WEBVTT

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I will now show you my preferred
way of finding an

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inverse of a 3 by 3 matrix.

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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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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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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.

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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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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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0, 2, 1.

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And 0, 1, 0.

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Fair enough.

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Now what do I want to do?

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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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So how could I get as 0 here?

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Well what if I subtracted 2
times row two from row one?

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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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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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1, 0, 1, 1, 0, 0.

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And the second row's not
changing for now.

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Minus 1, 0, 1.

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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.

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So this is 0 minus
2 times 0 is 0.

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2 minus 2 times 1,
well that's 0.

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1 minus 2 times 0 is 1.

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0 minus 2 times negative 1 is--
so let's remember 0 minus

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2 times negative 1.

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So that's 0 minus negative
2, so that's positive 2.

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1 minus 2 times 0.

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Well that's just still 1.

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0 minus 2 times 1.

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So that's minus 2.

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Have I done that right?

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I just want to make sure.

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0 minus 2 times-- right, 2
times minus 1 is minus 2.

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And I'm subtracting
it, so it's plus.

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OK, so I'm close.

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This almost looks like the
identity matrix or reduced row

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echelon form.

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Except for this 1 right here.

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So I'm finally going to have
to touch the top row.

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And what can I do?

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well how about I replace the top
row with the top row minus

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the bottom row?

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Because if I subtract
this from that,

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this'll get a 0 there.

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So let's do that.

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So I'm replacing the top
row with the top row

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minus the third row.

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So 1 minus 0 is 1.

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0 minus 0 is 0.

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1 minus 1 is 0.

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That was our whole goal.

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And then 1 minus 2
is negative 1.

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0 minus 1 is negative 1.

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0 minus negative 2., well
that's positive 2.

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And then the other rows
stay the same.

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0, 1, 0, minus 1, 0, 1.

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And then 0, 0, 1, 2,
1, negative 2.

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And there you have it.

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We have performed a series
of operations on

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the left hand side.

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And we've performed the
same operations on

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the right hand side.

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This became the identity
matrix, or

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reduced row echelon form.

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And we did this using
Gauss-Jordan elimination.

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And what is this?

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Well this is the inverse of
this original matrix.

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This times this will equal
the identity matrix.

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So if this is a, than
this is a inverse.

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And that's all you have to do.

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And as you could see, this took
me half the amount of

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time, and required a lot less
hairy mathematics than when I

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did it using the adjoint and
the cofactors and the

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determinant.

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And if you think about it, I'll
give you a little hint of

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why this worked.

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Every one of these operations
I did on the left hand side,

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you could kind of view them as
multiplying-- you know, to get

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from here to here,
I multiplied.

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You can kind of say that
there's a matrix.

00:10:14.500 --> 00:10:16.240
That if I multiplied by that
matrix, it would have

00:10:16.240 --> 00:10:17.670
performed this operation.

00:10:17.670 --> 00:10:20.250
And then I would have had to
multiply by another matrix to

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do this operation.

00:10:21.550 --> 00:10:24.250
So essentially what we did is
we multiplied by a series of

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matrices to get here.

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And if you multiplied all
of those, what we call

00:10:28.500 --> 00:10:31.410
elimination matrices, together,
you essentially

00:10:31.410 --> 00:10:34.070
multiply this times
the inverse.

00:10:34.070 --> 00:10:35.590
So what am I saying?

00:10:35.590 --> 00:10:43.470
So if we have a, to go from
here to here, we have to

00:10:43.470 --> 00:10:47.300
multiply a times the
elimination matrix.

00:10:47.300 --> 00:10:49.630
And this might be completely
confusing for you, so ignore

00:10:49.630 --> 00:10:51.990
it if it is, but it might
be insightful.

00:10:51.990 --> 00:10:55.250
So what did we eliminate
in this?

00:10:55.250 --> 00:10:58.470
We eliminated 3, 1.

00:10:58.470 --> 00:11:01.120
We multiplied by the
elimination matrix

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3, 1, to get here.

00:11:03.670 --> 00:11:05.740
And then, to go from
here to here, we've

00:11:05.740 --> 00:11:07.220
multiplied by some matrix.

00:11:07.220 --> 00:11:07.970
And I'll tell you more.

00:11:07.970 --> 00:11:09.160
I'll show you how
we can construct

00:11:09.160 --> 00:11:10.940
these elimination matrices.

00:11:10.940 --> 00:11:12.830
We multiply by an elimination
matrix.

00:11:12.830 --> 00:11:16.150
Well actually, we had
a row swap here.

00:11:16.150 --> 00:11:17.070
I don't know what you
want to call that.

00:11:17.070 --> 00:11:21.240
You could call that
the swap matrix.

00:11:21.240 --> 00:11:24.730
We swapped row two for three.

00:11:24.730 --> 00:11:28.830
And then here, we multiplied
by elimination

00:11:28.830 --> 00:11:31.110
matrix-- what did we do?

00:11:31.110 --> 00:11:34.030
We eliminated this, so
this was row three,

00:11:34.030 --> 00:11:36.270
column two, 3, 2.

00:11:36.270 --> 00:11:39.320
And then finally, to get here,
we had to multiply by

00:11:39.320 --> 00:11:40.470
elimination matrix.

00:11:40.470 --> 00:11:41.740
We had to eliminate
this right here.

00:11:41.740 --> 00:11:44.220
So we eliminated row
one, column three.

00:11:44.220 --> 00:11:47.200


00:11:47.200 --> 00:11:49.590
And I want you to know right
now that it's not important

00:11:49.590 --> 00:11:51.420
what these matrices are.

00:11:51.420 --> 00:11:53.210
I'll show you how we can
construct these matrices.

00:11:53.210 --> 00:11:55.530
But I just want you to have kind
of a leap of faith that

00:11:55.530 --> 00:11:58.600
each of these operations could
have been done by multiplying

00:11:58.600 --> 00:12:01.040
by some matrix.

00:12:01.040 --> 00:12:03.510
But what we do know is by
multiplying by all of these

00:12:03.510 --> 00:12:06.760
matrices, we essentially got
the identity matrix.

00:12:06.760 --> 00:12:07.930
Back here.

00:12:07.930 --> 00:12:11.450
So the combination of all of
these matrices, when you

00:12:11.450 --> 00:12:13.600
multiply them by each
other, this must

00:12:13.600 --> 00:12:15.370
be the inverse matrix.

00:12:15.370 --> 00:12:18.420
If I were to multiply each of
these elimination and row swap

00:12:18.420 --> 00:12:22.420
matrices, this must be the
inverse matrix of a.

00:12:22.420 --> 00:12:23.680
Because if you multiply
all them times

00:12:23.680 --> 00:12:26.130
a, you get the inverse.

00:12:26.130 --> 00:12:28.630
Well what happened?

00:12:28.630 --> 00:12:31.780
If these matrices are
collectively the inverse

00:12:31.780 --> 00:12:36.400
matrix, if I do them, if I
multiply the identity matrix

00:12:36.400 --> 00:12:40.620
times them-- the elimination
matrix, this one times that

00:12:40.620 --> 00:12:41.270
equals that.

00:12:41.270 --> 00:12:42.970
This one times that
equals that.

00:12:42.970 --> 00:12:44.510
This one times that
equals that.

00:12:44.510 --> 00:12:45.360
And so forth.

00:12:45.360 --> 00:12:48.870
I'm essentially multiplying--
when you combine all of

00:12:48.870 --> 00:12:53.050
these-- a inverse times
the identity matrix.

00:12:53.050 --> 00:12:55.520
So if you think about it just
very big picture-- and I don't

00:12:55.520 --> 00:12:56.470
want to confuse you.

00:12:56.470 --> 00:12:57.910
It's good enough at this
point if you just

00:12:57.910 --> 00:13:00.370
understood what I did.

00:13:00.370 --> 00:13:03.500
But what I'm doing from all of
these steps, I'm essentially

00:13:03.500 --> 00:13:07.800
multiplying both sides of this
augmented matrix, you could

00:13:07.800 --> 00:13:10.450
call it, by a inverse.

00:13:10.450 --> 00:13:13.080
So I multiplied this by a
inverse, to get to the

00:13:13.080 --> 00:13:14.300
identity matrix.

00:13:14.300 --> 00:13:16.740
But of course, if I multiplied
the inverse matrix times the

00:13:16.740 --> 00:13:19.130
identity matrix, I'll get
the inverse matrix.

00:13:19.130 --> 00:13:20.990
But anyway, I don't want
to confuse you.

00:13:20.990 --> 00:13:22.410
Hopefully that'll give you
a little intuition.

00:13:22.410 --> 00:13:25.130
I'll do this later with some
more concrete examples.

00:13:25.130 --> 00:13:27.850
But hopefully you see that this
is a lot less hairy than

00:13:27.850 --> 00:13:30.115
the way we did it with the
adjoint and the cofactors and

00:13:30.115 --> 00:13:32.540
the minor matrices and the
determinants, et cetera.

00:13:32.540 --> 00:13:35.290
Anyway, I'll see you
in the next video.