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I will now show you my preferred
way of finding an

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00:00:04,100 --> 00:00:05,770
inverse of a 3 by 3 matrix.

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00:00:05,770 --> 00:00:07,220
And I actually think it's
a lot more fun.

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And you're less likely to
make careless mistakes.

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But if I remember correctly from
Algebra 2, they didn't

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00:00:11,020 --> 00:00:12,760
teach it this way
in Algebra 2.

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And that's why I taught the
other way initially.

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But let's go through this.

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And in a future video, I will
teach you why it works.

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Because that's always
important.

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But in linear algebra, this is
one of the few subjects where

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I think it's very important
learn how to do the operations

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first. And then later,
we'll learn the why.

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Because the how is
very mechanical.

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And it really just involves
some basic arithmetic

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for the most part.

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But the why tends to
be quite deep.

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So I'll leave that
to later videos.

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And you can often think about
the depth of things when you

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have confidence that you at
least understand the hows.

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So anyway, let's go back
to our original matrix.

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And what was that original
matrix that I

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did in the last video?

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It was 1, 0, 1, 0,
2, 1, 1, 1, 1.

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And we wanted to find the
inverse of this matrix.

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So this is what we're
going to do.

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It's called Gauss-Jordan
elimination, to find the

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inverse of the matrix.

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And the way you do it-- and it
might seem a little bit like

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magic, it might seem a little
bit like voodoo, but I think

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you'll see in future videos that
it makes a lot of sense.

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What we do is we augment
this matrix.

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What does augment mean?

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It means we just add
something to it.

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So I draw a dividing line.

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Some people don't.

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So if I put a dividing
line here.

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And what do I put on the other
side of the dividing line?

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I put the identity matrix
of the same size.

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This is 3 by 3, so I put a
3 by 3 identity matrix.

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So that's 1, 0, 0,
0, 1, 0, 0, 0, 1.

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All right, so what are
we going to do?

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What I'm going to do is perform
a series of elementary

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row operations.

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And I'm about to tell you what
are valid elementary row

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operations on this matrix.

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But whatever I do to any of
these rows here, I have to do

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00:02:07,440 --> 00:02:09,360
to the corresponding
rows here.

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And my goal is essentially to
perform a bunch of operations

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on the left hand side.

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And of course, the same
operations will be applied to

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the right hand side, so that I
eventually end up with the

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identity matrix on the
left hand side.

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And then when I have the
identity matrix on the left

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hand side, what I have left on
the right hand side will be

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the inverse of this
original matrix.

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And when this becomes an
identity matrix, that's

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actually called reduced
row echelon form.

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And I'll talk more about that.

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There's a lot of names and
labels in linear algebra.

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But they're really just fairly
simple concepts.

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But anyway, let's get started
and this should become a

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little clear.

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At least the process
will become clear.

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Maybe not why it works.

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So first of all, I said I'm
going to perform a bunch of

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operations here.

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What are legitimate
operations?

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They're called elementary
row operations.

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So there's a couple
things I can do.

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I can replace any row
with that row

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multiplied by some number.

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So I could do that.

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I can swap any two rows.

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And of course if I swap say the
first and second row, I'd

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have to do it here as well.

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And I can add or subtract one
row from another row.

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So when I do that-- so for
example, I could take this row

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and replace it with this
row added to this row.

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And you'll see what I
mean in the second.

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And you know, if you combine it,
you could you could say,

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well I'm going to multiple this
row times negative 1, and

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add it to this row, and replace
this row with that.

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So if you start to feel like
this is something like what

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you learned when you learned
solving systems of linear

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equations, that's
no coincidence.

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Because matrices are actually
a very good way to represent

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that, and I will show
you that soon.

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But anyway, let's do some
elementary row operations to

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get this left hand side into
reduced row echelon form.

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Which is really just a fancy way
of saying, let's turn it

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into the identity matrix.

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So let's see what
we want to do.

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We want to have 1's
all across here.

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We want these to be 0's.

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Let's see how we can do
this efficiently.

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Let me draw the matrix again.

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So let's get a 0 here.

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That would be convenient.

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So I'm going to keep the
top two rows the same.

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1, 0, 1.

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I have my dividing line.

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1, 0, 0.

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I didn't do anything there.

106
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I'm not doing anything
to the second row.

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0, 2, 1.

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0, 1, 0.

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And what I'm going to do, I'm
going to replace this row--

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And just so you know my
motivation, my goal

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is to get a 0 here.

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So I'm a little bit closer
to having the

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identity matrix here.

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So how do I get a 0 here?

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What I could do is I can replace
this row with this row

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minus this row.

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So I can replace the third
row with the third row

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minus the first row.

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So what's the third row
minus the first row?

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1 minus 1 is 0.

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1 minus 0 is 1.

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1 minus 1 is 0.

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Well I did it on the left hand
side, so I have to do it on

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the right hand side.

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I have to replace this
with this minus this.

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So 0 minus 1 is minus 1.

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0 minus 0 is 0.

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And 1 minus 0 is 1.

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Fair enough.

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Now what can I do?

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Well this row right here, this
third row, it has 0 and 0-- it

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looks a lot like what I want
for my second row in the

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identity matrix.

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So why don't I just swap
these two rows?

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Why don't I just swap the
first and second rows?

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So let's do that.

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I'm going to swap the first
and second rows.

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So the first row
stays the same.

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1, 0, 1.

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And then the other side stays
the same as well.

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And I'm swapping the second
and third rows.

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So now my second row
is now 0, 1, 0.

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And I have to swap it on
the right hand side.

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So it's minus 1, 0, 1.

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00:06:09,520 --> 00:06:12,540
I'm just swapping these two.

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So then my third row now
becomes what the

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second row was here.

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00:06:15,450 --> 00:06:17,920
0, 2, 1.

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And 0, 1, 0.

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Fair enough.

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00:06:23,160 --> 00:06:24,770
Now what do I want to do?

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00:06:24,770 --> 00:06:26,910
Well it would be nice if
I had a 0 right here.

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That would get me that much
closer to the identity matrix.

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00:06:30,070 --> 00:06:32,260
So how could I get as 0 here?

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00:06:32,260 --> 00:06:37,390
Well what if I subtracted 2
times row two from row one?

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00:06:37,390 --> 00:06:40,360
Because this would be,
1 times 2 is 2.

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And if I subtracted that from
this, I'll get a 0 here.

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00:06:44,920 --> 00:06:47,140
So let's do that.

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So the first row has
been very lucky.

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It hasn't had to do anything.

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It's just sitting there.

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00:06:52,580 --> 00:06:58,670
1, 0, 1, 1, 0, 0.

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00:06:58,670 --> 00:07:02,120
And the second row's not
changing for now.

165
00:07:02,120 --> 00:07:05,430
Minus 1, 0, 1.

166
00:07:05,430 --> 00:07:07,110
Now what did I say I
was going to do?

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I'm going to subtract 2 times
row two from row three.

168
00:07:13,240 --> 00:07:18,960
So this is 0 minus
2 times 0 is 0.

169
00:07:18,960 --> 00:07:23,990
2 minus 2 times 1,
well that's 0.

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00:07:23,990 --> 00:07:29,150
1 minus 2 times 0 is 1.

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00:07:29,150 --> 00:07:38,210
0 minus 2 times negative 1 is--
so let's remember 0 minus

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00:07:38,210 --> 00:07:39,880
2 times negative 1.

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00:07:39,880 --> 00:07:44,520
So that's 0 minus negative
2, so that's positive 2.

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00:07:44,520 --> 00:07:47,970
1 minus 2 times 0.

175
00:07:47,970 --> 00:07:49,810
Well that's just still 1.

176
00:07:49,810 --> 00:07:53,240
0 minus 2 times 1.

177
00:07:53,240 --> 00:07:54,490
So that's minus 2.

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00:07:54,490 --> 00:07:57,190


179
00:07:57,190 --> 00:07:58,130
Have I done that right?

180
00:07:58,130 --> 00:07:58,810
I just want to make sure.

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00:07:58,810 --> 00:08:04,800
0 minus 2 times-- right, 2
times minus 1 is minus 2.

182
00:08:04,800 --> 00:08:06,910
And I'm subtracting
it, so it's plus.

183
00:08:06,910 --> 00:08:08,150
OK, so I'm close.

184
00:08:08,150 --> 00:08:11,140
This almost looks like the
identity matrix or reduced row

185
00:08:11,140 --> 00:08:11,680
echelon form.

186
00:08:11,680 --> 00:08:12,950
Except for this 1 right here.

187
00:08:12,950 --> 00:08:16,740
So I'm finally going to have
to touch the top row.

188
00:08:16,740 --> 00:08:18,450
And what can I do?

189
00:08:18,450 --> 00:08:23,170
well how about I replace the top
row with the top row minus

190
00:08:23,170 --> 00:08:24,060
the bottom row?

191
00:08:24,060 --> 00:08:25,480
Because if I subtract
this from that,

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00:08:25,480 --> 00:08:26,550
this'll get a 0 there.

193
00:08:26,550 --> 00:08:27,790
So let's do that.

194
00:08:27,790 --> 00:08:29,720
So I'm replacing the top
row with the top row

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00:08:29,720 --> 00:08:31,790
minus the third row.

196
00:08:31,790 --> 00:08:35,570
So 1 minus 0 is 1.

197
00:08:35,570 --> 00:08:38,659
0 minus 0 is 0.

198
00:08:38,659 --> 00:08:41,000
1 minus 1 is 0.

199
00:08:41,000 --> 00:08:43,559
That was our whole goal.

200
00:08:43,559 --> 00:08:48,000
And then 1 minus 2
is negative 1.

201
00:08:48,000 --> 00:08:53,490
0 minus 1 is negative 1.

202
00:08:53,490 --> 00:08:58,950
0 minus negative 2., well
that's positive 2.

203
00:08:58,950 --> 00:09:02,460
And then the other rows
stay the same.

204
00:09:02,460 --> 00:09:07,590
0, 1, 0, minus 1, 0, 1.

205
00:09:07,590 --> 00:09:15,550
And then 0, 0, 1, 2,
1, negative 2.

206
00:09:15,550 --> 00:09:16,640
And there you have it.

207
00:09:16,640 --> 00:09:18,650
We have performed a series
of operations on

208
00:09:18,650 --> 00:09:19,720
the left hand side.

209
00:09:19,720 --> 00:09:21,380
And we've performed the
same operations on

210
00:09:21,380 --> 00:09:22,960
the right hand side.

211
00:09:22,960 --> 00:09:25,670
This became the identity
matrix, or

212
00:09:25,670 --> 00:09:27,410
reduced row echelon form.

213
00:09:27,410 --> 00:09:30,530
And we did this using
Gauss-Jordan elimination.

214
00:09:30,530 --> 00:09:32,180
And what is this?

215
00:09:32,180 --> 00:09:36,570
Well this is the inverse of
this original matrix.

216
00:09:36,570 --> 00:09:38,960
This times this will equal
the identity matrix.

217
00:09:38,960 --> 00:09:46,750
So if this is a, than
this is a inverse.

218
00:09:46,750 --> 00:09:47,580
And that's all you have to do.

219
00:09:47,580 --> 00:09:49,700
And as you could see, this took
me half the amount of

220
00:09:49,700 --> 00:09:53,260
time, and required a lot less
hairy mathematics than when I

221
00:09:53,260 --> 00:09:56,310
did it using the adjoint and
the cofactors and the

222
00:09:56,310 --> 00:09:58,110
determinant.

223
00:09:58,110 --> 00:09:59,990
And if you think about it, I'll
give you a little hint of

224
00:09:59,990 --> 00:10:01,420
why this worked.

225
00:10:01,420 --> 00:10:06,910
Every one of these operations
I did on the left hand side,

226
00:10:06,910 --> 00:10:10,570
you could kind of view them as
multiplying-- you know, to get

227
00:10:10,570 --> 00:10:12,370
from here to here,
I multiplied.

228
00:10:12,370 --> 00:10:14,500
You can kind of say that
there's a matrix.

229
00:10:14,500 --> 00:10:16,240
That if I multiplied by that
matrix, it would have

230
00:10:16,240 --> 00:10:17,670
performed this operation.

231
00:10:17,670 --> 00:10:20,250
And then I would have had to
multiply by another matrix to

232
00:10:20,250 --> 00:10:21,550
do this operation.

233
00:10:21,550 --> 00:10:24,250
So essentially what we did is
we multiplied by a series of

234
00:10:24,250 --> 00:10:26,440
matrices to get here.

235
00:10:26,440 --> 00:10:28,500
And if you multiplied all
of those, what we call

236
00:10:28,500 --> 00:10:31,410
elimination matrices, together,
you essentially

237
00:10:31,410 --> 00:10:34,070
multiply this times
the inverse.

238
00:10:34,070 --> 00:10:35,590
So what am I saying?

239
00:10:35,590 --> 00:10:43,470
So if we have a, to go from
here to here, we have to

240
00:10:43,470 --> 00:10:47,300
multiply a times the
elimination matrix.

241
00:10:47,300 --> 00:10:49,630
And this might be completely
confusing for you, so ignore

242
00:10:49,630 --> 00:10:51,990
it if it is, but it might
be insightful.

243
00:10:51,990 --> 00:10:55,250
So what did we eliminate
in this?

244
00:10:55,250 --> 00:10:58,470
We eliminated 3, 1.

245
00:10:58,470 --> 00:11:01,120
We multiplied by the
elimination matrix

246
00:11:01,120 --> 00:11:03,670
3, 1, to get here.

247
00:11:03,670 --> 00:11:05,740
And then, to go from
here to here, we've

248
00:11:05,740 --> 00:11:07,220
multiplied by some matrix.

249
00:11:07,220 --> 00:11:07,970
And I'll tell you more.

250
00:11:07,970 --> 00:11:09,160
I'll show you how
we can construct

251
00:11:09,160 --> 00:11:10,940
these elimination matrices.

252
00:11:10,940 --> 00:11:12,830
We multiply by an elimination
matrix.

253
00:11:12,830 --> 00:11:16,150
Well actually, we had
a row swap here.

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I don't know what you
want to call that.

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You could call that
the swap matrix.

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We swapped row two for three.

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And then here, we multiplied
by elimination

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matrix-- what did we do?

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We eliminated this, so
this was row three,

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column two, 3, 2.

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And then finally, to get here,
we had to multiply by

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elimination matrix.

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We had to eliminate
this right here.

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So we eliminated row
one, column three.

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And I want you to know right
now that it's not important

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what these matrices are.

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I'll show you how we can
construct these matrices.

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But I just want you to have kind
of a leap of faith that

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each of these operations could
have been done by multiplying

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by some matrix.

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But what we do know is by
multiplying by all of these

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matrices, we essentially got
the identity matrix.

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Back here.

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So the combination of all of
these matrices, when you

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multiply them by each
other, this must

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be the inverse matrix.

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If I were to multiply each of
these elimination and row swap

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matrices, this must be the
inverse matrix of a.

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00:12:22,420 --> 00:12:23,680
Because if you multiply
all them times

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a, you get the inverse.

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Well what happened?

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00:12:28,630 --> 00:12:31,780
If these matrices are
collectively the inverse

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00:12:31,780 --> 00:12:36,400
matrix, if I do them, if I
multiply the identity matrix

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times them-- the elimination
matrix, this one times that

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00:12:40,620 --> 00:12:41,270
equals that.

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00:12:41,270 --> 00:12:42,970
This one times that
equals that.

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00:12:42,970 --> 00:12:44,510
This one times that
equals that.

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00:12:44,510 --> 00:12:45,360
And so forth.

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00:12:45,360 --> 00:12:48,870
I'm essentially multiplying--
when you combine all of

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00:12:48,870 --> 00:12:53,050
these-- a inverse times
the identity matrix.

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00:12:53,050 --> 00:12:55,520
So if you think about it just
very big picture-- and I don't

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want to confuse you.

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00:12:56,470 --> 00:12:57,910
It's good enough at this
point if you just

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00:12:57,910 --> 00:13:00,370
understood what I did.

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00:13:00,370 --> 00:13:03,500
But what I'm doing from all of
these steps, I'm essentially

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00:13:03,500 --> 00:13:07,800
multiplying both sides of this
augmented matrix, you could

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00:13:07,800 --> 00:13:10,450
call it, by a inverse.

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00:13:10,450 --> 00:13:13,080
So I multiplied this by a
inverse, to get to the

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00:13:13,080 --> 00:13:14,300
identity matrix.

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00:13:14,300 --> 00:13:16,740
But of course, if I multiplied
the inverse matrix times the

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00:13:16,740 --> 00:13:19,130
identity matrix, I'll get
the inverse matrix.

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00:13:19,130 --> 00:13:20,990
But anyway, I don't want
to confuse you.

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00:13:20,990 --> 00:13:22,410
Hopefully that'll give you
a little intuition.

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00:13:22,410 --> 00:13:25,130
I'll do this later with some
more concrete examples.

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00:13:25,130 --> 00:13:27,850
But hopefully you see that this
is a lot less hairy than

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00:13:27,850 --> 00:13:30,115
the way we did it with the
adjoint and the cofactors and

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00:13:30,115 --> 00:13:32,540
the minor matrices and the
determinants, et cetera.

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00:13:32,540 --> 00:13:35,290
Anyway, I'll see you
in the next video.