1
00:00:08,150 --> 00:00:13,310
OK, this is the second
lecture on determinants.

2
00:00:13,310 --> 00:00:15,260
There are only three.

3
00:00:15,260 --> 00:00:19,270
With determinants it's a
fascinating, small topic

4
00:00:19,270 --> 00:00:21,480
inside linear algebra.

5
00:00:21,480 --> 00:00:24,000
Used to be determinants
were the big thing,

6
00:00:24,000 --> 00:00:27,780
and linear algebra was the
little thing, but they --

7
00:00:27,780 --> 00:00:30,510
those changed, that
situation changed.

8
00:00:30,510 --> 00:00:36,170
Now determinants is one specific
part, very neat little part.

9
00:00:36,170 --> 00:00:41,740
And my goal today is to find
a formula for the determinant.

10
00:00:41,740 --> 00:00:45,480
It'll be a messy formula.

11
00:00:45,480 --> 00:00:49,570
So that's why you didn't
see it right away.

12
00:00:49,570 --> 00:00:52,090
But if I'm given
this n by n matrix

13
00:00:52,090 --> 00:00:55,890
then I use those
entries to create

14
00:00:55,890 --> 00:00:57,450
this number, the determinant.

15
00:00:57,450 --> 00:00:58,620
So there's a formula for it.

16
00:00:58,620 --> 00:01:03,250
In fact, there's another
formula, a second formula using

17
00:01:03,250 --> 00:01:05,129
something called cofactors.

18
00:01:05,129 --> 00:01:07,910
So you'll -- you have to
know what cofactors are.

19
00:01:07,910 --> 00:01:10,590
And then I'll apply
those formulas

20
00:01:10,590 --> 00:01:15,410
for some, some matrices
that have a lot of zeros

21
00:01:15,410 --> 00:01:18,430
away from the three diagonals.

22
00:01:18,430 --> 00:01:19,220
OK.

23
00:01:19,220 --> 00:01:24,690
So I'm shooting now for a
formula for the determinant.

24
00:01:24,690 --> 00:01:30,470
You remember we started with
these three properties, three

25
00:01:30,470 --> 00:01:32,610
simple properties,
but out of that we

26
00:01:32,610 --> 00:01:37,440
got all these amazing facts,
like the determinant of A B

27
00:01:37,440 --> 00:01:42,210
equals determinant of A
times determinant of B.

28
00:01:42,210 --> 00:01:45,280
But the three facts were --

29
00:01:45,280 --> 00:01:48,950
oh, how about I just
take two by twos.

30
00:01:48,950 --> 00:01:52,320
I know, because everybody here
knows, the determinant of a two

31
00:01:52,320 --> 00:01:57,770
by two matrix, but let's get
it out of these three formulas.

32
00:01:57,770 --> 00:02:00,940
OK, so here's my, my
two by two matrix.

33
00:02:00,940 --> 00:02:04,160
I'm looking for a formula
for this determinant.

34
00:02:04,160 --> 00:02:07,810
a b c d, OK.

35
00:02:07,810 --> 00:02:14,220
So property one, I know what
to do with the identity.

36
00:02:14,220 --> 00:02:14,900
Right?

37
00:02:14,900 --> 00:02:19,190
Property two allows
me to exchange rows,

38
00:02:19,190 --> 00:02:20,940
and I know what to do then.

39
00:02:20,940 --> 00:02:23,570
So I know that that
determinant is one.

40
00:02:23,570 --> 00:02:27,280
Property two allows me
to exchange rows and know

41
00:02:27,280 --> 00:02:32,400
that this determinant
is minus one.

42
00:02:32,400 --> 00:02:37,970
And now I want to use property
three to get everybody,

43
00:02:37,970 --> 00:02:39,210
to get everybody.

44
00:02:39,210 --> 00:02:40,850
And how will I do that?

45
00:02:40,850 --> 00:02:41,500
OK.

46
00:02:41,500 --> 00:02:46,600
So remember that if I keep
the second row the same,

47
00:02:46,600 --> 00:02:52,590
I'm allowed to use
linearity in the first row.

48
00:02:52,590 --> 00:02:54,750
And I'll just use
it in a simple way.

49
00:02:54,750 --> 00:03:02,020
I'll write this vector
a b as a 0 + 0 b.

50
00:03:04,590 --> 00:03:09,750
So that's one step using
property three, linearity

51
00:03:09,750 --> 00:03:12,361
in the first row when the
second row's the same.

52
00:03:12,361 --> 00:03:12,860
OK.

53
00:03:12,860 --> 00:03:15,650
But now you can guess
what I'm going to do next.

54
00:03:15,650 --> 00:03:17,870
I'll -- because I'd like to --

55
00:03:17,870 --> 00:03:20,040
if I can make the
matrices diagonal,

56
00:03:20,040 --> 00:03:22,730
then I'm clearly there.

57
00:03:22,730 --> 00:03:24,810
So I'll take this one.

58
00:03:24,810 --> 00:03:28,470
Now I'll keep the first row
fixed and split the second row,

59
00:03:28,470 --> 00:03:32,510
so that'll be an a 0
and I'll split that

60
00:03:32,510 --> 00:03:39,660
into a c 0 and, keeping that
first row the same, a 0 d.

61
00:03:39,660 --> 00:03:42,920
I used, for this
part, linearity.

62
00:03:42,920 --> 00:03:47,600
And now I'll -- whoops, that's
plus because I've got more

63
00:03:47,600 --> 00:03:48,710
coming.

64
00:03:48,710 --> 00:03:50,530
This one I'll do the same.

65
00:03:50,530 --> 00:03:53,710
I'll keep this
first row the same

66
00:03:53,710 --> 00:03:59,320
and I'll split c d
into c 0 and 0 d.

67
00:03:59,320 --> 00:04:00,170
OK.

68
00:04:00,170 --> 00:04:03,310
Now I've got four
easy determinants,

69
00:04:03,310 --> 00:04:05,250
and two of them are --

70
00:04:05,250 --> 00:04:07,460
well, all four are
extremely easy.

71
00:04:07,460 --> 00:04:12,020
Two of them are so easy as
to turn into zero, right?

72
00:04:12,020 --> 00:04:17,500
Which two of these determinants
are zero right away?

73
00:04:17,500 --> 00:04:20,670
The first guy is zero.

74
00:04:20,670 --> 00:04:21,720
Why is he zero?

75
00:04:21,720 --> 00:04:26,520
Why is that determinant
nothing, forget him?

76
00:04:26,520 --> 00:04:30,090
Well, it has a column of zeros.

77
00:04:30,090 --> 00:04:33,630
And by the -- well, so
one way to think is, well,

78
00:04:33,630 --> 00:04:35,170
it's a singular matrix.

79
00:04:35,170 --> 00:04:38,150
Oh, for, for like forty-eight
different reasons,

80
00:04:38,150 --> 00:04:40,060
that determinant is zero.

81
00:04:40,060 --> 00:04:43,100
It's a singular matrix
that has a column of zeros.

82
00:04:43,100 --> 00:04:44,570
It's, it's dead.

83
00:04:44,570 --> 00:04:47,900
And this one is
about as dead too.

84
00:04:47,900 --> 00:04:49,191
Column of zeros.

85
00:04:49,191 --> 00:04:49,690
OK.

86
00:04:49,690 --> 00:04:51,650
So that's leaving
us with this one.

87
00:04:51,650 --> 00:04:54,370
Now what do I -- how do
I know its determinant,

88
00:04:54,370 --> 00:04:56,910
following the rules?

89
00:04:56,910 --> 00:05:00,660
Well, I guess one of the
properties that we actually got

90
00:05:00,660 --> 00:05:07,120
to was the determinant of that
-- diagonal matrix, then --

91
00:05:07,120 --> 00:05:12,340
so I, I'm finally getting to
that determinant is the a d.

92
00:05:12,340 --> 00:05:16,030
And this determinant is what?

93
00:05:16,030 --> 00:05:18,010
What's this one?

94
00:05:18,010 --> 00:05:23,120
Minus -- because I would use
property two to do a flip

95
00:05:23,120 --> 00:05:29,220
to make it c b, then property
three to factor out the b,

96
00:05:29,220 --> 00:05:31,370
property c to
factor out the c --

97
00:05:31,370 --> 00:05:35,320
the property again to factor
out the c, and that minus,

98
00:05:35,320 --> 00:05:39,860
and of course finally I got the
answer that we knew we would

99
00:05:39,860 --> 00:05:42,110
get.

100
00:05:42,110 --> 00:05:44,810
But you see the method.

101
00:05:44,810 --> 00:05:48,250
You see the method, because it's
method I'm looking for here,

102
00:05:48,250 --> 00:05:52,190
not just a two by two answer
but the method of doing --

103
00:05:52,190 --> 00:05:59,220
now I can do three by threes
and four by fours and any size.

104
00:05:59,220 --> 00:06:05,470
So if you can see the method
of taking each row at a time --

105
00:06:05,470 --> 00:06:07,880
so let's -- what would
happen with three by threes?

106
00:06:07,880 --> 00:06:10,190
Can we mentally do
it rather than I

107
00:06:10,190 --> 00:06:13,730
write everything on the
board for three by threes?

108
00:06:13,730 --> 00:06:16,900
So what would we do if
I had three by threes?

109
00:06:16,900 --> 00:06:20,360
I would keep rows two
and three the same

110
00:06:20,360 --> 00:06:25,540
and I would split the first
row into how many pieces?

111
00:06:25,540 --> 00:06:26,570
Three pieces.

112
00:06:26,570 --> 00:06:30,050
I'd have an A zero
zero and a zero B zero

113
00:06:30,050 --> 00:06:35,670
and a zero zero C or
something for the first row.

114
00:06:35,670 --> 00:06:40,250
So I would instead of going from
one piece to two pieces to four

115
00:06:40,250 --> 00:06:48,560
pieces, I would go from one
piece to three pieces to --

116
00:06:48,560 --> 00:06:49,650
what would it be?

117
00:06:49,650 --> 00:06:54,290
Each of those three,
would, would it be nine?

118
00:06:54,290 --> 00:06:56,280
Or twenty-seven?

119
00:06:56,280 --> 00:06:58,940
Oh yeah, I've actually
got more steps,

120
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right.

121
00:06:59,510 --> 00:07:02,520
I'd go to nine but then I'd have
another row to straighten out,

122
00:07:02,520 --> 00:07:03,200
twenty-seven.

123
00:07:03,200 --> 00:07:04,630
Yes, oh God.

124
00:07:04,630 --> 00:07:06,610
OK, let me say this again then.

125
00:07:06,610 --> 00:07:10,170
If I -- if it was three
by three, I would --

126
00:07:10,170 --> 00:07:12,830
separating out one
row into three pieces

127
00:07:12,830 --> 00:07:16,820
would give me three, separating
out the second row into three

128
00:07:16,820 --> 00:07:20,230
pieces, then I'd be up to nine,
separating out the third row

129
00:07:20,230 --> 00:07:24,560
into its three pieces, I'd be
up to twenty-seven, three cubed,

130
00:07:24,560 --> 00:07:25,290
pieces.

131
00:07:25,290 --> 00:07:28,470
But a lot of them would be zero.

132
00:07:28,470 --> 00:07:31,350
So now when would
they not be zero?

133
00:07:31,350 --> 00:07:35,480
Tell me the pieces
that would not be zero.

134
00:07:35,480 --> 00:07:37,450
Now I will write
the non-zero ones.

135
00:07:37,450 --> 00:07:39,790
OK, so I have this matrix.

136
00:07:39,790 --> 00:07:43,520
I think I have to
use these, start

137
00:07:43,520 --> 00:07:50,590
using these double symbols here
because otherwise I could never

138
00:07:50,590 --> 00:07:52,970
do n by n.

139
00:07:52,970 --> 00:07:54,080
OK.

140
00:07:54,080 --> 00:07:55,220
OK.

141
00:07:55,220 --> 00:07:57,075
So I split this up like crazy.

142
00:08:00,200 --> 00:08:01,730
A bunch of pieces are zero.

143
00:08:01,730 --> 00:08:06,790
Whenever I have a column of
zeros, I know I've got zero.

144
00:08:06,790 --> 00:08:08,620
When do I not have zero?

145
00:08:08,620 --> 00:08:12,950
When do I have -- what is
it that's like these guys?

146
00:08:12,950 --> 00:08:16,330
These are the survivors,
two survivors there.

147
00:08:16,330 --> 00:08:18,080
So my question
for three by three

148
00:08:18,080 --> 00:08:20,990
is going to be what
are the survivors?

149
00:08:20,990 --> 00:08:22,760
How many survivors are there?

150
00:08:22,760 --> 00:08:24,290
What are they?

151
00:08:24,290 --> 00:08:28,010
And when do I get a survivor.

152
00:08:28,010 --> 00:08:30,780
Well, I would get a survivor --

153
00:08:30,780 --> 00:08:32,730
for example, one
survivor will be

154
00:08:32,730 --> 00:08:35,960
that one times that one times
that one, with all zeros

155
00:08:35,960 --> 00:08:37,340
everywhere else.

156
00:08:37,340 --> 00:08:39,320
That would be one survivor.

157
00:08:39,320 --> 00:08:44,330
a one one zero zero
zero a two two zero zero

158
00:08:44,330 --> 00:08:47,030
zero a three three.

159
00:08:47,030 --> 00:08:50,990
That's like the a d survivor.

160
00:08:50,990 --> 00:08:53,970
Tell me another survivor.

161
00:08:53,970 --> 00:08:58,560
What other thing -- oh,
now here you see the clue.

162
00:08:58,560 --> 00:09:00,750
Now can -- shall I just
say the whole clue?

163
00:09:00,750 --> 00:09:02,920
That I'm having --

164
00:09:02,920 --> 00:09:10,460
the survivors have one entry
from each row and each column.

165
00:09:10,460 --> 00:09:14,430
One entry from each
row and column.

166
00:09:14,430 --> 00:09:16,760
Because if some
column is missing,

167
00:09:16,760 --> 00:09:20,600
then I get a singular matrix.

168
00:09:20,600 --> 00:09:22,960
And that, that's
one of these guys.

169
00:09:22,960 --> 00:09:25,470
See, you see what
happened with --

170
00:09:25,470 --> 00:09:27,480
this guy?

171
00:09:27,480 --> 00:09:32,950
Column one never
got used in 0 b 0 d.

172
00:09:32,950 --> 00:09:35,620
So its determinant was
zero and I forget it.

173
00:09:35,620 --> 00:09:37,510
So I'm going to forget
those and just put --

174
00:09:37,510 --> 00:09:42,750
so tell me one more that
would be a survivor?

175
00:09:42,750 --> 00:09:44,960
Well -- well,
here's another one.

176
00:09:44,960 --> 00:09:50,690
a one one zero zero -- now OK,
that's used up row -- row one

177
00:09:50,690 --> 00:09:51,800
is used.

178
00:09:51,800 --> 00:09:54,420
Column one is already
used so it better be zero.

179
00:09:57,510 --> 00:09:58,850
What else could I have?

180
00:09:58,850 --> 00:10:03,050
Where could I pick the guy --
which column shall I use in row

181
00:10:03,050 --> 00:10:04,360
two?

182
00:10:04,360 --> 00:10:08,070
Use column three, because
here if I use column --

183
00:10:08,070 --> 00:10:10,140
here I used column
one and row one.

184
00:10:10,140 --> 00:10:12,710
This was like the column --

185
00:10:12,710 --> 00:10:15,950
numbers were one two
three, right in order.

186
00:10:15,950 --> 00:10:23,140
Now the column numbers are going
to be one three, column three,

187
00:10:23,140 --> 00:10:26,070
and column two.

188
00:10:26,070 --> 00:10:29,620
So the row numbers are
one two three, of course.

189
00:10:29,620 --> 00:10:32,100
The column numbers are some --

190
00:10:32,100 --> 00:10:35,550
OK, some permutation
of one two three,

191
00:10:35,550 --> 00:10:38,880
and here they come in
the order one three two.

192
00:10:38,880 --> 00:10:42,060
It's just like having
a permutation matrix

193
00:10:42,060 --> 00:10:46,330
with, instead of the
ones, with numbers.

194
00:10:46,330 --> 00:10:50,940
And actually, it's very close
to having a permutation matrix,

195
00:10:50,940 --> 00:10:55,550
because I, what I do eventually
is I factor out these numbers

196
00:10:55,550 --> 00:10:57,630
and then I have got.

197
00:10:57,630 --> 00:10:59,520
So what is that
determinant equal?

198
00:10:59,520 --> 00:11:01,120
I factor those
numbers out and I've

199
00:11:01,120 --> 00:11:05,400
got a one one times a two
two times a three three.

200
00:11:05,400 --> 00:11:08,020
And what does this
determinant equal?

201
00:11:08,020 --> 00:11:09,530
Yeah, now tell me the, this --

202
00:11:09,530 --> 00:11:13,140
I mean, we're really getting
to the heart of these formulas

203
00:11:13,140 --> 00:11:13,690
now.

204
00:11:13,690 --> 00:11:16,800
What is that determinant?

205
00:11:16,800 --> 00:11:20,400
By the laws of -- by,
by our three properties,

206
00:11:20,400 --> 00:11:24,920
I can factor these out, I
can factor out the a one one,

207
00:11:24,920 --> 00:11:27,780
the a two three,
and the a three two.

208
00:11:27,780 --> 00:11:28,980
They're in separate rows.

209
00:11:28,980 --> 00:11:31,960
I can do each row separately.

210
00:11:31,960 --> 00:11:35,230
And then I just have to
decide is that plus sign

211
00:11:35,230 --> 00:11:37,530
or is that a minus sign?

212
00:11:37,530 --> 00:11:42,640
And the answer is it's a minus.

213
00:11:42,640 --> 00:11:43,490
Why minus?

214
00:11:43,490 --> 00:11:47,350
Because these is
one row exchange

215
00:11:47,350 --> 00:11:49,780
to get it back to the identity.

216
00:11:49,780 --> 00:11:52,740
So that's a minus.

217
00:11:52,740 --> 00:11:53,960
Now I through?

218
00:11:53,960 --> 00:11:55,455
No, because there
are other ways.

219
00:11:59,020 --> 00:12:02,060
What I'm really
through with, what

220
00:12:02,060 --> 00:12:04,510
I've done, what I've,
what I've completed

221
00:12:04,510 --> 00:12:08,110
is only the part where
the a one one is there.

222
00:12:08,110 --> 00:12:11,810
But now I've got parts
where it's a one two.

223
00:12:15,010 --> 00:12:18,820
And now if it's a one two that
row is used, that column is

224
00:12:18,820 --> 00:12:19,610
used.

225
00:12:19,610 --> 00:12:21,260
You see that idea?

226
00:12:21,260 --> 00:12:25,380
I could use this row and column.

227
00:12:25,380 --> 00:12:28,070
Now that column is used,
that column is used,

228
00:12:28,070 --> 00:12:31,300
and this guy has to be
here, a three three.

229
00:12:31,300 --> 00:12:33,190
And what's that determinant?

230
00:12:33,190 --> 00:12:38,510
That's an a one two times an a
two one times an a three three,

231
00:12:38,510 --> 00:12:42,510
and does it have
a plus or a minus?

232
00:12:42,510 --> 00:12:43,820
A minus is right.

233
00:12:43,820 --> 00:12:45,600
It has a minus.

234
00:12:45,600 --> 00:12:47,690
Because it's one
flip away from an id-

235
00:12:47,690 --> 00:12:51,730
from the, regular, the right
order, the diagonal order.

236
00:12:51,730 --> 00:12:54,200
And now what's the other
guy with a -- with,

237
00:12:54,200 --> 00:12:57,470
a one two up there?

238
00:12:57,470 --> 00:12:59,240
I could have used this row.

239
00:12:59,240 --> 00:13:05,080
I could have put this guy
here and this guy here.

240
00:13:05,080 --> 00:13:05,580
Right?

241
00:13:05,580 --> 00:13:07,730
You see the whole deal?

242
00:13:07,730 --> 00:13:14,030
Now that's an a one two,
a two three, a three one,

243
00:13:14,030 --> 00:13:17,970
and does that go with
a plus or a minus?

244
00:13:17,970 --> 00:13:19,720
Yeah, now that takes
a minute of thinking,

245
00:13:19,720 --> 00:13:23,210
doesn't it, because one
row exchange doesn't get it

246
00:13:23,210 --> 00:13:24,760
in line.

247
00:13:24,760 --> 00:13:26,300
So what is the answer for this?

248
00:13:26,300 --> 00:13:28,090
Plus or minus?

249
00:13:28,090 --> 00:13:32,120
Plus, because it
takes two exchanges.

250
00:13:32,120 --> 00:13:35,920
I could exchange rows one and
three and then two and three.

251
00:13:35,920 --> 00:13:40,360
Two exchanges makes
this thing a plus.

252
00:13:40,360 --> 00:13:43,300
And then finally we have --
we're going to have two more.

253
00:13:43,300 --> 00:13:43,800
OK.

254
00:13:43,800 --> 00:13:52,770
Zero zero a one three, a two
one zero zero, zero a three two

255
00:13:52,770 --> 00:13:54,160
zero.

256
00:13:54,160 --> 00:13:55,720
And one more guy.

257
00:13:55,720 --> 00:14:00,820
Zero zero a one
three, zero a two

258
00:14:00,820 --> 00:14:06,750
two zero, A three one zero zero.

259
00:14:06,750 --> 00:14:08,870
And let's put down
what we get from those.

260
00:14:08,870 --> 00:14:14,160
An a one three, an a two one,
and an a three two, and I

261
00:14:14,160 --> 00:14:16,650
think that one is a plus.

262
00:14:16,650 --> 00:14:21,510
And this guys is a minus because
one exchange would put it --

263
00:14:21,510 --> 00:14:24,864
would order it.

264
00:14:24,864 --> 00:14:25,655
And that's a minus.

265
00:14:29,230 --> 00:14:32,510
All right, that has
taken one whole board

266
00:14:32,510 --> 00:14:35,030
just to do the three by three.

267
00:14:35,030 --> 00:14:37,630
But do you agree
that we now have

268
00:14:37,630 --> 00:14:42,590
a formula for the
determinant which

269
00:14:42,590 --> 00:14:44,080
came from the three properties?

270
00:14:46,840 --> 00:14:50,440
And it must be it.

271
00:14:50,440 --> 00:14:53,090
And I'm going to
keep that formula.

272
00:14:53,090 --> 00:14:57,870
That's a famous -- that three
by three formula is one that

273
00:14:57,870 --> 00:15:01,570
if, if the cameras will follow
me back to the beginning here,

274
00:15:01,570 --> 00:15:07,130
I, I get the ones with the plus
sign are the ones that go down

275
00:15:07,130 --> 00:15:08,800
like down this way.

276
00:15:08,800 --> 00:15:10,390
And the ones with
the minus signs

277
00:15:10,390 --> 00:15:13,940
are sort of the ones
that go this way.

278
00:15:13,940 --> 00:15:17,440
I won't make that precise.

279
00:15:17,440 --> 00:15:20,700
For two reasons, one,
it would clutter up

280
00:15:20,700 --> 00:15:24,970
the board, and second reason,
it wouldn't be right for four

281
00:15:24,970 --> 00:15:26,300
by fours.

282
00:15:26,300 --> 00:15:29,540
For four by four, let
me just say right away,

283
00:15:29,540 --> 00:15:33,360
four by four matrix --
the, the cross diagonal,

284
00:15:33,360 --> 00:15:38,190
the wrong diagonal happens
to come out with a plus sign.

285
00:15:38,190 --> 00:15:39,800
Why is that?

286
00:15:39,800 --> 00:15:43,600
If I have a four by
four matrix with ones

287
00:15:43,600 --> 00:15:50,080
coming on the counter diagonal,
that determinant is plus.

288
00:15:50,080 --> 00:15:50,980
Why?

289
00:15:50,980 --> 00:15:54,980
Why plus for that guy?

290
00:15:54,980 --> 00:15:59,720
Because if I exchange
rows one and four and then

291
00:15:59,720 --> 00:16:02,850
I exchange rows two and
three, I've got the identity,

292
00:16:02,850 --> 00:16:05,590
and I did two exchanges.

293
00:16:05,590 --> 00:16:10,880
So this down to this, like,
you know, down toward Miami

294
00:16:10,880 --> 00:16:16,330
and down toward LA stuff is,
like, three by three only.

295
00:16:16,330 --> 00:16:16,830
OK.

296
00:16:19,510 --> 00:16:25,185
But I do want to get now --

297
00:16:25,185 --> 00:16:27,310
I don't want to go through
this for a four by four.

298
00:16:29,990 --> 00:16:34,280
I do want to get now
the general formula.

299
00:16:34,280 --> 00:16:39,570
So this is what I refer to in
the book as the big formula.

300
00:16:39,570 --> 00:16:43,140
So now this is the big
formula for the determinant.

301
00:16:43,140 --> 00:16:47,380
I'm asking you to make a jump
from two by two and three

302
00:16:47,380 --> 00:16:50,260
by three to n by n.

303
00:16:50,260 --> 00:16:52,220
OK, so this will
be the big formula.

304
00:17:00,250 --> 00:17:07,310
That the determinant of A is
the sum of a whole lot of terms.

305
00:17:07,310 --> 00:17:09,550
And what are those terms?

306
00:17:09,550 --> 00:17:12,490
And, and is it a
plus or a minus sign,

307
00:17:12,490 --> 00:17:14,869
and I have to tell you
which, which it is,

308
00:17:14,869 --> 00:17:18,520
because this came in -- in
the three by three case,

309
00:17:18,520 --> 00:17:20,140
I had how many terms?

310
00:17:20,140 --> 00:17:21,859
Six.

311
00:17:21,859 --> 00:17:25,630
And half were plus
and half were minus.

312
00:17:25,630 --> 00:17:30,570
How many terms are you
figuring for four by four?

313
00:17:30,570 --> 00:17:36,520
If I get two terms in the
two by two case, three --

314
00:17:36,520 --> 00:17:41,760
six terms in the three by three
case, what's that pattern?

315
00:17:41,760 --> 00:17:44,210
How many terms in the
four by four case?

316
00:17:46,830 --> 00:17:48,320
Twenty-four.

317
00:17:48,320 --> 00:17:49,715
Four factorial.

318
00:17:52,430 --> 00:17:53,750
Why four factorial?

319
00:17:53,750 --> 00:17:56,200
This will be a sum
of n factorial terms.

320
00:18:00,000 --> 00:18:01,530
Twenty-four, a
hundred and twenty,

321
00:18:01,530 --> 00:18:05,570
seven hundred and twenty,
whatever's after that.

322
00:18:05,570 --> 00:18:06,580
OK.

323
00:18:06,580 --> 00:18:08,945
Half plus and half minus.

324
00:18:12,110 --> 00:18:14,720
And where do those n
factorial -- terms come from?

325
00:18:14,720 --> 00:18:17,310
This is the moment to
listen to this lecture.

326
00:18:17,310 --> 00:18:20,110
Where do those n
factorial terms come from?

327
00:18:20,110 --> 00:18:23,690
They come because the first,
the guy in the first row

328
00:18:23,690 --> 00:18:26,940
can be chosen n ways.

329
00:18:26,940 --> 00:18:33,390
And after he's chosen, that's
used up that, that column.

330
00:18:33,390 --> 00:18:38,440
So the one in the second row
can be chosen n minus one ways.

331
00:18:38,440 --> 00:18:42,360
And after she's chosen,
that second column has been

332
00:18:42,360 --> 00:18:43,160
used.

333
00:18:43,160 --> 00:18:46,650
And then the one in the third
row can be chosen n minus two

334
00:18:46,650 --> 00:18:49,140
ways, and after it's chosen --

335
00:18:49,140 --> 00:18:52,330
notice how I'm getting
these personal pronouns.

336
00:18:52,330 --> 00:18:53,510
But I've run out.

337
00:18:53,510 --> 00:18:59,570
And I'm not willing to
stop with three by three,

338
00:18:59,570 --> 00:19:02,370
so I'm just going to
write the formula down.

339
00:19:02,370 --> 00:19:07,470
So the one in the first row
comes from some column alpha.

340
00:19:07,470 --> 00:19:10,670
I don't know what alpha is.

341
00:19:10,670 --> 00:19:11,600
And the one in the --

342
00:19:11,600 --> 00:19:14,410
I multiply that by somebody
in the second row that comes

343
00:19:14,410 --> 00:19:16,530
from some different column.

344
00:19:16,530 --> 00:19:19,470
And I multiply that by somebody
in the third row who comes

345
00:19:19,470 --> 00:19:21,850
from some yet different column.

346
00:19:21,850 --> 00:19:25,269
And then in the n-th
row, I don't know what --

347
00:19:25,269 --> 00:19:26,310
I don't know how to draw.

348
00:19:26,310 --> 00:19:29,570
Maybe omega, for last.

349
00:19:29,570 --> 00:19:32,170
And the whole point
is then that --

350
00:19:32,170 --> 00:19:34,510
that those column
numbers are different,

351
00:19:34,510 --> 00:19:40,990
that alpha, beta, gamma, omega,
that set of column numbers

352
00:19:40,990 --> 00:19:50,100
is some permutation,
permutation of one to n.

353
00:19:50,100 --> 00:19:54,830
It, it, the n column
numbers are each used once.

354
00:19:54,830 --> 00:19:57,570
And that gives us
n factorial terms.

355
00:19:57,570 --> 00:20:02,130
And when I choose
a term, that means

356
00:20:02,130 --> 00:20:04,850
I'm choosing somebody
from every row and column.

357
00:20:04,850 --> 00:20:10,300
And then I just -- like the way
I had this from row and column

358
00:20:10,300 --> 00:20:14,160
one, row and column two, row
and column three, so that --

359
00:20:14,160 --> 00:20:19,080
what was the alpha beta stuff
in that, for that term here?

360
00:20:19,080 --> 00:20:22,360
Alpha was one, beta was
two, gamma was three.

361
00:20:22,360 --> 00:20:26,180
The permutation was, was
the trivial permutation, one

362
00:20:26,180 --> 00:20:28,140
two three, everybody
in the right order.

363
00:20:30,730 --> 00:20:31,750
You see that formula?

364
00:20:34,420 --> 00:20:37,120
It's -- do you see why I
didn't want to start with that

365
00:20:37,120 --> 00:20:39,930
the first day, Friday?

366
00:20:39,930 --> 00:20:42,470
I'd rather we understood
the properties.

367
00:20:42,470 --> 00:20:44,920
Because out of this
formula, presumably I

368
00:20:44,920 --> 00:20:47,720
could figure out all
these properties.

369
00:20:47,720 --> 00:20:50,900
How would I know that the
determinant of the identity

370
00:20:50,900 --> 00:20:56,630
matrix was one, for example,
out of this formula?

371
00:20:56,630 --> 00:20:59,680
Why is -- if A is
the identity matrix,

372
00:20:59,680 --> 00:21:04,080
how does this formula
give me a plus one?

373
00:21:04,080 --> 00:21:05,160
You see it, right?

374
00:21:05,160 --> 00:21:10,440
Because, because almost
all the terms are zeros.

375
00:21:10,440 --> 00:21:15,450
Which term isn't zero, if,
if A is the identity matrix?

376
00:21:15,450 --> 00:21:18,501
Almost all the terms are zero
because almost all the As are

377
00:21:18,501 --> 00:21:19,000
zero.

378
00:21:19,000 --> 00:21:21,070
It's only, the only
time I'll get something

379
00:21:21,070 --> 00:21:25,270
is if it's a one one times a
two two times a three three.

380
00:21:25,270 --> 00:21:28,590
Only, only the,
only the permutation

381
00:21:28,590 --> 00:21:32,650
that's in the right order
will, will give me something.

382
00:21:32,650 --> 00:21:34,400
It'll come with a plus sign.

383
00:21:34,400 --> 00:21:37,490
And the determinant of
the identity is one.

384
00:21:37,490 --> 00:21:40,910
So, so we could go back
from this formula and prove

385
00:21:40,910 --> 00:21:41,710
everything.

386
00:21:41,710 --> 00:21:45,460
We could even try to prove
that the determinant of A B

387
00:21:45,460 --> 00:21:48,860
was the determinant of A
times the determinant of B.

388
00:21:48,860 --> 00:21:51,070
But like next week we would
still be working on it,

389
00:21:51,070 --> 00:21:54,600
because it's not --

390
00:21:54,600 --> 00:21:56,500
clear from -- if I took A B,

391
00:21:56,500 --> 00:21:57,030
my God.

392
00:21:57,030 --> 00:21:57,530
You know --.

393
00:21:57,530 --> 00:22:02,210
The entries of A B would
be all these pieces.

394
00:22:02,210 --> 00:22:06,630
Well, probably, it's probably
-- historically it's been done,

395
00:22:06,630 --> 00:22:09,131
but it won't be repeated
in eighteen oh six.

396
00:22:09,131 --> 00:22:09,630
OK.

397
00:22:09,630 --> 00:22:16,480
It would be possible probably
to see, why the determinant of A

398
00:22:16,480 --> 00:22:18,190
equals the determinant
of A transpose.

399
00:22:18,190 --> 00:22:21,154
That was another, like,
miracle property at the end.

400
00:22:21,154 --> 00:22:22,820
That would, that
would, that's an easier

401
00:22:22,820 --> 00:22:25,290
one, which we could find.

402
00:22:25,290 --> 00:22:26,200
OK.

403
00:22:26,200 --> 00:22:30,460
Is that all right
for the big formula?

404
00:22:30,460 --> 00:22:33,160
I could take you
then a, a typical --

405
00:22:33,160 --> 00:22:36,070
let me do an example.

406
00:22:36,070 --> 00:22:39,280
Which I'll just create.

407
00:22:39,280 --> 00:22:42,410
I'll take a four by four matrix.

408
00:22:42,410 --> 00:22:46,820
I'll put some, I'll put some
ones in and some zeros in.

409
00:22:46,820 --> 00:22:47,350
OK.

410
00:22:47,350 --> 00:22:48,530
Let me --

411
00:22:48,530 --> 00:22:52,690
I don't know how many to
put in, to tell the truth.

412
00:22:52,690 --> 00:22:54,100
I've never done this before.

413
00:22:58,840 --> 00:23:02,260
I don't know the
determinant of that matrix.

414
00:23:02,260 --> 00:23:05,380
So like mathematics is being
done for the first time

415
00:23:05,380 --> 00:23:07,772
in, in front of your eyes.

416
00:23:07,772 --> 00:23:08,730
What's the determinant?

417
00:23:12,090 --> 00:23:15,210
Well, a lot of -- there
are twenty-four terms,

418
00:23:15,210 --> 00:23:17,700
because it's four by four.

419
00:23:17,700 --> 00:23:19,460
Many of them will be
zero, because I've

420
00:23:19,460 --> 00:23:22,430
got all those zeros there.

421
00:23:22,430 --> 00:23:25,350
Maybe the whole
determinant is zero.

422
00:23:25,350 --> 00:23:29,390
I mean, I -- is that
a singular matrix?

423
00:23:29,390 --> 00:23:33,470
That possibility
definitely exists.

424
00:23:33,470 --> 00:23:37,620
I could, I could, So one way
to do it would be elimination.

425
00:23:37,620 --> 00:23:42,090
Actually, that would probably
be a fairly reasonable way.

426
00:23:42,090 --> 00:23:44,440
I could use elimination,
so I could use --

427
00:23:44,440 --> 00:23:47,760
go back to those properties,
that -- and use elimination,

428
00:23:47,760 --> 00:23:50,990
get down, eliminate it down,
do I have a row of zeros

429
00:23:50,990 --> 00:23:53,020
at the end of elimination?

430
00:23:53,020 --> 00:23:54,230
The answer is zero.

431
00:23:54,230 --> 00:23:58,320
I was thinking, shall
I try this big formula?

432
00:23:58,320 --> 00:23:59,390
OK.

433
00:23:59,390 --> 00:24:00,810
Let's try the big formula.

434
00:24:00,810 --> 00:24:08,190
How -- tell me one way I can go
down the matrix, taking a one,

435
00:24:08,190 --> 00:24:13,250
taking a one from every row and
column, and make it to the end?

436
00:24:13,250 --> 00:24:15,640
So it's -- I get
something that isn't zero.

437
00:24:15,640 --> 00:24:18,280
Well, one way to do it, I could
take that times that times

438
00:24:18,280 --> 00:24:20,360
that times that times that.

439
00:24:20,360 --> 00:24:23,510
That would be one and,
and, and I just said,

440
00:24:23,510 --> 00:24:25,990
that comes in with what sign?

441
00:24:25,990 --> 00:24:26,820
Plus.

442
00:24:26,820 --> 00:24:28,690
That comes with a plus sign.

443
00:24:28,690 --> 00:24:32,600
Because, because
that permutation --

444
00:24:32,600 --> 00:24:35,160
I've just written
the permutation

445
00:24:35,160 --> 00:24:38,530
about four three two
one, and one exchange

446
00:24:38,530 --> 00:24:40,900
and a second exchange,
two exchanges

447
00:24:40,900 --> 00:24:42,540
puts it in the correct order.

448
00:24:46,750 --> 00:24:51,620
Keep walking away, don't....

449
00:24:51,620 --> 00:24:54,550
OK, we're executing a
determinant formula here.

450
00:24:54,550 --> 00:25:07,900
Uh as long as it's not
periodic, of course.

451
00:25:07,900 --> 00:25:11,290
If he comes back I'm in --

452
00:25:11,290 --> 00:25:11,790
no.

453
00:25:11,790 --> 00:25:13,620
All right, all right.

454
00:25:13,620 --> 00:25:16,260
OK, so that would
give me a plus one.

455
00:25:21,260 --> 00:25:22,700
All right.

456
00:25:22,700 --> 00:25:24,240
Are there any others?

457
00:25:24,240 --> 00:25:26,610
Well, of course we
see another one here.

458
00:25:26,610 --> 00:25:29,840
This times this times this
times this strikes us right

459
00:25:29,840 --> 00:25:30,340
away.

460
00:25:30,340 --> 00:25:34,470
So that's the order
three, the order --

461
00:25:34,470 --> 00:25:37,450
let me make a little
different mark here.

462
00:25:37,450 --> 00:25:41,520
Three two one four.

463
00:25:41,520 --> 00:25:45,330
And is that a plus or a
minus, three two one four?

464
00:25:48,030 --> 00:25:52,950
Is that, is that permutation
a plus or a minus permutation?

465
00:25:52,950 --> 00:25:53,827
It's a minus.

466
00:25:53,827 --> 00:25:54,660
How do you see that?

467
00:25:54,660 --> 00:25:59,640
What exchange shall I do to
get it in the right order?

468
00:25:59,640 --> 00:26:02,480
If I exchange the one and the
three I'm in the right orders,

469
00:26:02,480 --> 00:26:05,120
took one exchange to do it,
so that would be a plus --

470
00:26:05,120 --> 00:26:07,040
that would be a minus one.

471
00:26:07,040 --> 00:26:09,700
And now I don't know if
there're any more here.

472
00:26:09,700 --> 00:26:10,280
Let's see.

473
00:26:10,280 --> 00:26:15,670
Let me try again
starting with this.

474
00:26:15,670 --> 00:26:18,360
Now I've got to pick somebody
from -- oh yeah, see,

475
00:26:18,360 --> 00:26:20,400
you see what's happening.

476
00:26:20,400 --> 00:26:24,900
If I I start there, OK,
column three is used.

477
00:26:24,900 --> 00:26:27,680
So then when I go to next
row, I can't use that,

478
00:26:27,680 --> 00:26:28,850
I must use that.

479
00:26:28,850 --> 00:26:30,810
Now columns two
and three are used.

480
00:26:30,810 --> 00:26:33,370
When I come to this
row I must use that.

481
00:26:33,370 --> 00:26:34,700
And then I must use that.

482
00:26:34,700 --> 00:26:38,280
So if I start there, this
is the only one I get.

483
00:26:38,280 --> 00:26:42,220
And similarly, if I start there,
that's the only one I get.

484
00:26:42,220 --> 00:26:45,280
So what's the determinant?

485
00:26:45,280 --> 00:26:47,420
What's the determinant?

486
00:26:47,420 --> 00:26:47,920
Zero.

487
00:26:47,920 --> 00:26:51,940
The determinant is
zero for that case.

488
00:26:51,940 --> 00:26:56,080
Because we, we were able to
check the twenty-four terms.

489
00:26:56,080 --> 00:26:57,850
Twenty-two of them were zero.

490
00:26:57,850 --> 00:26:59,290
One of them was plus one.

491
00:26:59,290 --> 00:27:01,100
One of them was minus one.

492
00:27:01,100 --> 00:27:04,600
Add up the twenty-four
terms, zero is the answer.

493
00:27:04,600 --> 00:27:05,120
OK.

494
00:27:05,120 --> 00:27:07,136
Well, I didn't know
it would be zero, I --

495
00:27:07,136 --> 00:27:08,760
because I wasn't,
like, thinking ahead.

496
00:27:08,760 --> 00:27:10,700
I was a little scared, actually.

497
00:27:13,300 --> 00:27:18,360
I said, that,
apparition went by.

498
00:27:18,360 --> 00:27:22,420
So and I don't know if
the camera caught that.

499
00:27:22,420 --> 00:27:23,970
So whether the rest
of the world will

500
00:27:23,970 --> 00:27:27,450
realize that I was in danger
or not, we don't know.

501
00:27:27,450 --> 00:27:29,674
But anyway, I guess
he just wanted

502
00:27:29,674 --> 00:27:31,590
to be sure that we got
the right answer, which

503
00:27:31,590 --> 00:27:33,260
is determinant zero.

504
00:27:33,260 --> 00:27:36,060
And then that makes me
think, OK, the matrix

505
00:27:36,060 --> 00:27:41,110
must be, the matrix
must be singular.

506
00:27:41,110 --> 00:27:43,020
And then if the
matrix is singular,

507
00:27:43,020 --> 00:27:46,000
maybe there's another way to see
that it's singular, like find

508
00:27:46,000 --> 00:27:47,225
something in its null space.

509
00:27:50,690 --> 00:27:54,190
Or find a combination of
the rows that gives zero.

510
00:27:54,190 --> 00:27:58,320
And like what d- what, what
combination of those rows

511
00:27:58,320 --> 00:28:01,070
does give zero.

512
00:28:01,070 --> 00:28:05,980
Suppose I add rows
one and rows three.

513
00:28:05,980 --> 00:28:08,660
If I add rows one and
rows three, what do I get?

514
00:28:08,660 --> 00:28:10,990
I get a row of all ones.

515
00:28:10,990 --> 00:28:15,340
Then if I add rows two and rows
four I get a row of all ones.

516
00:28:15,340 --> 00:28:19,370
So row one minus row two
plus row three minus row four

517
00:28:19,370 --> 00:28:21,280
is probably the zero row.

518
00:28:21,280 --> 00:28:24,490
It's a singular matrix.

519
00:28:24,490 --> 00:28:27,330
And I could find something in
its null space the same way.

520
00:28:27,330 --> 00:28:29,720
That would be a combination
of columns that gives zero.

521
00:28:29,720 --> 00:28:32,040
OK, there's an example.

522
00:28:32,040 --> 00:28:32,950
All right.

523
00:28:32,950 --> 00:28:36,840
So that's, well, that
shows two things.

524
00:28:36,840 --> 00:28:39,280
That shows how we get
the twenty-four terms

525
00:28:39,280 --> 00:28:41,700
and it shows the great
advantage of having

526
00:28:41,700 --> 00:28:43,810
a lot of zeros in there.

527
00:28:43,810 --> 00:28:45,830
OK.

528
00:28:45,830 --> 00:28:48,970
So we'll use this big
formula, but I want to pick --

529
00:28:48,970 --> 00:28:53,710
I want to go onward
now to cofactors.

530
00:28:53,710 --> 00:28:56,030
Onward to cofactors.

531
00:28:56,030 --> 00:29:03,330
Cofactors is a way of breaking
up this big formula that

532
00:29:03,330 --> 00:29:08,910
connects this n by n -- this
is an n by n determinant that

533
00:29:08,910 --> 00:29:13,880
we've just have a formula
for, the big formula.

534
00:29:13,880 --> 00:29:18,450
So cofactors is a way to connect
this n by n determinant to,

535
00:29:18,450 --> 00:29:22,270
determinants one smaller.

536
00:29:22,270 --> 00:29:24,170
One smaller.

537
00:29:24,170 --> 00:29:29,430
And the way we want to do it
is actually going to show up in

538
00:29:29,430 --> 00:29:30,260
this.

539
00:29:30,260 --> 00:29:34,470
Since the three by three is the
one that we wrote out in full,

540
00:29:34,470 --> 00:29:38,060
let's, let me do
this three by --

541
00:29:38,060 --> 00:29:43,350
so I'm talking about cofactors,
and I'm going to start again

542
00:29:43,350 --> 00:29:44,420
with three by three.

543
00:29:48,180 --> 00:29:51,220
And I'm going to take
the, the exact formula,

544
00:29:51,220 --> 00:29:55,690
and I'm just going to
write it as a one one --

545
00:29:55,690 --> 00:29:59,540
this is the determinant
I'm writing.

546
00:29:59,540 --> 00:30:03,360
I'm just going to say
a one one times what?

547
00:30:03,360 --> 00:30:04,760
A one one times what?

548
00:30:04,760 --> 00:30:08,830
And it's a one one times
a two two a three three

549
00:30:08,830 --> 00:30:12,230
minus a two three a three two.

550
00:30:15,410 --> 00:30:22,640
Then I've got the a one
two stuff times something.

551
00:30:22,640 --> 00:30:26,910
And I've got the a one
three stuff times something.

552
00:30:26,910 --> 00:30:29,500
Do you see what I'm doing?

553
00:30:29,500 --> 00:30:33,990
I'm taking our big formula
and I'm saying, OK,

554
00:30:33,990 --> 00:30:37,320
choose column --

555
00:30:37,320 --> 00:30:40,380
out of the first row,
choose column one.

556
00:30:40,380 --> 00:30:43,520
And take all the possibilities.

557
00:30:43,520 --> 00:30:46,240
And those extra
factors will be what

558
00:30:46,240 --> 00:30:51,800
we'll call the cofactor, co
meaning going with a one one.

559
00:30:51,800 --> 00:30:55,890
So this in parenthesis
are, these are in,

560
00:30:55,890 --> 00:30:57,930
the cofactors are in parens.

561
00:31:02,210 --> 00:31:05,660
A one one times something.

562
00:31:05,660 --> 00:31:09,820
And I figured out what that
something was by just looking

563
00:31:09,820 --> 00:31:14,290
back -- if I can walk back
here to the, to the a one one,

564
00:31:14,290 --> 00:31:17,440
the one that comes down the
diagonal minus the one that

565
00:31:17,440 --> 00:31:19,830
comes that way.

566
00:31:19,830 --> 00:31:24,570
That's, those are the two,
only two that used a one one.

567
00:31:24,570 --> 00:31:26,970
So there they are, one
with a plus and one with a

568
00:31:26,970 --> 00:31:28,240
minus.

569
00:31:28,240 --> 00:31:30,800
And now I can write in the --

570
00:31:30,800 --> 00:31:33,100
I could look back and
see what used a one two

571
00:31:33,100 --> 00:31:34,740
and I can see what
used a one three,

572
00:31:34,740 --> 00:31:36,770
and those will give
me the cofactors

573
00:31:36,770 --> 00:31:38,920
of a one two and a one

574
00:31:38,920 --> 00:31:41,650
three.

575
00:31:41,650 --> 00:31:45,770
Before I do that, what's this
number, what is this cofactor?

576
00:31:48,310 --> 00:31:52,080
What is it there that's
multiplying a one one?

577
00:31:52,080 --> 00:31:55,360
Tell me what a two two a three
three minus a two three a three

578
00:31:55,360 --> 00:31:59,310
two is, for this --

579
00:31:59,310 --> 00:32:00,340
do you recognize that?

580
00:32:03,600 --> 00:32:06,500
Do you recognize --

581
00:32:06,500 --> 00:32:08,780
let's see, I can --
and I'll put it here.

582
00:32:08,780 --> 00:32:11,520
There's the a one one.

583
00:32:11,520 --> 00:32:14,160
That's used column one.

584
00:32:14,160 --> 00:32:17,645
Then there's -- the other
factors involved these other

585
00:32:17,645 --> 00:32:18,145
columns.

586
00:32:25,070 --> 00:32:27,360
This row is used.

587
00:32:27,360 --> 00:32:29,140
This column is used.

588
00:32:29,140 --> 00:32:32,870
So this the only things
left to use are these.

589
00:32:32,870 --> 00:32:36,250
And this formula
uses them, and what's

590
00:32:36,250 --> 00:32:39,860
the, what's the cofactor?

591
00:32:39,860 --> 00:32:42,420
Tell me what it is because
you see it, and then --

592
00:32:42,420 --> 00:32:47,150
I'll be happy you see what
the idea of cofactors.

593
00:32:47,150 --> 00:32:50,010
It's the determinant
of this smaller guy.

594
00:32:52,920 --> 00:32:55,400
A one one multiplies
the determinant

595
00:32:55,400 --> 00:32:56,590
of this smaller guy.

596
00:32:56,590 --> 00:33:02,840
That gives me all the a one
one part of the big formula.

597
00:33:02,840 --> 00:33:03,540
You see that?

598
00:33:03,540 --> 00:33:05,500
This, the determinant
of this smaller guy

599
00:33:05,500 --> 00:33:10,540
is a two two a three three
minus a two three a three two.

600
00:33:10,540 --> 00:33:14,050
In other words, once I've
used column one and row

601
00:33:14,050 --> 00:33:20,850
one, what's left is all the
ways to use the other n-1

602
00:33:20,850 --> 00:33:25,130
columns and n-1
rows, one of each.

603
00:33:25,130 --> 00:33:28,970
All the other -- and that's the
determinant of the smaller guy

604
00:33:28,970 --> 00:33:30,880
of size n-1.

605
00:33:30,880 --> 00:33:33,710
So that's the whole
idea of cofactors.

606
00:33:33,710 --> 00:33:37,150
And we just have to remember
that with determinants we've

607
00:33:37,150 --> 00:33:39,900
got pluses and minus
signs to keep straight.

608
00:33:39,900 --> 00:33:43,260
Can we keep this
next one straight?

609
00:33:43,260 --> 00:33:45,090
Let's do the next one.

610
00:33:45,090 --> 00:33:51,640
OK, the next one will
be when I use a one two.

611
00:33:51,640 --> 00:33:55,330
I'll have left -- so I can't
use that column any more,

612
00:33:55,330 --> 00:34:02,710
but I can use a two one and a
two three and I can use a three

613
00:34:02,710 --> 00:34:06,330
one and a three three.

614
00:34:06,330 --> 00:34:08,850
So this one gave me a one
times that determinant.

615
00:34:08,850 --> 00:34:13,760
This will give me a one two
times this determinant, a two

616
00:34:13,760 --> 00:34:24,690
one a three three minus
a two three a three one.

617
00:34:24,690 --> 00:34:27,969
So that's all the stuff
involving a one two.

618
00:34:27,969 --> 00:34:32,360
But have I got the sign right?

619
00:34:32,360 --> 00:34:35,280
Is the determinant of that
correctly given by that

620
00:34:35,280 --> 00:34:37,510
or is there a minus sign?

621
00:34:37,510 --> 00:34:39,520
There is a minus sign.

622
00:34:39,520 --> 00:34:40,929
I can follow one of these.

623
00:34:40,929 --> 00:34:43,080
If I do that times
that times that,

624
00:34:43,080 --> 00:34:45,046
that was one that's
showing up here,

625
00:34:45,046 --> 00:34:47,420
but it should have showed --
it should have been a minus.

626
00:34:52,969 --> 00:34:55,675
So I'm going to build that
minus sign into the cofactor.

627
00:34:58,580 --> 00:35:01,390
So, so the cofactor
-- so I'll put,

628
00:35:01,390 --> 00:35:04,110
put that minus sign in here.

629
00:35:04,110 --> 00:35:07,180
So because the cofactor
is going to be strictly

630
00:35:07,180 --> 00:35:09,900
the thing that multiplies
the, the factor.

631
00:35:09,900 --> 00:35:12,570
The factor is a one two,
the cofactor is this,

632
00:35:12,570 --> 00:35:15,620
is the parens, the
stuff in parentheses.

633
00:35:15,620 --> 00:35:18,590
So it's got the
minus sign built in.

634
00:35:18,590 --> 00:35:23,960
And if I did -- if I went on
to the third guy, there w-

635
00:35:23,960 --> 00:35:26,920
there'll be this and
this, this and this.

636
00:35:26,920 --> 00:35:28,410
And it would take
its determinant.

637
00:35:28,410 --> 00:35:31,790
It would come out
plus the determinant.

638
00:35:31,790 --> 00:35:34,960
So now I'm ready to
say what cofactors are.

639
00:35:34,960 --> 00:35:40,620
So this would be a plus and a
one three times its cofactor.

640
00:35:40,620 --> 00:35:45,940
And over here we had plus a
one one times this determinant.

641
00:35:45,940 --> 00:35:49,400
But and there we had the a
one two times its cofactor,

642
00:35:49,400 --> 00:35:53,310
but the -- so the point is
the cofactor is either plus

643
00:35:53,310 --> 00:35:56,320
or minus the determinant.

644
00:35:56,320 --> 00:35:57,990
So let me write that
underneath them.

645
00:35:57,990 --> 00:36:00,750
What is the, what are cofactors?

646
00:36:00,750 --> 00:36:09,650
The cofactor if any
number aij, let's say.

647
00:36:13,500 --> 00:36:18,630
This is, this is all the terms
in the, in the big formula that

648
00:36:18,630 --> 00:36:20,260
involve aij.

649
00:36:20,260 --> 00:36:25,490
We're especially interested
in a1j, the first row, that's

650
00:36:25,490 --> 00:36:28,790
what I've been talking about,
but any row would be all right.

651
00:36:28,790 --> 00:36:30,650
All right, so --

652
00:36:30,650 --> 00:36:32,900
what terms involve aij?

653
00:36:32,900 --> 00:36:42,563
So -- it's the determinant
of the n minus one matrix --

654
00:36:46,430 --> 00:36:51,470
with row i, column j erased.

655
00:36:56,460 --> 00:37:00,900
So it's the, it's a
matrix of size n-1 with --

656
00:37:00,900 --> 00:37:05,710
of course, because I can't use
this row or this column again.

657
00:37:05,710 --> 00:37:08,120
So I have the matrix all there.

658
00:37:08,120 --> 00:37:11,400
But now it's multiplied
by a plus or a minus.

659
00:37:11,400 --> 00:37:14,165
This is the cofactor, and
I'm going to call that cij.

660
00:37:17,750 --> 00:37:19,990
Capital, I use
capital c just to,

661
00:37:19,990 --> 00:37:22,570
just to emphasize that
these are important

662
00:37:22,570 --> 00:37:25,850
and emphasize that
they're, they're, they're

663
00:37:25,850 --> 00:37:29,591
different from the (a)s.

664
00:37:29,591 --> 00:37:30,090
OK.

665
00:37:30,090 --> 00:37:34,340
So now is it a plus
or is it a minus?

666
00:37:34,340 --> 00:37:36,340
Because we see
that in this case,

667
00:37:36,340 --> 00:37:41,260
for a one one it was a plus,
for a one two I -- this is ij --

668
00:37:41,260 --> 00:37:43,360
it was a minus.

669
00:37:43,360 --> 00:37:46,530
For this ij it was a plus.

670
00:37:46,530 --> 00:37:50,550
So any any guess on the
rule for plus or minus

671
00:37:50,550 --> 00:37:55,650
when we see those examples,
ij equal one one or one three

672
00:37:55,650 --> 00:37:58,070
was a plus?

673
00:37:58,070 --> 00:38:04,400
It sounds very like
i+j odd or even.

674
00:38:04,400 --> 00:38:06,170
That, that's
doesn't surprise us,

675
00:38:06,170 --> 00:38:07,600
and that's the right answer.

676
00:38:07,600 --> 00:38:17,950
So it's a plus if i+j is even
and it's a minus if i+j is odd.

677
00:38:24,450 --> 00:38:28,170
So if I go along row one
and look at the cofactors,

678
00:38:28,170 --> 00:38:32,430
I just take those determinants,
those one smaller determinants,

679
00:38:32,430 --> 00:38:36,830
and they come in order plus
minus plus minus plus minus.

680
00:38:36,830 --> 00:38:42,120
But if I go along row two and,
and, and take the cofactors

681
00:38:42,120 --> 00:38:46,310
of sub-determinants, they
would start with a minus,

682
00:38:46,310 --> 00:38:52,520
because the two one entry,
two plus one is odd, so the --

683
00:38:52,520 --> 00:38:57,560
like there's a pattern plus
minus plus minus plus if it was

684
00:38:57,560 --> 00:39:01,180
five by five, but then if I was
doing a cofactor then this sign

685
00:39:01,180 --> 00:39:06,087
would be minus plus minus
plus minus, plus minus plus --

686
00:39:06,087 --> 00:39:07,170
it's sort of checkerboard.

687
00:39:12,540 --> 00:39:13,040
OK.

688
00:39:17,551 --> 00:39:18,050
OK.

689
00:39:18,050 --> 00:39:22,220
Those are the signs that,
that are given by this rule,

690
00:39:22,220 --> 00:39:24,540
i+j even or odd.

691
00:39:24,540 --> 00:39:27,760
And those are built
into the cofactors.

692
00:39:27,760 --> 00:39:31,030
The thing is called
a minor without th-

693
00:39:31,030 --> 00:39:34,400
before you've built in the sign,
but I don't care about those.

694
00:39:34,400 --> 00:39:39,700
Build in that sign and
call it a cofactor.

695
00:39:39,700 --> 00:39:41,950
So what's the cofactor formula?

696
00:39:41,950 --> 00:39:42,450
OK.

697
00:39:42,450 --> 00:39:44,240
What's the cofactor
formula then?

698
00:39:44,240 --> 00:39:49,110
Let me come back to
this board and say,

699
00:39:49,110 --> 00:39:50,560
what's the cofactor formula?

700
00:39:58,150 --> 00:40:02,840
Determinant of A is --

701
00:40:02,840 --> 00:40:04,670
let's go along the first row.

702
00:40:04,670 --> 00:40:11,480
It's a one one
times its cofactor,

703
00:40:11,480 --> 00:40:15,730
and then the second guy is a
one two times its cofactor,

704
00:40:15,730 --> 00:40:19,070
and you just keep going
to the end of the row,

705
00:40:19,070 --> 00:40:22,750
a1n times its cofactor.

706
00:40:22,750 --> 00:40:24,840
So that's cofactor for --

707
00:40:24,840 --> 00:40:30,780
along row one.

708
00:40:30,780 --> 00:40:39,030
And if I went along row I, I
would -- those ones would be

709
00:40:39,030 --> 00:40:39,530
Is.

710
00:40:39,530 --> 00:40:43,520
That's worth putting a box over.

711
00:40:43,520 --> 00:40:47,470
That's the cofactor formula.

712
00:40:47,470 --> 00:40:51,180
Do you see that --

713
00:40:51,180 --> 00:40:53,740
actually, this would
give me another way

714
00:40:53,740 --> 00:41:00,150
I could have started the
whole topic of determinants.

715
00:41:00,150 --> 00:41:02,000
And some, some people
might do it this --

716
00:41:02,000 --> 00:41:04,440
choose to do it this way.

717
00:41:04,440 --> 00:41:06,540
Because the cofactor
formula would

718
00:41:06,540 --> 00:41:09,740
allow me to build up an
n by n determinant out

719
00:41:09,740 --> 00:41:14,790
of n-1 sized determinants, build
those out of n-2, and so on.

720
00:41:14,790 --> 00:41:17,740
I could boil all the
way down to one by ones.

721
00:41:17,740 --> 00:41:21,470
So what's the cofactor formula
for two by two matrices?

722
00:41:21,470 --> 00:41:23,300
Yeah, tell me that.

723
00:41:23,300 --> 00:41:24,580
What's the cofactor for us?

724
00:41:24,580 --> 00:41:28,620
Here is the, here is the world's
smallest example, practically,

725
00:41:28,620 --> 00:41:32,580
of a cofactor formula.

726
00:41:32,580 --> 00:41:33,170
OK.

727
00:41:33,170 --> 00:41:35,570
Let's go along row one.

728
00:41:35,570 --> 00:41:39,500
I take this first guy
times its cofactor.

729
00:41:39,500 --> 00:41:44,350
What's the cofactor
of the one one entry?

730
00:41:44,350 --> 00:41:48,460
d, because you strike out
the one one row and column

731
00:41:48,460 --> 00:41:50,460
and you're left with d.

732
00:41:50,460 --> 00:41:54,330
Then I take this guy,
b, times its cofactor.

733
00:41:54,330 --> 00:41:57,740
What's the cofactor of b?

734
00:41:57,740 --> 00:42:00,070
Is it c or it's --

735
00:42:00,070 --> 00:42:03,330
minus c, because I
strike out this guy,

736
00:42:03,330 --> 00:42:08,360
I take that determinant, and
then I follow the i+j rule

737
00:42:08,360 --> 00:42:11,730
and I get a minus, I get an odd.

738
00:42:11,730 --> 00:42:13,450
So it's b times minus c.

739
00:42:17,230 --> 00:42:18,120
OK, it worked.

740
00:42:18,120 --> 00:42:20,430
Of course it, it worked.

741
00:42:20,430 --> 00:42:23,100
And the three by three works.

742
00:42:23,100 --> 00:42:28,200
So that's the cofactor formula,
and that is, that's an --

743
00:42:28,200 --> 00:42:33,230
that's a good formula to know,
and now I'm feeling like, wow,

744
00:42:33,230 --> 00:42:38,560
I'm giving you a lot of
algebra to swallow here.

745
00:42:38,560 --> 00:42:41,955
Last lecture gave
you ten properties.

746
00:42:44,900 --> 00:42:46,750
Now I'm giving you --

747
00:42:46,750 --> 00:42:50,390
and by the way, those ten
properties led us to a formula

748
00:42:50,390 --> 00:42:52,480
for the determinant
which was very important,

749
00:42:52,480 --> 00:42:55,850
and I haven't
repeated it till now.

750
00:42:55,850 --> 00:42:56,820
What was that?

751
00:42:56,820 --> 00:43:01,140
The, the determinant is
the product of the pivots.

752
00:43:01,140 --> 00:43:03,980
So the pivot formula
is, is very important.

753
00:43:03,980 --> 00:43:07,440
The pivots have all this
complicated mess already

754
00:43:07,440 --> 00:43:08,810
built in.

755
00:43:08,810 --> 00:43:11,460
As you did elimination
to get the pivots,

756
00:43:11,460 --> 00:43:17,670
you built in all this horrible
stuff, quite efficiently.

757
00:43:17,670 --> 00:43:20,680
Then the big formula with
the n factorial terms,

758
00:43:20,680 --> 00:43:24,180
that's got all the
horrible stuff spread out.

759
00:43:24,180 --> 00:43:28,870
And the cofactor formula
is like in between.

760
00:43:28,870 --> 00:43:35,460
It's got easy stuff times
horrible stuff, basically.

761
00:43:35,460 --> 00:43:39,840
But it's, it shows you,
how to get determinants

762
00:43:39,840 --> 00:43:42,780
from smaller determinants, and
that's the application that I

763
00:43:42,780 --> 00:43:45,290
now want to make.

764
00:43:45,290 --> 00:43:51,030
So may I do one more example?

765
00:43:51,030 --> 00:43:54,670
So I remember the general idea.

766
00:43:54,670 --> 00:43:59,360
But I'm going to use this
cofactor formula for a matrix

767
00:43:59,360 --> 00:44:01,230
--

768
00:44:01,230 --> 00:44:03,640
so here is going
to be my example.

769
00:44:03,640 --> 00:44:07,890
It's -- I promised in
the, in the lecture,

770
00:44:07,890 --> 00:44:11,090
outline at the very
beginning to do an example.

771
00:44:11,090 --> 00:44:12,650
And let me do --

772
00:44:12,650 --> 00:44:18,280
I'm going to pick
tri-diagonal matrix of ones.

773
00:44:21,700 --> 00:44:26,340
I could, I'm drawing
here the four by four.

774
00:44:26,340 --> 00:44:27,880
So this will be the matrix.

775
00:44:27,880 --> 00:44:29,260
I could call that A4.

776
00:44:34,830 --> 00:44:40,590
But my real idea
is to do n by n.

777
00:44:40,590 --> 00:44:43,100
To do them all.

778
00:44:43,100 --> 00:44:44,930
So A -- I could --

779
00:44:44,930 --> 00:44:49,010
everybody understands
what A1 and A2 are.

780
00:44:49,010 --> 00:44:49,510
Yeah.

781
00:44:49,510 --> 00:44:54,900
Maybe we should just do A1
and A2 and A3 just for --

782
00:44:54,900 --> 00:44:55,400
so this is

783
00:44:55,400 --> 00:44:57,591
What's the determinant of A1?

784
00:44:57,591 --> 00:44:58,090
A4.

785
00:44:58,090 --> 00:45:01,120
What's the determinant of A1?

786
00:45:01,120 --> 00:45:04,470
So, so what's the matrix
A1 in this formula?

787
00:45:04,470 --> 00:45:06,310
It's just got that.

788
00:45:06,310 --> 00:45:08,850
So the determinant is one.

789
00:45:08,850 --> 00:45:10,950
What's the determinant of A2?

790
00:45:10,950 --> 00:45:14,831
So it's just got this two by
two, and its determinant is --

791
00:45:17,780 --> 00:45:19,690
zero.

792
00:45:19,690 --> 00:45:22,350
And then the three by three.

793
00:45:22,350 --> 00:45:23,800
Can we see its determinant?

794
00:45:23,800 --> 00:45:27,740
Can you take the determinant
of that three by three?

795
00:45:27,740 --> 00:45:32,910
Well, that's not quite so
obvious, at least not to me.

796
00:45:32,910 --> 00:45:35,060
Being three by three,
I don't know --

797
00:45:35,060 --> 00:45:36,710
so here's a, here's
a good example.

798
00:45:36,710 --> 00:45:39,590
How would you do that
three by three determinant?

799
00:45:39,590 --> 00:45:43,160
We've got, like, n
factorial different ways.

800
00:45:43,160 --> 00:45:44,080
Well, three factorial.

801
00:45:44,080 --> 00:45:45,260
So we've got six ways.

802
00:45:45,260 --> 00:45:46,220
OK.

803
00:45:46,220 --> 00:45:49,440
I mean, one way to do it --

804
00:45:49,440 --> 00:45:51,040
actually the way
I would probably

805
00:45:51,040 --> 00:45:52,990
do it, being three
by three, I would use

806
00:45:52,990 --> 00:45:55,250
the complete the big formula.

807
00:45:55,250 --> 00:45:57,120
I would say, I've
got a one from that,

808
00:45:57,120 --> 00:46:00,710
I've got a zero from that, I've
got a zero from that, a zero

809
00:46:00,710 --> 00:46:03,140
from that, and this
direction is a minus one,

810
00:46:03,140 --> 00:46:04,410
that direction's a minus one.

811
00:46:04,410 --> 00:46:06,200
I believe the
answer is minus one.

812
00:46:06,200 --> 00:46:14,220
Would you do it another way?

813
00:46:14,220 --> 00:46:16,400
Here's another way
to do it, look.

814
00:46:16,400 --> 00:46:18,065
Subtract row three from --

815
00:46:18,065 --> 00:46:19,440
I'm just looking
at this three by

816
00:46:19,440 --> 00:46:19,939
three.

817
00:46:19,939 --> 00:46:22,400
Everybody's looking
at the three by three.

818
00:46:22,400 --> 00:46:25,630
Subtract row three from row two.

819
00:46:25,630 --> 00:46:26,930
Determinant doesn't change.

820
00:46:26,930 --> 00:46:29,710
So those become zeros.

821
00:46:29,710 --> 00:46:31,710
OK, now use the
cofactor formula.

822
00:46:31,710 --> 00:46:33,160
How's that?

823
00:46:33,160 --> 00:46:36,310
How can, how -- if this was now
zeros and I'm looking at this

824
00:46:36,310 --> 00:46:39,530
three by three, use
the cofactor formula.

825
00:46:39,530 --> 00:46:42,875
Why not use the cofactor
formula along that row?

826
00:46:45,890 --> 00:46:49,360
Because then I take that
number times its cofactor,

827
00:46:49,360 --> 00:46:52,800
so I take this number -- let
me put a box around it --

828
00:46:52,800 --> 00:46:56,030
times its cofactor, which is the
determinant of that and that,

829
00:46:56,030 --> 00:46:56,830
which is what?

830
00:47:02,310 --> 00:47:07,230
That two by two matrix
has determinant one.

831
00:47:07,230 --> 00:47:08,625
So what's the cofactor?

832
00:47:11,410 --> 00:47:15,150
What's the cofactor
of this guy here?

833
00:47:15,150 --> 00:47:17,220
Looking just at
this three by three.

834
00:47:17,220 --> 00:47:21,470
The cofactor of that
one is this determinant,

835
00:47:21,470 --> 00:47:26,490
which is one times negative.

836
00:47:26,490 --> 00:47:30,370
So that's why the answer
came out minus one.

837
00:47:30,370 --> 00:47:31,280
OK.

838
00:47:31,280 --> 00:47:32,910
So I did the three by three.

839
00:47:32,910 --> 00:47:35,310
I don't know if we want
to try the four by four.

840
00:47:35,310 --> 00:47:38,090
Yeah, let's -- I guess that
was the point of my example,

841
00:47:38,090 --> 00:47:41,250
of course, so I have to try it.

842
00:47:41,250 --> 00:47:44,120
Sorry, I'm in a good
mood today, so you have

843
00:47:44,120 --> 00:47:45,840
to stand for all the bad jokes.

844
00:47:45,840 --> 00:47:46,340
OK.

845
00:47:46,340 --> 00:47:47,200
OK.

846
00:47:47,200 --> 00:47:50,830
So what was the matrix?

847
00:47:50,830 --> 00:47:51,330
Ah.

848
00:47:55,580 --> 00:47:57,660
OK, now I'm ready
for four by four.

849
00:47:57,660 --> 00:48:00,720
Who wants to -- who wants
to guess the, the --

850
00:48:00,720 --> 00:48:04,270
I don't know, frankly,
this four by four,

851
00:48:04,270 --> 00:48:07,280
what's, what's the determinant.

852
00:48:07,280 --> 00:48:08,575
I plan to use cofactors.

853
00:48:12,640 --> 00:48:14,310
OK, let's use cofactors.

854
00:48:14,310 --> 00:48:17,960
The determinant of A4 is --

855
00:48:17,960 --> 00:48:20,420
OK, let's use cofactors
on the first row.

856
00:48:20,420 --> 00:48:21,630
Those are easy.

857
00:48:21,630 --> 00:48:25,720
So I multiply this number,
which is a convenient one,

858
00:48:25,720 --> 00:48:28,240
times this determinant.

859
00:48:28,240 --> 00:48:31,790
So it's, it's one times
the, this three by three

860
00:48:31,790 --> 00:48:32,380
determinant.

861
00:48:32,380 --> 00:48:36,600
Now what is -- do you
recognize that matrix?

862
00:48:36,600 --> 00:48:37,900
It's A3.

863
00:48:37,900 --> 00:48:41,940
So it's one times the
determinant of A3.

864
00:48:41,940 --> 00:48:46,830
Coming along this row is a
one times this determinant,

865
00:48:46,830 --> 00:48:49,430
and it goes with a plus, right?

866
00:48:49,430 --> 00:48:50,665
And then we have this one.

867
00:48:53,170 --> 00:48:55,290
And what is its cofactor?

868
00:48:55,290 --> 00:48:58,780
Now I'm looking at, now
I'm looking at this three

869
00:48:58,780 --> 00:49:00,990
by three, this three
by three, so I'm

870
00:49:00,990 --> 00:49:04,270
looking at the three by three
that I haven't X-ed out.

871
00:49:04,270 --> 00:49:08,630
What is that -- oh, now
it, we did a plus or a --

872
00:49:08,630 --> 00:49:10,880
is it plus this determinant,
this three by three

873
00:49:10,880 --> 00:49:13,730
determinant, or minus it?

874
00:49:13,730 --> 00:49:16,150
It's minus it, right,
because this is --

875
00:49:16,150 --> 00:49:20,180
I'm starting in a one two
position, and that's a minus.

876
00:49:20,180 --> 00:49:22,390
So I want minus
this determinant.

877
00:49:22,390 --> 00:49:25,180
But these guys are X-ed out.

878
00:49:25,180 --> 00:49:25,680
OK.

879
00:49:25,680 --> 00:49:27,260
So I've got a three by three.

880
00:49:27,260 --> 00:49:31,170
Well, let's use cofactors again.

881
00:49:31,170 --> 00:49:33,740
Use cofactors of the
column, because actually we

882
00:49:33,740 --> 00:49:35,620
could use cofactors
of columns just

883
00:49:35,620 --> 00:49:39,930
as well as rows, because,
because the transpose rule.

884
00:49:39,930 --> 00:49:43,750
So I'll take this one, which
is now sitting in the plus

885
00:49:43,750 --> 00:49:46,440
position, times its
determinant -- oh!

886
00:49:46,440 --> 00:49:47,773
Oh, hell.

887
00:49:50,870 --> 00:49:53,990
What -- oh yeah, I
shouldn't have said hell,

888
00:49:53,990 --> 00:49:55,260
because it's all right.

889
00:49:55,260 --> 00:49:55,760
OK.

890
00:49:55,760 --> 00:49:58,230
One times the determinant.

891
00:49:58,230 --> 00:50:00,410
What is that matrix
now that I'm taking

892
00:50:00,410 --> 00:50:01,920
the, this smaller one of?

893
00:50:01,920 --> 00:50:03,420
Oh, but there's a minus, right?

894
00:50:03,420 --> 00:50:05,630
The minus came
from, from the fact

895
00:50:05,630 --> 00:50:10,570
that this was in the one
two position and that's odd.

896
00:50:10,570 --> 00:50:14,750
So this is a minus one
times -- and what's --

897
00:50:14,750 --> 00:50:17,130
and then this one
is the upper left,

898
00:50:17,130 --> 00:50:21,880
that's the one one position
in its matrix, so plus.

899
00:50:21,880 --> 00:50:23,910
And what's this matrix here?

900
00:50:23,910 --> 00:50:26,460
Do you recognize that?

901
00:50:26,460 --> 00:50:28,840
That matrix is --

902
00:50:28,840 --> 00:50:30,275
yes, please say it --

903
00:50:30,275 --> 00:50:30,775
A2.

904
00:50:36,320 --> 00:50:39,110
And we -- that's our
formula for any case.

905
00:50:39,110 --> 00:50:46,790
A of any size n is equal to the
determinant of A n minus one,

906
00:50:46,790 --> 00:50:49,310
that's what came from taking
the one in the upper corner,

907
00:50:49,310 --> 00:50:55,520
the first cofactor, minus the
determinant of A n minus two.

908
00:50:55,520 --> 00:51:02,020
What we discovered
there is true for all n.

909
00:51:02,020 --> 00:51:04,680
I didn't even mention
it, but I stopped taking

910
00:51:04,680 --> 00:51:06,610
cofactors when I got this one.

911
00:51:06,610 --> 00:51:08,920
Why did I stop?

912
00:51:08,920 --> 00:51:12,660
Why didn't I take the
cofactor of this guy?

913
00:51:12,660 --> 00:51:16,490
Because he's going to get
multiplied by zero, and no,

914
00:51:16,490 --> 00:51:18,130
no use wasting time.

915
00:51:18,130 --> 00:51:20,260
Or this one too.

916
00:51:20,260 --> 00:51:22,570
The cofactor, her
cofactor will be

917
00:51:22,570 --> 00:51:24,400
whatever that
determinant is, but it'll

918
00:51:24,400 --> 00:51:26,980
be multiplied by zero,
so I won't bother.

919
00:51:26,980 --> 00:51:30,010
OK, there is the formula.

920
00:51:30,010 --> 00:51:32,160
And that now tells us --

921
00:51:32,160 --> 00:51:34,620
I could figure out
what A4 is now.

922
00:51:34,620 --> 00:51:37,570
Oh yeah, finally I can get A4.

923
00:51:37,570 --> 00:51:43,770
Because it's A3, which is minus
one, minus A2, which is zero,

924
00:51:43,770 --> 00:51:44,880
so it's minus one.

925
00:51:47,750 --> 00:51:50,090
You see how we're
getting kind of numbers

926
00:51:50,090 --> 00:51:51,710
that you might not have guessed.

927
00:51:51,710 --> 00:51:55,800
So our sequence right now is
one zero minus one minus one.

928
00:51:55,800 --> 00:52:00,760
What's the next one
in the sequence, A5?

929
00:52:00,760 --> 00:52:06,300
A5 is this minus
this, so it is zero.

930
00:52:06,300 --> 00:52:09,260
What's A6?

931
00:52:09,260 --> 00:52:15,140
A6 is this minus
this, which is one.

932
00:52:15,140 --> 00:52:18,150
What's A7?

933
00:52:18,150 --> 00:52:21,640
I'm, I'm going to be stopped
by either the time runs out

934
00:52:21,640 --> 00:52:23,230
or the board runs out.

935
00:52:23,230 --> 00:52:27,740
OK, A7 is this minus
this, which is one.

936
00:52:27,740 --> 00:52:30,790
I'll stop here, because time
is out, but let me tell you

937
00:52:30,790 --> 00:52:32,180
what we've got.

938
00:52:32,180 --> 00:52:35,850
What -- these determinants
have this series,

939
00:52:35,850 --> 00:52:39,730
one zero minus one
minus one zero one,

940
00:52:39,730 --> 00:52:42,760
and then it starts repeating.

941
00:52:42,760 --> 00:52:44,600
It's pretty fantastic.

942
00:52:44,600 --> 00:52:48,480
These determinants
have period six.

943
00:52:48,480 --> 00:52:51,880
So the determinant
of A sixty-one

944
00:52:51,880 --> 00:52:55,771
would be the determinant
of A1, which would be one.

945
00:52:55,771 --> 00:52:56,270
OK.

946
00:52:56,270 --> 00:52:58,190
I hope you liked that example.

947
00:52:58,190 --> 00:53:04,640
A non-trivial example of a
tri-diagonal determinant.

948
00:53:04,640 --> 00:53:05,520
Thanks.

949
00:53:05,520 --> 00:53:08,260
See you on Wednesday.