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We will now embark on what is probably my least favorite

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exercise or computation in mathematics-- and I think

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00:00:08,300 --> 00:00:12,750
you'll see why-- where we will invert a 3 by 3 matrix.

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And in my mind, the only thing less pleasant than inverting a

6
00:00:16,199 --> 00:00:19,989
3 by 3 matrix is inverting a 4 by 4 matrix.

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It very quickly becomes obvious to you that it's

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probably better for a computer to do this.

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But you need to learn how to do it.

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And it's a good exercise for me to do.

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And if I keep doing it my whole life, it'll prevent my

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brain from degrading.

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But as you'll see, this is almost an exercise in not

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making careless mistakes.

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So let's start with a 3 by 3 matrix and

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try to take the inverse.

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So let's say I have matrix a.

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I think I'm going to need a lot of space here, so I'll try

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to do this small, without being confusing.

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Matrix a.

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Let's say it's 1, 0-- and I'm specifically choosing this

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matrix because the numbers are reasonably non-hairy-- 0, 2,

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1, 1, 1, 1.

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So the first thing that I do when I take an inverse of a 3

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by 3 matrix, I create what I call-- or not what I call,

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what everyone calls-- a matrix of minors.

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So let me write that down.

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Matrix of minors.

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So what's a minor?

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Let me draw that out.

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So it's going to be another 3 by 3 matrix.

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And what it is, so this element, this top left

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element, is essentially going to be the determinant.

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If I were to take my original matrix, and I were to cross

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out this position's row and column.

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So for example, this 1, 1 position, row 1, column 1.

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I cross out row 1 and column 1.

40
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What numbers do I have left?

41
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I have this 2, 1, 1, 1.

42
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I have this right here.

43
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So it's the determinant of 2, 1, 1, 1.

44
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And actually maybe I'll write that down.

45
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So it's the determinant of 2, 1, 1, 1.

46
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So I'm gonna run out of space I'm sure.

47
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It's going to be 2, 1, 1, 1.

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The determinant.

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The absolute value sign says it's the determinant.

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Remember, all I did is, I said, OK, in position 1,1, let

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me cross out the column and the row 1,1, and take the

52
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determinant of what's left.

53
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Or the minor of this matrix.

54
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And then I will take the determinant-- so when I go to

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this position, I'm in row 1, column 2.

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I'm essentially going to take the determinant.

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If I were to cross out row 1 and column 2, what

58
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do I have left over?

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I have 0, 1, 1, 1.

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It might be confusing, but just remember-- I wish I had

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something I could cover this with.

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Unfortunately my fingers can't show up on this video.

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But if you cross this row and this column out, you're just

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left with this 0, this 1, this 1, and this 1.

66
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And you take the determinant of that minor.

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And we'll keep going.

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I'm probably going to run out of space here, but I will try

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my best. And so when you go to this position-- row 1, column

70
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3-- what do you do?

71
00:03:26,949 --> 00:03:29,609
Well you cross out row 1, column 3.

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And then the determinant, or the minor that you have to do,

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is 0, 2, 1, 1.

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75
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So the determinant of that two by two matrix.

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And then you keep doing that, so forth and so on.

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And I'm going to run out of space.

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But what I'm going to do is I'm just going

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to calculate it.

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I think you understand how to do it.

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Well you don't understand how to do it, but I think when we

82
00:03:53,400 --> 00:03:54,939
calculate it it'll make a little more sense.

83
00:03:54,939 --> 00:03:58,579
Let me actually just calculate it out.

84
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Because if I were to write these 2 by 2 matrices, I would

85
00:04:00,919 --> 00:04:02,109
run out of space.

86
00:04:02,110 --> 00:04:05,260
But anyway, let's go back to this position 1, 1.

87
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Cross out the first row, first column, I want the determinant

88
00:04:08,389 --> 00:04:10,329
of this thing right here.

89
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So what's the determinant of this 2 by 2?

90
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That's not too hard.

91
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It's 2 times 1, minus 1 times 1.

92
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So what's 2 times 1, minus 1 times 1?

93
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Well it's just 1.

94
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Then when we go to row 1, column 2, I want the

95
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determinant of 0, 1, 1, 1.

96
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So it's 0 times 1, minus 1 times 1.

97
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So 0 times 1 is 0, minus 1 times 1 is minus 1.

98
00:04:40,290 --> 00:04:42,290
And that's just this determinant right here.

99
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I'm just kind of reshowing you how I visualize when I cross

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out the rows and columns.

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So it's 0 times 1, minus 1 times 1.

102
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And in this position of course, you cross out this

103
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row, this column, and 0 times 1 minus 1 times 2.

104
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So that's minus 2.

105
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106
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Let's keep going.

107
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All right.

108
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So now, when we're in row 2, column 1, we cross out row 2,

109
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cross out column 1.

110
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And we're left with this 0, this 1, this 1 and this 1.

111
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So it's 0 times 1, which is 0.

112
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Minus 1 times 1.

113
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So we're at minus 1.

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115
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Then when we get row 2, column 2, we cross those two out, and

116
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we take the matrix of the minor that's left.

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So that's 1 times 1, minus 1 times 1.

118
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So that's 0.

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120
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Almost. We're halfway done.

121
00:05:41,620 --> 00:05:45,069
OK, so then we're in row 2, column 3.

122
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So we cross out row 2, column 3.

123
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And what we have left is 1 times 1, minus 1 time 0.

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So that is just 1.

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Last row.

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OK, so we're in row 3, column 1.

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So we cross out row 3, column 1.

128
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You're left with 0 times 1 is 0.

129
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Minus 2 times 1.

130
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So that's minus 2.

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132
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Then we're in row 3, column 2.

133
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So we cross out row 3, column 2.

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And you have 1 times 1, minus 0 times 1.

135
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So that's just 1.

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Last one.

137
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Row 3, column 3.

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So we cross out row 3, we cross out column 3.

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And you're just left with 1 times 2, minus 0 times 0.

140
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So that is 2.

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And if I haven't made any careless mistakes, that is our

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matrix of minors.

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144
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Now we now have to convert this to what we call the

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matrix of cofactors.

146
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And actually this step is fairly straightforward.

147
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So to convert from a matrix of minors to a matrix of

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cofactors, you just have to remember this pattern.

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This pattern applies to any 3 by 3 matrix.

150
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Plus, minus, plus, minus, plus, minus,

151
00:07:08,629 --> 00:07:11,269
plus, minus, plus.

152
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And so you can kind of just imagine this as kind of a

153
00:07:13,050 --> 00:07:14,960
checkerboard of pluses and minuses.

154
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And you apply that to this.

155
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So what do I mean by that?

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Well that means, when you start, it's a checkerboard,

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and you start with a plus at the top left.

158
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And then you just keep alternating plus, minus.

159
00:07:25,800 --> 00:07:28,470
So if you applied this to this, you get

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the matrix of cofactors.

161
00:07:31,019 --> 00:07:32,269
Let me write that down.

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163
00:07:43,040 --> 00:07:45,800
You can imagine this is a bit of a marathon of computation.

164
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OK, so the matrix of cofactors is essentially applying this

165
00:07:51,839 --> 00:07:54,879
pattern to the matrix of minors.

166
00:07:54,879 --> 00:07:55,430
So what do you do?

167
00:07:55,430 --> 00:07:59,860
You say this plus 1 times 1 is 1.

168
00:07:59,860 --> 00:08:01,680
But now we have a minus.

169
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So that's minus times minus 1 is positive 1.

170
00:08:06,240 --> 00:08:09,812
Then you have plus times minus 2 is minus 2.

171
00:08:09,812 --> 00:08:11,500
Then you have a minus here.

172
00:08:11,500 --> 00:08:14,990
Minus times minus 1 is positive 1.

173
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Plus times 0 is still 0.

174
00:08:17,790 --> 00:08:22,819
Minus times 1 is minus 1.

175
00:08:22,819 --> 00:08:27,199
Plus times minus 2 is minus 2.

176
00:08:27,199 --> 00:08:30,689
Minus applied to 1 is minus 1.

177
00:08:30,689 --> 00:08:33,460
And then plus applied to 2 is just 2.

178
00:08:33,460 --> 00:08:36,509
And we have our matrix of cofactors.

179
00:08:36,509 --> 00:08:38,298
And we are more than halfway done with

180
00:08:38,298 --> 00:08:39,000
inverting this matrix.

181
00:08:39,000 --> 00:08:40,720
And I just want to take a note here.

182
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What we're doing is kind of just a magic formula.

183
00:08:44,200 --> 00:08:46,600
It might seem a little bit like voodoo for you.

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But I just want you to keep in mind that in future videos, I

185
00:08:49,740 --> 00:08:51,090
will show you where this comes from.

186
00:08:51,090 --> 00:08:52,629
Although it will be quite hairy to prove

187
00:08:52,629 --> 00:08:53,500
it for a 3 by 3.

188
00:08:53,500 --> 00:08:56,750
But I'll definitely show it to you for a 2 by 2.

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00:08:56,750 --> 00:08:59,029
And actually, I'll show you other algorithms that might

190
00:08:59,029 --> 00:09:00,889
make a little bit more intuitive sense for doing it

191
00:09:00,889 --> 00:09:02,039
for a 3 by 3.

192
00:09:02,039 --> 00:09:04,679
But I just wanted to show you how to do it this way, so that

193
00:09:04,679 --> 00:09:07,529
at least when you see it on your Algebra 2 exam-- because

194
00:09:07,529 --> 00:09:10,429
I think they actually teach this in Algebra 2-- you could

195
00:09:10,429 --> 00:09:13,469
at least, if the teacher asks you, solve for the matrix of

196
00:09:13,470 --> 00:09:15,730
minors or the cofactors or solve for the determinant of

197
00:09:15,730 --> 00:09:17,740
the inverse, you can do it.

198
00:09:17,740 --> 00:09:19,430
And then we'll worry about getting the intuition, which

199
00:09:19,429 --> 00:09:22,009
is not how I normally like to teach things.

200
00:09:22,009 --> 00:09:23,629
But this is an exception.

201
00:09:23,629 --> 00:09:25,909
But anyway, back to the problem.

202
00:09:25,909 --> 00:09:28,100
This is the matrix of cofactors.

203
00:09:28,100 --> 00:09:34,409
Now from this, we take the adjoint of matrix a-- or I

204
00:09:34,409 --> 00:09:36,569
learned from Wikipedia, the correct term is the adjugate

205
00:09:36,570 --> 00:09:38,629
of matrix a.

206
00:09:38,629 --> 00:09:46,659
But this is determined the notation is the adjugate of a.

207
00:09:46,659 --> 00:09:50,029
And all this is is the transpose of

208
00:09:50,029 --> 00:09:51,509
the matrix of cofactors.

209
00:09:51,509 --> 00:09:53,970
And I know I'm throwing out a lot of weird terminology here.

210
00:09:53,970 --> 00:09:58,300
But the transpose, all that means is that you switch the

211
00:09:58,299 --> 00:09:59,549
rows and the columns.

212
00:09:59,549 --> 00:10:02,209


213
00:10:02,210 --> 00:10:08,009
So this one right here is in row 1, column 1.

214
00:10:08,009 --> 00:10:10,100
But you know, so the rows and columns are the same, so that

215
00:10:10,100 --> 00:10:10,950
just stays the same.

216
00:10:10,950 --> 00:10:13,170
So actually anything on the diagonal stays the same.

217
00:10:13,169 --> 00:10:14,259
Because this is row 2, column 2.

218
00:10:14,259 --> 00:10:15,909
This is row 3, column 3.

219
00:10:15,909 --> 00:10:19,480
So the diagonals stay the same.

220
00:10:19,480 --> 00:10:20,879
And then you switch places.

221
00:10:20,879 --> 00:10:23,129
You kind of flip across the diagonal.

222
00:10:23,129 --> 00:10:24,470
And what do I mean by that?

223
00:10:24,470 --> 00:10:29,879
Well this 1 was in row 1, column 2.

224
00:10:29,879 --> 00:10:33,509


225
00:10:33,509 --> 00:10:37,350
So it'll then be moved to row 2, column 1.

226
00:10:37,350 --> 00:10:41,680
So this 1 right here will go here.

227
00:10:41,679 --> 00:10:46,399
So you can kind of say that it flipped over the diagonal.

228
00:10:46,399 --> 00:10:50,909
And similarly, this right here is in row 1, column 3.

229
00:10:50,909 --> 00:10:54,839
It's going to be switched to row 3, column 1.

230
00:10:54,840 --> 00:10:56,090
So it's going to go here.

231
00:10:56,090 --> 00:10:59,960


232
00:10:59,960 --> 00:11:02,450
And you can kind of see that it just flipped over that end.

233
00:11:02,450 --> 00:11:03,890
So this minus 2 isn't this one.

234
00:11:03,889 --> 00:11:06,039
It's this one over here.

235
00:11:06,039 --> 00:11:08,399
And actually, we see that this matrix is symmetric.

236
00:11:08,399 --> 00:11:10,329
When you flip it, you actually get the same thing.

237
00:11:10,330 --> 00:11:11,490
So maybe it was a bad example.

238
00:11:11,490 --> 00:11:14,230
But I want you to understand that the transpose is where--

239
00:11:14,230 --> 00:11:19,830
if something like this number, if it's in a row 1, column 2,

240
00:11:19,830 --> 00:11:22,180
then it moves to row 2, column 1.

241
00:11:22,179 --> 00:11:24,079
So you're switching the rows and columns.

242
00:11:24,080 --> 00:11:25,480
But anyway, we could keep doing that.

243
00:11:25,480 --> 00:11:28,080
But essentially you're just flipping over the diagonal.

244
00:11:28,080 --> 00:11:28,710
So let's see.

245
00:11:28,710 --> 00:11:32,040
So then this number will be flipped to this position, so

246
00:11:32,039 --> 00:11:33,969
it goes there.

247
00:11:33,970 --> 00:11:36,340
This is in row 2, column 1.

248
00:11:36,340 --> 00:11:40,820
So it will go to column 2, row 1, which is that.

249
00:11:40,820 --> 00:11:43,610
And then if we go here, that's going to be flipped down here,

250
00:11:43,610 --> 00:11:44,870
flipped across the diagonal.

251
00:11:44,870 --> 00:11:48,060
So that's minus 1.

252
00:11:48,059 --> 00:11:49,869
This is going to be flipped all the way up there.

253
00:11:49,870 --> 00:11:51,679
So that's minus 2.

254
00:11:51,679 --> 00:11:53,245
And then this will be flipped there.

255
00:11:53,245 --> 00:11:54,730
This is minus 1.

256
00:11:54,730 --> 00:11:56,779
We are almost done.

257
00:11:56,779 --> 00:12:00,529
So this is the adjoint of matrix a.

258
00:12:00,529 --> 00:12:03,629
So to get the inverse of a-- and let me actually erase some

259
00:12:03,629 --> 00:12:05,879
of this, because we're going to run out of space otherwise.

260
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And as you can see, I'll be very impressed if I have not

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introduced a careless mistake yet.

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So let me erase all of this.

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I'm building an appetite just doing this problem.

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It's so taxing on me.

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So the inverse of matrix a is equal to 1 over the

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determinant of a times the adjugate, or

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adjoint, of matrix a.

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So we solved for this part.

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So now let's solve for the determinant.

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So the determinant of a-- and I kept the matrix of cofactors

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here for a reason-- the determinant of a is-- if you

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go across-- you can actually go across any row-- but just

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for simplicity, just remember it this way.

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You go across the top row, and you multiply each term times

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its corresponding cofactor, and you add them.

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So in this case, it'll be 1 times its corresponding

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cofactor, which is 1.

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Plus 0 times its corresponding cofactor, which is 1.

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Plus 1 times its corresponding cofactor plus minus 2.

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So this is 1 plus 0 minus 2.

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It equals minus 1.

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And thank God it was a relatively straightforward

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determinant.

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And if you didn't have this matrix of cofactors, the other

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way you could think about it-- and this is good because it

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gives you an intuition of how we even got to the matrix of

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cofactors-- you could view this as the same thing as 1

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times the determinant of its minor.

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So if you cross out the row and the column, it's this

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determinant.

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So it's 2, 1, 1, 1.

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And remember there was that pattern.

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You have plus, and then you go minus.

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So minus 0 times the determinant of its minor.

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So you cross out that row, that column.

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So 0, 1, 1, 1.

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And then we switch again.

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We go back to plus.

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Plus 1 times the determinant of its minor.

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So you cross out that row, that column.

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You get 0, 2, 1, 1.

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And you could compute this out.

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And this is this cofactor.

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This, with a minor sign, this is just a minor.

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And then when you apply the minus sign, it

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becomes this cofactor.

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And then this is that minor.

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And since it's a plus sign there, that's that cofactor.

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But anyway, I just wanted to explain that, and hopefully it

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hasn't confused you.

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But we're ready now to solve the inverse of a.

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We know that the determinant of a is equal to minus 1.

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We know that the adjugate of a is this number here.

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So we now can solve for the inverse.

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And let's do that.

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Let me erase all of this stuff.

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00:15:16,110 --> 00:15:18,300
Cause actually, after I solve for the inverse, I want to

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prove to you that it is the inverse-- maybe.

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If I have enough time.

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Because I just realized I'm running pretty long.

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That might be a good exercise for you.

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OK.

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So the inverse of a is equal to 1 over the determinant.

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We figured out the determinant is negative 1 times the

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00:15:42,850 --> 00:15:45,639
adjugate of a.

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1, 1, minus 2.

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1, 0, minus 1.

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Minus 2, minus 1, 2.

335
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So this is just minus 1, right?

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So we just apply minus 1 times everything.

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00:16:02,250 --> 00:16:05,269
So we get-- if I haven't made any careless mistakes-- minus

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00:16:05,269 --> 00:16:18,289
1, minus 1, plus 2, minus 1, 0, 1, 2, 1, minus 2.

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00:16:18,289 --> 00:16:23,860
I think that I have-- let's see, I just did a minus times

340
00:16:23,860 --> 00:16:25,610
everything.

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00:16:25,610 --> 00:16:26,730
That looks right.

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And so that is a inverse.

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And it only took us 17 minutes.

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And I will leave you there, because it will probably take

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me another 5 or 10 minutes to [? seed ?]

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00:16:34,539 --> 00:16:35,459
and prove.

347
00:16:35,460 --> 00:16:36,800
But that might be a good exercise for you.

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To multiply this matrix times this matrix, and make sure

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that you get the identity matrix.

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I will see you in the next video.