,650 333 00:19:03,650 --> 00:19:09,060 Well, the -- course, the solution is A inverse b. 334 00:19:09,060 --> 00:19:11,820 But now I have a formula for A inverse. 335 00:19:11,820 --> 00:19:17,170 A inverse is one over the determinant 336 00:19:17,170 --> 00:19:22,030 times this C transpose times B. 337 00:19:22,030 --> 00:19:24,510 I now know what A inverse is. 338 00:19:24,510 --> 00:19:27,270 So now I just have to say, what have I got here? 339 00:19:27,270 --> 00:19:33,600 Is there any way to, to make this formula, this answer, 340 00:19:33,600 --> 00:19:36,550 which is the one and only answer -- 341 00:19:36,550 --> 00:19:40,980 it's the very same answer we got on the first day of the class 342 00:19:40,980 --> 00:19:42,620 by elimination. 343 00:19:42,620 --> 00:19:47,290 Now I'm -- now I've got a formula for the answer. 344 00:19:47,290 --> 00:19:51,890 Can I play with it further to see what's going on? 345 00:19:51,890 --> 00:20:01,640 And Cramer's, this Cramer's Rule is exactly, that -- 346 00:20:01,640 --> 00:20:07,010 a way of looking at this formula. 347 00:20:07,010 --> 00:20:07,510 OK. 348 00:20:07,510 --> 00:20:11,620 So this is a formula for x. 349 00:20:11,620 --> 00:20:13,620 Here's my formula. 350 00:20:13,620 --> 00:20:14,350 Well, of course. 351 00:20:14,350 --> 00:20:16,050 The first thing I see from the formula 352 00:20:16,050 --> 00:20:21,100 is that the answer x always has that in the determinant. 353 00:20:21,100 --> 00:20:22,590 I'm not surprised. 354 00:20:22,590 --> 00:20:25,240 There's a division by the determinant. 355 00:20:25,240 --> 00:20:29,190 But then I have to say a little more carefully what's going on 356 00:20:29,190 --> 00:20:31,610 And let me tell you what Cramer's Rule is. up here. 357 00:20:31,610 --> 00:20:35,610 Let, let me take x1, the first component. 358 00:20:35,610 --> 00:20:37,740 So this is the first component of the answer. 359 00:20:37,740 --> 00:20:40,870 Then there'll be a second component and a, 360 00:20:40,870 --> 00:20:42,900 all the other components. 361 00:20:42,900 --> 00:20:46,310 Can I take just the first component of this formula? 362 00:20:46,310 --> 00:20:52,040 Well, I certainly have determinant of A down under. 363 00:20:52,040 --> 00:20:56,400 And what the heck is the first -- 364 00:20:56,400 --> 00:21:00,300 so what do I get in C transpose b? 365 00:21:00,300 --> 00:21:03,440 What's the first entry of C transpose b? 366 00:21:03,440 --> 00:21:06,060 That's what I have to answer myself. 367 00:21:06,060 --> 00:21:10,420 Well, what's the first entry of C transpose b? 368 00:21:10,420 --> 00:21:13,820 369 00:21:13,820 --> 00:21:17,670 This B is -- let me tell you what it is. 370 00:21:17,670 --> 00:21:18,170 OK. 371 00:21:18,170 --> 00:21:22,290 Somehow I'm multiplying cofactors 372 00:21:22,290 --> 00:21:26,970 by the entries of B, right, in this product. 373 00:21:26,970 --> 00:21:30,430 Cofactors from the matrix times entries of b. 374 00:21:30,430 --> 00:21:33,650 So any time I'm multiplying cofactors by numbers, 375 00:21:33,650 --> 00:21:36,980 I think, I'm getting the determinant of something. 376 00:21:36,980 --> 00:21:39,990 And let me call that something B1. 377 00:21:39,990 --> 00:21:46,730 So this is a matrix, the matrix whose determinant is coming out 378 00:21:46,730 --> 00:21:47,230 of that. 379 00:21:47,230 --> 00:21:49,530 And we'll, we'll see what it is. 380 00:21:49,530 --> 00:21:54,550 x2 will be the determinant of some other matrix B2, also 381 00:21:54,550 --> 00:21:57,650 divided by determinant of A. 382 00:21:57,650 --> 00:21:59,190 So now I just -- 383 00:21:59,190 --> 00:22:00,970 Cramer just had a good idea. 384 00:22:00,970 --> 00:22:06,920 He realized what matrix it was, what these B1 and B2 and B3 385 00:22:06,920 --> 00:22:08,470 and so on matrices were. 386 00:22:08,470 --> 00:22:10,520 Let me write them on the board underneath. 387 00:22:10,520 --> 00:22:14,610 388 00:22:14,610 --> 00:22:15,260 OK. 389 00:22:15,260 --> 00:22:17,470 So what is this B1? 390 00:22:17,470 --> 00:22:24,430 This B1 is the matrix that has b in its first column 391 00:22:24,430 --> 00:22:27,900 and otherwise the rest of it is A. 392 00:22:27,900 --> 00:22:38,740 So it otherwise it has the rest, the, the n-1 columns of A. 393 00:22:38,740 --> 00:22:42,880 It's the matrix with -- 394 00:22:42,880 --> 00:22:51,410 it's just the matrix A with column one replaced 395 00:22:51,410 --> 00:22:58,520 by the right-hand side, by the right-hand side b. 396 00:22:58,520 --> 00:23:04,490 Because somehow when I take the determinant of this guy, 397 00:23:04,490 --> 00:23:09,140 it's giving me this matrix multiplication. 398 00:23:09,140 --> 00:23:10,570 Well, how could that be? 399 00:23:10,570 --> 00:23:13,180 400 00:23:13,180 --> 00:23:16,490 How could -- so what's, what's the determinant formula 401 00:23:16,490 --> 00:23:18,050 I'll use here? 402 00:23:18,050 --> 00:23:21,910 I'll use cofactors, of course. 403 00:23:21,910 --> 00:23:25,240 And I might as well use cofactors down column one. 404 00:23:25,240 --> 00:23:27,910 So when I use cofactors down column one, 405 00:23:27,910 --> 00:23:32,840 I'm taking the first entry of b times what? 406 00:23:32,840 --> 00:23:35,650 Times the cofactor c11. 407 00:23:35,650 --> 00:23:38,410 Do you see that? 408 00:23:38,410 --> 00:23:40,830 When I, when I use cofactors here, 409 00:23:40,830 --> 00:23:43,530 I take the first entry here, B one 410 00:23:43,530 --> 00:23:47,120 let's call it, times the cofactor there. 411 00:23:47,120 --> 00:23:51,390 But what's the cofactor in -- my little hand-waving is meant 412 00:23:51,390 --> 00:23:54,650 to indicate that it's a matrix of one size smaller, 413 00:23:54,650 --> 00:23:56,030 the cofactor. 414 00:23:56,030 --> 00:23:58,730 And it's exactly c11. 415 00:23:58,730 --> 00:24:01,130 Well, that's just what we wanted. 416 00:24:01,130 --> 00:24:06,200 This first entry is c11 times b1. 417 00:24:06,200 --> 00:24:13,970 And then the next entry is whatever, is c21 times b2, 418 00:24:13,970 --> 00:24:14,860 and so on. 419 00:24:14,860 --> 00:24:17,310 And sure enough, if I look here, when 420 00:24:17,310 --> 00:24:19,720 I'm, when I do the cofactor expansion, 421 00:24:19,720 --> 00:24:22,790 b2 is getting multiplied by the right thing, and so on. 422 00:24:22,790 --> 00:24:25,450 423 00:24:25,450 --> 00:24:27,400 So there's Cramer's Rule. 424 00:24:27,400 --> 00:24:31,560 And the book gives another kind of cute proof 425 00:24:31,560 --> 00:24:36,960 without, without building so much on, on cofactors. 426 00:24:36,960 --> 00:24:38,950 But here we've got cofactors, so I thought 427 00:24:38,950 --> 00:24:40,650 I'd just give you this proof. 428 00:24:40,650 --> 00:24:42,700 So what is B -- 429 00:24:42,700 --> 00:24:45,430 in general, what is Bj? 430 00:24:45,430 --> 00:24:58,970 This is the, this is A with column j replaced by, by b. 431 00:24:58,970 --> 00:25:02,920 432 00:25:02,920 --> 00:25:08,870 So that's -- the determinant of that matrix that you divide 433 00:25:08,870 --> 00:25:11,700 by determinant of A to get xj. 434 00:25:11,700 --> 00:25:14,980 So x -- let me change this general formula. 435 00:25:14,980 --> 00:25:18,190 xj, the j-th one, is the determinant 436 00:25:18,190 --> 00:25:22,510 of Bj divided by the determinant of A. 437 00:25:22,510 --> 00:25:24,360 And now we've said what Bj is. 438 00:25:24,360 --> 00:25:30,160 439 00:25:30,160 --> 00:25:33,720 Well, so Cramer found a rule. 440 00:25:33,720 --> 00:25:39,320 And we could ask him, OK, great, good work, Cramer. 441 00:25:39,320 --> 00:25:44,180 But is your rule any good in practice? 442 00:25:44,180 --> 00:25:49,960 So he says, well, you couldn't ask about a rule in mine, 443 00:25:49,960 --> 00:25:52,220 right, because it's just -- 444 00:25:52,220 --> 00:25:56,850 all you have to do is find the determinant of A and these 445 00:25:56,850 --> 00:26:01,220 other determinants, so I guess -- oh, he just says, well, 446 00:26:01,220 --> 00:26:04,140 all you have to do is find n+1 determinants, 447 00:26:04,140 --> 00:26:06,170 the, the n Bs and the A. 448 00:26:06,170 --> 00:26:17,530 And actually, I remember reading -- there was a book, 449 00:26:17,530 --> 00:26:22,530 popular book that, that kids interested in math read when I 450 00:26:22,530 --> 00:26:25,980 was a kid interested in math called Mathematics 451 00:26:25,980 --> 00:26:29,840 for the Million or something, by a guy named Bell. 452 00:26:29,840 --> 00:26:34,830 And it had a little page about linear algebra. 453 00:26:34,830 --> 00:26:39,000 And it said,-- so it explained elimination 454 00:26:39,000 --> 00:26:41,250 in a very complicated way. 455 00:26:41,250 --> 00:26:43,200 I certainly didn't understand it. 456 00:26:43,200 --> 00:26:47,950 And, and it made it, you know, it sort of said, well, 457 00:26:47,950 --> 00:26:51,120 there is this formula for elimination, 458 00:26:51,120 --> 00:26:55,770 but look at this great formula, Cramer's Rule. 459 00:26:55,770 --> 00:27:00,410 So it certainly said Cramer's Rule was the way to go. 460 00:27:00,410 --> 00:27:05,180 But actually, Cramer's Rule is a disastrous way to go, 461 00:27:05,180 --> 00:27:07,920 because to compute these determinants, 462 00:27:07,920 --> 00:27:12,060 it takes, like, approximately forever. 463 00:27:12,060 --> 00:27:17,020 So actually I now think of that book title 464 00:27:17,020 --> 00:27:18,980 as being Mathematics for the Millionaire, 465 00:27:18,980 --> 00:27:22,160 because you'd have to be able to pay for, 466 00:27:22,160 --> 00:27:26,570 a hopelessly long calculation where elimination, of course, 467 00:27:26,570 --> 00:27:30,450 produced the x-s, in an instant. 468 00:27:30,450 --> 00:27:36,060 But having a formula allows you to, with, with letters, you 469 00:27:36,060 --> 00:27:40,010 know, allows you to do algebra instead of, algorithms. 470 00:27:40,010 --> 00:27:44,220 So the, there's some value in the Cramer's Rule formula 471 00:27:44,220 --> 00:27:52,000 for x and in the explicit formula for, for A inverse. 472 00:27:52,000 --> 00:27:54,890 They're nice formulas, but I just 473 00:27:54,890 --> 00:27:57,710 don't want you to use them. 474 00:27:57,710 --> 00:27:59,230 That'ss what it comes to. 475 00:27:59,230 --> 00:28:02,960 If you had to -- and Matlab would never, never do it. 476 00:28:02,960 --> 00:28:05,480 I mean, it would use elimination. 477 00:28:05,480 --> 00:28:06,720 OK. 478 00:28:06,720 --> 00:28:11,380 Now I'm ready for number three in today's list 479 00:28:11,380 --> 00:28:17,790 of amazing connections coming through the determinant. 480 00:28:17,790 --> 00:28:21,900 And that number three is the fact that the determinant gives 481 00:28:21,900 --> 00:28:22,950 a volume. 482 00:28:22,950 --> 00:28:24,480 OK. 483 00:28:24,480 --> 00:28:28,860 So now -- so that's my final topic for -- 484 00:28:28,860 --> 00:28:33,070 among these -- this my number three application, 485 00:28:33,070 --> 00:28:37,700 that the determinant is actually equals the volume of something. 486 00:28:37,700 --> 00:28:42,720 Can I use this little space to consider a special case, 487 00:28:42,720 --> 00:28:46,050 and then I'll use the far board to think 488 00:28:46,050 --> 00:28:47,840 about the general rule. 489 00:28:47,840 --> 00:28:50,400 So what I going to prove? 490 00:28:50,400 --> 00:28:51,970 Or claim. 491 00:28:51,970 --> 00:28:56,950 I claim that the determinant of the matrix 492 00:28:56,950 --> 00:29:02,800 is the volume of a box. 493 00:29:02,800 --> 00:29:05,930 OK, and you say, which box? 494 00:29:05,930 --> 00:29:06,970 Fair enough. 495 00:29:06,970 --> 00:29:08,570 OK. 496 00:29:08,570 --> 00:29:11,670 So let's see. 497 00:29:11,670 --> 00:29:16,170 I'm in -- shall we say we're in, say three by three? 498 00:29:16,170 --> 00:29:18,780 Shall we suppose -- let's, let's say three by three. 499 00:29:18,780 --> 00:29:22,680 So, so we can really -- we're, we're talking about boxes 500 00:29:22,680 --> 00:29:26,400 in three dimensions, and three by three matrices. 501 00:29:26,400 --> 00:29:30,990 And so all I do -- you could guess what the box is. 502 00:29:30,990 --> 00:29:34,700 Here is, here is, three dimensions. 503 00:29:34,700 --> 00:29:35,450 OK. 504 00:29:35,450 --> 00:29:41,880 Now I take the first row of the matrix, a11, a22, A -- 505 00:29:41,880 --> 00:29:44,360 sorry. a11, a12, a13. 506 00:29:44,360 --> 00:29:48,090 That row is a vector. 507 00:29:48,090 --> 00:29:50,190 It goes to some point. 508 00:29:50,190 --> 00:29:53,220 That point will be -- and that edge going to it, 509 00:29:53,220 --> 00:29:56,650 will be an edge of the box, and that point will be a corner 510 00:29:56,650 --> 00:29:57,740 of the box. 511 00:29:57,740 --> 00:30:01,010 So here is zero zero zero, of course. 512 00:30:01,010 --> 00:30:09,500 And here's the first row of the matrix: a11, a12, a13. 513 00:30:09,500 --> 00:30:14,840 So that's one edge of the box. 514 00:30:14,840 --> 00:30:19,700 Another edge of the box is to the second row 515 00:30:19,700 --> 00:30:21,820 of the matrix, row two. 516 00:30:21,820 --> 00:30:24,510 Can I just call it there row two? 517 00:30:24,510 --> 00:30:28,760 And a third row of the box will be to -- 518 00:30:28,760 --> 00:30:32,200 a third row -- a third edge of the box will be given by row 519 00:30:32,200 --> 00:30:33,300 three. 520 00:30:33,300 --> 00:30:36,470 So, so there's row three. 521 00:30:36,470 --> 00:30:38,780 That, the coordinates, what are the coordinates 522 00:30:38,780 --> 00:30:41,030 of that corner of the box? 523 00:30:41,030 --> 00:30:47,650 a31, a32, a33. 524 00:30:47,650 --> 00:30:51,220 So I've got that edge of the box, that edge of the box, 525 00:30:51,220 --> 00:30:53,830 that edge of the box, and that's all I need. 526 00:30:53,830 --> 00:30:59,820 Now I just finish out the box, right? 527 00:30:59,820 --> 00:31:03,650 I just -- the proper word, of course, is parallelepiped. 528 00:31:03,650 --> 00:31:08,920 But for obvious reasons, I wrote box. 529 00:31:08,920 --> 00:31:09,550 OK. 530 00:31:09,550 --> 00:31:11,360 So, OK. 531 00:31:11,360 --> 00:31:14,550 So there's the, there's the bottom of the box. 532 00:31:14,550 --> 00:31:19,660 There're the four edge sides of the box. 533 00:31:19,660 --> 00:31:23,570 There's the top of the box. 534 00:31:23,570 --> 00:31:24,540 Cute, right? 535 00:31:24,540 --> 00:31:28,470 It's the box that has these three edges 536 00:31:28,470 --> 00:31:33,080 and then it's completed to a, to a, each, you know, 537 00:31:33,080 --> 00:31:35,080 each side is a, is a parallelogram. 538 00:31:35,080 --> 00:31:37,730 539 00:31:37,730 --> 00:31:43,050 And it's that box whose volume is given by the determinant. 540 00:31:43,050 --> 00:31:45,610 541 00:31:45,610 --> 00:31:50,650 That's -- now it's -- possible that the determinant is 542 00:31:50,650 --> 00:31:52,920 negative. 543 00:31:52,920 --> 00:31:56,110 So we have to just say what's going on in that case. 544 00:31:56,110 --> 00:32:01,840 If the determinant is negative, then the volume, we, 545 00:32:01,840 --> 00:32:04,030 we should take the absolute value really. 546 00:32:04,030 --> 00:32:07,030 So the volume, if we, if we think of volume as positive, 547 00:32:07,030 --> 00:32:11,310 we should take the absolute value of the determinant. 548 00:32:11,310 --> 00:32:14,400 But the, the sign, what does the sign of the determinant -- 549 00:32:14,400 --> 00:32:16,560 it always must tell us something. 550 00:32:16,560 --> 00:32:20,380 And somehow it, it will tell us whether these three is a -- 551 00:32:20,380 --> 00:32:23,270 whether it's a right-handed box or a left-handed box. 552 00:32:23,270 --> 00:32:30,300 If we, if we reversed two of the edges, 553 00:32:30,300 --> 00:32:32,250 we would go between a right-handed box 554 00:32:32,250 --> 00:32:33,390 and a left-handed box. 555 00:32:33,390 --> 00:32:35,440 We wouldn't change the volume, but we would 556 00:32:35,440 --> 00:32:40,500 change the, the cyclic, order. 557 00:32:40,500 --> 00:32:42,900 So I won't worry about that. 558 00:32:42,900 --> 00:32:47,020 And, so one special case is what? 559 00:32:47,020 --> 00:32:49,930 A equal identity matrix. 560 00:32:49,930 --> 00:32:52,070 So let's take that special case. 561 00:32:52,070 --> 00:32:55,520 A equal identity matrix. 562 00:32:55,520 --> 00:32:59,550 Is the formula determinant of identity matrix, 563 00:32:59,550 --> 00:33:01,230 does that equal the volume of the box? 564 00:33:01,230 --> 00:33:04,300 565 00:33:04,300 --> 00:33:06,460 Well, what is the box? 566 00:33:06,460 --> 00:33:07,370 What's the box? 567 00:33:07,370 --> 00:33:12,230 If A is the identity matrix, then these three rows are 568 00:33:12,230 --> 00:33:17,460 the three coordinate vectors, and the box is -- 569 00:33:17,460 --> 00:33:18,920 it's a cube. 570 00:33:18,920 --> 00:33:21,200 It's the unit cube. 571 00:33:21,200 --> 00:33:23,970 So if, if A is the identity matrix, of course 572 00:33:23,970 --> 00:33:26,080 our formula is 573 00:33:26,080 --> 00:33:29,850 Well, actually that proves property one -- 574 00:33:29,850 --> 00:33:31,740 that the volume right. has property one. 575 00:33:31,740 --> 00:33:35,440 Actually, we could, we could, we could get this thing if we -- 576 00:33:35,440 --> 00:33:39,210 if we can show that the box volume has the same three 577 00:33:39,210 --> 00:33:42,010 properties that define the determinant, 578 00:33:42,010 --> 00:33:44,060 then it must be the determinant. 579 00:33:44,060 --> 00:33:46,670 580 00:33:46,670 --> 00:33:51,080 So that's like the, the, the elegant way to prove this. 581 00:33:51,080 --> 00:33:53,860 To prove this amazing fact that the determinant equals 582 00:33:53,860 --> 00:33:58,540 the volume, first we'll check it for the identity matrix. 583 00:33:58,540 --> 00:34:00,470 That's fine. 584 00:34:00,470 --> 00:34:03,190 The box is a cube and its volume is one 585 00:34:03,190 --> 00:34:07,180 and the determinant is one and, and one agrees with one. 586 00:34:07,180 --> 00:34:11,590 Now let me take one -- let me go up one level to an orthogonal 587 00:34:11,590 --> 00:34:12,889 matrix. 588 00:34:12,889 --> 00:34:16,060 Because I'd like to take this chance to bring in chapter -- 589 00:34:16,060 --> 00:34:18,750 the, the previous chapter. 590 00:34:18,750 --> 00:34:21,100 Suppose I have an orthogonal matrix. 591 00:34:21,100 --> 00:34:22,110 What did that mean? 592 00:34:22,110 --> 00:34:25,500 I always called those things Q. 593 00:34:25,500 --> 00:34:27,909 What was the point of -- suppose I have, 594 00:34:27,909 --> 00:34:32,670 suppose instead of the identity matrix I'm now going to take A 595 00:34:32,670 --> 00:34:36,910 equal Q, an orthogonal matrix. 596 00:34:36,910 --> 00:34:42,940 597 00:34:42,940 --> 00:34:46,510 What was Q then? 598 00:34:46,510 --> 00:34:52,179 That was a matrix whose columns were orthonormal, right? 599 00:34:52,179 --> 00:34:54,610 Those were its columns were unit vectors, 600 00:34:54,610 --> 00:34:56,059 perpendicular unit vectors. 601 00:34:56,059 --> 00:34:59,220 602 00:34:59,220 --> 00:35:03,090 So what kind of a box have we got now? 603 00:35:03,090 --> 00:35:06,520 What kind of a box comes from the rows or the columns, 604 00:35:06,520 --> 00:35:08,530 I don't mind, because the determinant 605 00:35:08,530 --> 00:35:10,420 is the determinant of the transpose, 606 00:35:10,420 --> 00:35:11,850 so I'm never worried about 607 00:35:11,850 --> 00:35:12,730 that. 608 00:35:12,730 --> 00:35:15,850 What kind of a box, what shape box have we 609 00:35:15,850 --> 00:35:20,380 got if the matrix is an orthogonal matrix? 610 00:35:20,380 --> 00:35:23,040 It's another cube. 611 00:35:23,040 --> 00:35:24,710 It's a cube again. 612 00:35:24,710 --> 00:35:27,900 How is it different from the identity cube? 613 00:35:27,900 --> 00:35:30,510 614 00:35:30,510 --> 00:35:33,140 It's just rotated. 615 00:35:33,140 --> 00:35:36,830 It's just the orthogonal matrix Q doesn't 616 00:35:36,830 --> 00:35:38,860 have to be the identity matrix. 617 00:35:38,860 --> 00:35:43,240 It's just the unit cube but turned in space. 618 00:35:43,240 --> 00:35:48,490 So sure enough, it's the unit cube, and its volume is one. 619 00:35:48,490 --> 00:35:53,410 Now is the determinant one? 620 00:35:53,410 --> 00:35:55,410 What's the determinant of Q? 621 00:35:55,410 --> 00:35:58,980 We believe that the determinant of Q better be one or minus 622 00:35:58,980 --> 00:36:03,100 one, so that our formula is -- checks out in that -- 623 00:36:03,100 --> 00:36:06,800 if we can't check it in these easy cases where we got a cube, 624 00:36:06,800 --> 00:36:11,020 we're not going to get it in the general case. 625 00:36:11,020 --> 00:36:18,290 So why is the determinant of Q equal one or minus one? 626 00:36:18,290 --> 00:36:19,810 What do we know about Q? 627 00:36:19,810 --> 00:36:25,050 What's the one matrix statement of the properties of Q? 628 00:36:25,050 --> 00:36:29,020 A matrix with orthonormal columns has -- 629 00:36:29,020 --> 00:36:31,860 satisfies a certain equation. 630 00:36:31,860 --> 00:36:33,400 What, what is that? 631 00:36:33,400 --> 00:36:38,930 It's if we have this orthogonal matrix, then the fact -- 632 00:36:38,930 --> 00:36:45,550 the way to say what it, what its properties are is this. 633 00:36:45,550 --> 00:36:50,940 Q prime, u- u- Q transpose Q equals I. 634 00:36:50,940 --> 00:36:52,790 Right? 635 00:36:52,790 --> 00:36:57,450 That's what -- those are the matrices that get the name Q, 636 00:36:57,450 --> 00:37:01,241 the matrices that Q transpose Q is I. 637 00:37:01,241 --> 00:37:01,740 OK. 638 00:37:01,740 --> 00:37:07,180 Now from that, tell me why is the determinant one 639 00:37:07,180 --> 00:37:09,500 or minus one. 640 00:37:09,500 --> 00:37:12,930 How do I, out of this fact -- 641 00:37:12,930 --> 00:37:14,525 this may even be a homework problem. 642 00:37:14,525 --> 00:37:17,240 643 00:37:17,240 --> 00:37:20,890 It's there in the, in the list of exercises in the book, 644 00:37:20,890 --> 00:37:23,100 and let's just do it. 645 00:37:23,100 --> 00:37:27,340 How do I get, how do I discover that the determinant of Q 646 00:37:27,340 --> 00:37:31,530 is one or maybe minus one? 647 00:37:31,530 --> 00:37:34,180 I take determinants of both sides, everybody says, 648 00:37:34,180 --> 00:37:36,870 so I won't -- 649 00:37:36,870 --> 00:37:38,610 I take determinants of both sides. 650 00:37:38,610 --> 00:37:41,250 On the right-hand side -- so I, when I take determinants 651 00:37:41,250 --> 00:37:46,240 of both sides, let me just do it. 652 00:37:46,240 --> 00:37:48,315 Take the determinant of -- take determinants. 653 00:37:48,315 --> 00:37:51,680 654 00:37:51,680 --> 00:37:53,970 Determinant of the identity is one. 655 00:37:53,970 --> 00:37:55,840 What's the determinant of that product? 656 00:37:55,840 --> 00:37:58,690 657 00:37:58,690 --> 00:38:02,400 Rule nine is paying off now. 658 00:38:02,400 --> 00:38:07,920 The determinant of a product is the determinant of this guy -- 659 00:38:07,920 --> 00:38:11,110 maybe I'll put it, I'll use that symbol for determinant. 660 00:38:11,110 --> 00:38:14,152 It's the determinant of that guy times the determinant 661 00:38:14,152 --> 00:38:14,860 of the other guy. 662 00:38:14,860 --> 00:38:17,460 663 00:38:17,460 --> 00:38:21,400 And then what's the determinant of Q transpose? 664 00:38:21,400 --> 00:38:22,990 It's the same as the determinant of Q. 665 00:38:22,990 --> 00:38:24,810 Rule ten pays off. 666 00:38:24,810 --> 00:38:28,330 So this is just this thing squared. 667 00:38:28,330 --> 00:38:33,740 So that determinant squared is one and sure enough it's one 668 00:38:33,740 --> 00:38:35,290 or minus one. 669 00:38:35,290 --> 00:38:35,790 Great. 670 00:38:35,790 --> 00:38:38,370 671 00:38:38,370 --> 00:38:42,000 So in these special cases of cubes, 672 00:38:42,000 --> 00:38:48,650 we really do have determinant equals volume. 673 00:38:48,650 --> 00:38:54,530 Now can I just push that to non-cubes. 674 00:38:54,530 --> 00:39:02,930 Let me push it first to rectangles, rectangular boxes, 675 00:39:02,930 --> 00:39:07,820 where I'm just multiplying the e- the edges are -- 676 00:39:07,820 --> 00:39:10,440 let me keep all the ninety degree angles, 677 00:39:10,440 --> 00:39:13,510 because those are -- that, that makes my life easy. 678 00:39:13,510 --> 00:39:16,910 And just stretch the edges. 679 00:39:16,910 --> 00:39:21,910 Suppose I stretch that first edge, suppose this first edge 680 00:39:21,910 --> 00:39:23,230 I double. 681 00:39:23,230 --> 00:39:27,990 Suppose I double that first edge, 682 00:39:27,990 --> 00:39:31,060 keeping the other edges the same. 683 00:39:31,060 --> 00:39:34,650 What happens to the volume? 684 00:39:34,650 --> 00:39:36,640 It doubles, right? 685 00:39:36,640 --> 00:39:39,420 We know that the volume of a cube doubles. 686 00:39:39,420 --> 00:39:42,370 In fact, because we know that the new cube would sit right 687 00:39:42,370 --> 00:39:43,400 on top -- 688 00:39:43,400 --> 00:39:46,210 I mean, the new, the added cube would sit right on -- 689 00:39:46,210 --> 00:39:47,620 would fit -- 690 00:39:47,620 --> 00:39:50,120 probably a geometer would say congruent or something -- 691 00:39:50,120 --> 00:39:51,810 would go right in, in the other. 692 00:39:51,810 --> 00:39:52,830 We'd have two. 693 00:39:52,830 --> 00:39:54,540 We have two identical cubes. 694 00:39:54,540 --> 00:39:58,421 Total volume is now two. 695 00:39:58,421 --> 00:39:58,920 OK. 696 00:39:58,920 --> 00:40:02,392 So I want -- if I double an edge, the volume doubles. 697 00:40:02,392 --> 00:40:03,725 What happens to the determinant? 698 00:40:03,725 --> 00:40:07,580 699 00:40:07,580 --> 00:40:14,880 If I double, the first row of a matrix, what ch- 700 00:40:14,880 --> 00:40:18,010 ch- what's the effect on the determinant? 701 00:40:18,010 --> 00:40:21,320 It also doubles, right? 702 00:40:21,320 --> 00:40:25,910 And that was rule number 3a. 703 00:40:25,910 --> 00:40:29,500 Remember rule 3a was that if I, I could, 704 00:40:29,500 --> 00:40:36,780 if I had a factor in, in row one, T, I could factor it out. 705 00:40:36,780 --> 00:40:41,020 So if, if I have a factor two in that row one, 706 00:40:41,020 --> 00:40:43,340 I can factor it out of the determinant. 707 00:40:43,340 --> 00:40:48,860 It agrees with the -- the volume of the box has that factor two. 708 00:40:48,860 --> 00:40:53,140 So, so volume satisfies this property 3a. 709 00:40:53,140 --> 00:40:59,040 And now I really close, but I -- but to get to the very end 710 00:40:59,040 --> 00:41:01,980 of this proof, I have to get away from right 711 00:41:01,980 --> 00:41:02,650 angles. 712 00:41:02,650 --> 00:41:10,330 I have to allow the possibility of, other angles. 713 00:41:10,330 --> 00:41:13,090 And -- or what's saying the same thing, 714 00:41:13,090 --> 00:41:18,600 I have to check that the volume also satisfies 3b. 715 00:41:18,600 --> 00:41:20,600 So can I -- 716 00:41:20,600 --> 00:41:24,740 This is end of proof that the -- so I'm -- 717 00:41:24,740 --> 00:41:34,380 determinant of A equals volume of box, and where I right now? 718 00:41:34,380 --> 00:41:40,820 This volume has properties, properties one, no problem. 719 00:41:40,820 --> 00:41:44,030 If the box is the cube, everything is -- 720 00:41:44,030 --> 00:41:49,000 if the box is the unit cube, its volume is one. 721 00:41:49,000 --> 00:41:54,680 Property two was if I reverse two rows, 722 00:41:54,680 --> 00:41:57,960 but that doesn't change the box. 723 00:41:57,960 --> 00:42:00,520 And it doesn't change the absolute value, so no problem 724 00:42:00,520 --> 00:42:01,200 there. 725 00:42:01,200 --> 00:42:06,890 Property 3a was if I mul- you remember what 3a was? 726 00:42:06,890 --> 00:42:10,390 So property one was about the identity matrix. 727 00:42:10,390 --> 00:42:12,900 Property two was about a plus or minus sign 728 00:42:12,900 --> 00:42:14,560 that I don't care about. 729 00:42:14,560 --> 00:42:18,490 Property 3a was a factor T in a row. 730 00:42:18,490 --> 00:42:24,080 But now I've got property three B to deal with. 731 00:42:24,080 --> 00:42:25,520 What was property 3b? 732 00:42:25,520 --> 00:42:30,270 This is a great way to review these, properties. 733 00:42:30,270 --> 00:42:34,860 So that 3b, the property 3b said -- let's do, 734 00:42:34,860 --> 00:42:36,910 let's do two by two. 735 00:42:36,910 --> 00:42:42,620 So said that if I had a+a', b+b', c, 736 00:42:42,620 --> 00:42:47,590 d that this equaled what? 737 00:42:47,590 --> 00:42:49,120 So this is property 3b. 738 00:42:49,120 --> 00:42:53,840 This is the linearity in row one by itself. 739 00:42:53,840 --> 00:42:59,060 So c d is staying the same, and I can split this into a b 740 00:42:59,060 --> 00:43:01,340 and a' b'. 741 00:43:01,340 --> 00:43:06,160 742 00:43:06,160 --> 00:43:12,890 That's property 3b, at least in the two by two case. 743 00:43:12,890 --> 00:43:15,340 And what I -- 744 00:43:15,340 --> 00:43:20,540 I wanted now to show that the volume, which 745 00:43:20,540 --> 00:43:25,040 two, two by two, that means area, has this, 746 00:43:25,040 --> 00:43:25,790 has this property. 747 00:43:25,790 --> 00:43:28,920 748 00:43:28,920 --> 00:43:32,540 Let me just emphasize that we have got -- we're getting -- 749 00:43:32,540 --> 00:43:37,150 this is a formula, then, for the area of a parallelogram. 750 00:43:37,150 --> 00:43:40,310 The area of this parallelogram -- can I just draw it? 751 00:43:40,310 --> 00:43:42,160 OK, here's the, here's the parallelogram. 752 00:43:42,160 --> 00:43:45,370 I have the row a b. 753 00:43:45,370 --> 00:43:47,000 That's the first row. 754 00:43:47,000 --> 00:43:49,260 That's the point a b. 755 00:43:49,260 --> 00:43:54,060 And I tack on c d. 756 00:43:54,060 --> 00:43:57,690 c d, coming out of here. 757 00:43:57,690 --> 00:43:59,065 And I complete the parallelogram. 758 00:43:59,065 --> 00:44:02,560 759 00:44:02,560 --> 00:44:03,750 So this is -- 760 00:44:03,750 --> 00:44:08,140 well, I better make it look right. 761 00:44:08,140 --> 00:44:12,530 It's really this one that has coordinates c d and this has 762 00:44:12,530 --> 00:44:17,040 coordinates -- well, whatever the sum is. 763 00:44:17,040 --> 00:44:18,820 And of course starting at zero zero. 764 00:44:18,820 --> 00:44:22,240 765 00:44:22,240 --> 00:44:26,026 So we all know, this is a+c, b+d. 766 00:44:26,026 --> 00:44:29,140 767 00:44:29,140 --> 00:44:31,120 Rather than -- 768 00:44:31,120 --> 00:44:33,460 I'm pausing on that proof for a minute 769 00:44:33,460 --> 00:44:37,320 just to going back to our formula. 770 00:44:37,320 --> 00:44:40,650 Because I want you to see that unlike Cramer's Rule, 771 00:44:40,650 --> 00:44:44,060 that I wasn't that impressed by, I'm 772 00:44:44,060 --> 00:44:46,640 very impressed by this formula for the area 773 00:44:46,640 --> 00:44:48,540 of a parallelogram. 774 00:44:48,540 --> 00:44:50,930 And what's our formula? 775 00:44:50,930 --> 00:44:54,890 What, what's the area of that parallelogram? 776 00:44:54,890 --> 00:45:00,250 If I had asked you that last year, 777 00:45:00,250 --> 00:45:03,950 you would have said OK, the area of a parallelogram 778 00:45:03,950 --> 00:45:06,060 is the base times the height, 779 00:45:06,060 --> 00:45:07,190 right? 780 00:45:07,190 --> 00:45:10,890 So you would have figured out what this base, the -- 781 00:45:10,890 --> 00:45:12,850 how long that base was. 782 00:45:12,850 --> 00:45:16,182 It's like the square root of A squared plus b squared. 783 00:45:16,182 --> 00:45:17,640 And then you would have figured out 784 00:45:17,640 --> 00:45:20,360 how much is this height, whatever it is. 785 00:45:20,360 --> 00:45:21,920 It's horrible. 786 00:45:21,920 --> 00:45:28,330 This, I mean, we got square roots, and in that height 787 00:45:28,330 --> 00:45:32,040 there would be other revolting stuff. 788 00:45:32,040 --> 00:45:35,800 But now what's the formula that we now know for the area? 789 00:45:35,800 --> 00:45:42,290 790 00:45:42,290 --> 00:45:45,740 It's the determinant of our little matrix. 791 00:45:45,740 --> 00:45:48,540 792 00:45:48,540 --> 00:45:51,050 It's just ad-bc. 793 00:45:51,050 --> 00:45:56,990 794 00:45:56,990 --> 00:45:59,110 No square roots. 795 00:45:59,110 --> 00:46:03,000 Totally rememberable, because it's exactly a formula 796 00:46:03,000 --> 00:46:06,960 that we've been studying the whole, for three lectures. 797 00:46:06,960 --> 00:46:07,460 OK. 798 00:46:07,460 --> 00:46:11,090 799 00:46:11,090 --> 00:46:13,960 That's, you know, that's the most important point 800 00:46:13,960 --> 00:46:15,190 I'm making here. 801 00:46:15,190 --> 00:46:20,990 Is that if you know the coordinates of a box, 802 00:46:20,990 --> 00:46:24,750 of the corners, then you have a great formula 803 00:46:24,750 --> 00:46:28,620 for the volume, area or volume, that 804 00:46:28,620 --> 00:46:33,550 doesn't involve any lengths or any angles or any heights, 805 00:46:33,550 --> 00:46:37,850 but just involves the coordinates that you've got. 806 00:46:37,850 --> 00:46:40,390 And similarly, what's the area of this triangle? 807 00:46:40,390 --> 00:46:43,530 Suppose I chop that off and say what about -- 808 00:46:43,530 --> 00:46:45,810 because you might often be interested in a triangle 809 00:46:45,810 --> 00:46:47,330 instead of a parallelogram. 810 00:46:47,330 --> 00:46:48,725 What's the area of this triangle? 811 00:46:48,725 --> 00:46:52,040 812 00:46:52,040 --> 00:46:53,950 Now there again, everybody would have 813 00:46:53,950 --> 00:46:58,110 said the area of a triangle is half the base times the height. 814 00:46:58,110 --> 00:47:01,200 815 00:47:01,200 --> 00:47:04,300 And in some cases, if you know the base that a, that's -- 816 00:47:04,300 --> 00:47:06,300 and the height, that's fine. 817 00:47:06,300 --> 00:47:10,280 But here, we, what we know is the coordinates of the corners. 818 00:47:10,280 --> 00:47:11,980 We know the vertices. 819 00:47:11,980 --> 00:47:14,690 And so what's the area of that triangle? 820 00:47:14,690 --> 00:47:17,830 821 00:47:17,830 --> 00:47:22,550 If I know these, if I know a b, c d, and zero zero, 822 00:47:22,550 --> 00:47:25,060 what's the area? 823 00:47:25,060 --> 00:47:29,670 It's just half, so it's just half of this. 824 00:47:29,670 --> 00:47:33,040 So this is, this is a- a b -- a d - 825 00:47:33,040 --> 00:47:38,870 b c for the parallelogram and one half of that, 826 00:47:38,870 --> 00:47:43,585 one half of ad-bc for the triangle. 827 00:47:43,585 --> 00:47:47,140 828 00:47:47,140 --> 00:47:50,900 So I mean, this is a totally trivial remark, to say, well, 829 00:47:50,900 --> 00:47:52,630 divide by two. 830 00:47:52,630 --> 00:47:56,910 But it's just that you more often see triangles, 831 00:47:56,910 --> 00:48:01,530 and you feel you know the formula for the area 832 00:48:01,530 --> 00:48:04,950 but the good formula for the area is this one. 833 00:48:04,950 --> 00:48:06,760 And I'm just going to -- 834 00:48:06,760 --> 00:48:08,260 I'm just going to say one more thing 835 00:48:08,260 --> 00:48:09,750 about the area of a triangle. 836 00:48:09,750 --> 00:48:11,220 It's just because it's -- you know, 837 00:48:11,220 --> 00:48:15,760 it's so great to have a good formula for something. 838 00:48:15,760 --> 00:48:20,990 What if our triangle did not start at zero zero? 839 00:48:20,990 --> 00:48:25,630 What if our triangle, what if we had this -- 840 00:48:25,630 --> 00:48:28,390 what if we had -- so I'm coming back to triangles again. 841 00:48:28,390 --> 00:48:32,600 842 00:48:32,600 --> 00:48:40,320 But let me, let me put this triangle somewhere, it's -- 843 00:48:40,320 --> 00:48:43,800 I'm staying with triangles, I'm just in two dimensions, 844 00:48:43,800 --> 00:48:53,280 but I'm going to allow you to give me any three corners. 845 00:48:53,280 --> 00:48:57,420 846 00:48:57,420 --> 00:49:01,430 And in -- those six numbers must determine the area. 847 00:49:01,430 --> 00:49:03,750 And what's the formula? 848 00:49:03,750 --> 00:49:05,280 The area is going to be, it's going 849 00:49:05,280 --> 00:49:09,530 to be, there'll be that half of a parallelogram. 850 00:49:09,530 --> 00:49:13,580 I mean, basically this can't be completely new, right? 851 00:49:13,580 --> 00:49:17,230 We've got the area when -- we, we know the area when this is 852 00:49:17,230 --> 00:49:20,400 zero zero. 853 00:49:20,400 --> 00:49:24,810 Now we just want to lift our sight slightly and get the area 854 00:49:24,810 --> 00:49:27,820 when all th- so let me write down what it, what it 855 00:49:27,820 --> 00:49:29,190 comes out to be. 856 00:49:29,190 --> 00:49:37,390 It turns out that if you do this, x1 y1 and a 1, x2 y2 857 00:49:37,390 --> 00:49:44,150 and a 1, x3 y3 and a 1, that that works. 858 00:49:44,150 --> 00:49:47,320 That the determinant symbol, of course. 859 00:49:47,320 --> 00:49:51,640 It's just -- if I gave you that determinant to find, 860 00:49:51,640 --> 00:49:53,890 you might subtract this row from this. 861 00:49:53,890 --> 00:49:55,780 It would kill that one. 862 00:49:55,780 --> 00:49:59,170 Subtract this row from this, it would kill that one. 863 00:49:59,170 --> 00:50:02,470 Then you'd have a simple determinant to do with 864 00:50:02,470 --> 00:50:06,300 differences, and it would -- 865 00:50:06,300 --> 00:50:08,960 this little subtraction, what I did 866 00:50:08,960 --> 00:50:12,580 was equivalent to moving the triangle 867 00:50:12,580 --> 00:50:16,550 to start at the origin. 868 00:50:16,550 --> 00:50:19,850 I did it fast, because time is up. 869 00:50:19,850 --> 00:50:24,530 And I didn't complete that proof of 3b. 870 00:50:24,530 --> 00:50:28,930 I'll leave -- the book has a carefully drawn figure to show 871 00:50:28,930 --> 00:50:30,870 why that works. 872 00:50:30,870 --> 00:50:34,110 But I hope you saw the main point is 873 00:50:34,110 --> 00:50:36,760 that for area and volume, determinant 874 00:50:36,760 --> 00:50:39,540 gives a great formula. 875 00:50:39,540 --> 00:50:40,100 OK. 876 00:50:40,100 --> 00:50:44,410 And next lectures are about eigenvalues, 877 00:50:44,410 --> 00:50:47,780 so we're really into the big stuff. 878 00:50:47,780 --> 00:50:49,330 Thanks. 879 00:50:49,330 --> 00:51:00,621