WEBVTT
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CHRISTINE BREINER: Welcome
back to recitation.
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In this video, what
I want to work on
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is using what we know
about matrix multiplication
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and finding inverses of matrices
to solve a system of equations.
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So we've set up
the system already
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as if it's already
in matrix form.
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And what I'd like us to do
is, for this particular A--
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this 3-by-3 matrix A-- find a
vector x, so that A*x equals b.
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Where b is equal to
these two things.
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So you're going to
do two problems.
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You're going to do when b
equals 1, 2, negative 3.
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And you're going to do when
b is equal to [0, 0, 0].
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So you want to find vector x
so that A*x equals this value
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here.
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And what I'd like
you to do is I'd
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like you to use the
strategy that you
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saw in the lecture,
which is find A inverse,
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and then take A inverse b.
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So we really want to practice
understanding how to find
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the inverse of a matrix.
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So why don't you work on
this, pause the video.
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When you feel
comfortable, confident,
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that you have the right answer,
then bring the video back up,
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and you can compare
your work with mine.
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OK, welcome back.
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Well, hopefully you were
able to make some headway
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and you feel confident in
your answers for 1 and 2.
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I am going to find the
inverse of the matrix A first,
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and then solve the problem.
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And because there's
a lot of computation,
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I may make a mistake.
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So I'm going to have to
check every once in a while
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that I'm doing OK.
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So hopefully, it's-- too bad
you can't tell me if I've made
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a mistake, but hopefully my
studio audience will help me
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out.
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So the first thing
I need to do is
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I need to find the
determinant of A.
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So I'm going to do
that first, and then
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I'm going to find the cofactor
matrix and go from there.
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So if I want to find
the determinant of A--
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I guess I'll just use
the first row here,
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because it's pretty easy.
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So the determinant
of A is going to be
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3 times the determinant of this
matrix, this 2-by-2 matrix.
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So it's going to be
3 times-- and then I
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get a 2 times negative
1, which is negative 2,
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and then minus 0, so I
get a 3 times negative 2.
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Oops.
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And I was about to write plus,
but I should write minus.
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I take minus 1 times--
because this is my minus,
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I take negative of
this thing times
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the matrix that is these two
components in the first column
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and these two components
in the second column.
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Right?
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We take away the column and the
row that the 1 is contained in
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and we look at what remains,
the 2-by-2 matrix that remains.
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And we find the
determinant of that.
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So we get negative 1 times
negative 1, which gives me a 1.
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And then negative 1
times 0 gives me a 0.
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So I just have the negative 1
from the row 1, column 2 spot,
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and then the determinant
of the matrix that remains
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is 1, OK-- of the minor
matrix that remains.
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And then the last one
I should put a plus,
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but notice that it
is a minus already,
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so I'm going to put just
minus 1 times what remains.
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What's this minor?
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This one is this 2-by-2
matrix I'm looking at, right?
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So I need to take
the determinant
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of this 2-by-2 matrix and
multiply it by that negative 1
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to get the third component
here I have to add in.
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Negative 1 times
negative 1 is 1.
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And then I subtract
negative 1 times 2.
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So this is where I
have to be careful.
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It's 1 minus negative 2.
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So I'm going to get a 3.
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OK.
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1 here minus a negative 2-- so
1 plus 2-- I'm going to get a 3.
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OK.
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And so negative 6 minus
1 minus 3-- looks like I
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get a negative 10.
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That's good, because
I think that's what
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I'm supposed to get.
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OK.
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Now what I want to do is I want
to find the matrix of minors
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for A. And then
I'm going to find--
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so I'm going to find the
matrix of minors first,
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and then I'm going to switch
the signs appropriately
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so I get the cofactors correct.
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OK?
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So some of them I already have.
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But, the whole
matrix of minors, I'm
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going to just go through and do
it again, to be very careful.
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So the first one I delete.
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For the first row
and column spot,
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I delete row 1 and
column 1, and I
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look at the determinant
of that matrix.
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That's 2 times negative
1 is negative 2, minus 0,
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so I get a negative 2 there.
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For the first row, second
column I come back,
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and I'm now again looking-- I'm
deleting this column and row,
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and so I'm looking at the
determinant of this matrix.
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So I get negative
1 times negative 1
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is 1, minus 0, so I get a 1.
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Again, I'm going to change
all the signs later.
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So I'm going to do that
in the second step.
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Now I'm in row 1, column 3.
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So I'm going to delete
row 1, column 3 and look
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at the determinant
of that matrix.
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I get negative 1 times negative
1 is 1, minus the negative 2,
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so there's my 3.
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Those I already knew,
but I didn't want
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to just plop them in from here.
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But notice that is
what you get here.
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Negative 2, 1, and 3.
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That's exactly where
they come from, right?
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We got them by the same method.
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OK, and so now I want to find
the minors for the rest of it.
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So let's look at-- when
I delete row 2, column 1,
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I'm left with 1,
negative 1 here.
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Negative 1, negative 1 here.
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Right?
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So 1 times negative
1 is negative 1.
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And then negative and
negative is positive.
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So it's negative 1
minus negative 1,
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so I get negative 2.
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That one was a lot of signs,
so you might want to check.
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Maybe I should check.
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OK, maybe I should check.
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I'm deleting this
column and this row,
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so I get 1 times negative 1.
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That's a negative 1, right?
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Negative 1 minus--
negative 1 times negative 1
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is 1-- and so there's
negative 1 minus 1.
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That looks good.
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Negative 2.
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Right?
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Negative, negative, negative.
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Yeah.
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OK.
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And then I'm looking
at row 2, column 2.
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So now I'm deleting this
row and this column.
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All right.
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And so I have these
sort of diagonals here.
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That's what I'm
interested in, right?
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So I get 3 times negative 1.
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That's negative 3.
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And then minus 1, because I
have negative 1 times negative 1
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is positive 1.
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So negative 3 minus 1.
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So I should get negative 4.
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Right?
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And then I'm over here.
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So I need to delete this
column and this row.
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So I get 3 times
negative 1 is negative 3.
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Minus the negative
1, that's plus 1.
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So negative 3 plus
1 is negative 2.
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And before I go on, I'm going
to check those first 2 rows.
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Because if I made a mistake now,
it's only going to get worse.
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What did I have?
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Yes.
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OK.
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So far so good.
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Whew.
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All right.
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Next, final row.
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OK, final row is, I'm
going to delete this column
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and row here, and I'm
looking at this matrix.
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1 times 0 is 0.
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2 times negative 1 is negative
1, but I subtract that.
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So it's 0 minus
negative 2, so it's 2.
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And then row 3, column 2.
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So row 3, I delete
row 3 and column 2.
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3 times 0 is 0.
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0 minus-- negative
1 times negative 1
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is 1-- so 0 minus 1,
that's negative 1.
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And then the last
spot, I'm deleting
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this row and this column.
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So 3 times 2 is 6,
minus negative 1.
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I get 7.
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All right, let's check that row.
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2, negative 1, 7.
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OK.
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I have not done the
cofactor matrix yet,
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because now I need to change
the appropriate signs.
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OK, so if this is
the matrix of minors,
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then if I want to change it
to the cofactor matrix, what
00:07:56.010 --> 00:07:57.060
do I have to do?
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I'm going to scratch this
out and write the cofactor
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matrix so that we can just
change the signs appropriately.
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I'm going to do
it all right here.
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And how does it work?
00:08:05.570 --> 00:08:09.090
Well, remember I'm going to go
plus, minus, plus; minus, plus,
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minus; plus, minus, plus.
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I have to do this grid
that starts with plus
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and alternates minus.
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So this sign stays the
same, this sign switches,
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this sign stays the same.
00:08:19.090 --> 00:08:20.630
That's the plus, minus, plus.
00:08:20.630 --> 00:08:23.200
This one is going to
be minus, plus, minus.
00:08:23.200 --> 00:08:24.690
So the minus switches that.
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Plus keeps that the same.
00:08:26.490 --> 00:08:28.430
Minus switches that.
00:08:28.430 --> 00:08:30.010
And then I was at
minus, plus, minus.
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So I'm going to have
plus, minus, plus.
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And so these two stay the
same, and this one switches.
00:08:34.690 --> 00:08:37.220
So a lot of things that were
negative became positive.
00:08:37.220 --> 00:08:39.880
And I had to change-- maybe
I threw in one negative,
00:08:39.880 --> 00:08:41.090
maybe not.
00:08:41.090 --> 00:08:44.080
But, so all the signs I kept,
this one stayed the same,
00:08:44.080 --> 00:08:46.230
this one stayed the same,
this one stayed the same,
00:08:46.230 --> 00:08:47.890
these two stayed
the same, and then
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these four switched,
because it's
00:08:49.990 --> 00:08:53.010
the plus, minus,
plus sort of grid
00:08:53.010 --> 00:08:54.980
that I have to put
on top of this.
00:08:54.980 --> 00:08:56.870
OK, so that's the
cofactor matrix.
00:08:56.870 --> 00:08:58.620
We're getting closer.
00:08:58.620 --> 00:09:04.320
OK, now we need the
transpose of this, right?
00:09:04.320 --> 00:09:08.220
So if I look at the
transpose-- actually,
00:09:08.220 --> 00:09:09.430
know what I'm going to do?
00:09:09.430 --> 00:09:12.290
Because I'm also just going
to have to take the transpose
00:09:12.290 --> 00:09:14.330
and then multiply it by
1 over the determinant,
00:09:14.330 --> 00:09:16.190
I'm going to do
that all at once.
00:09:16.190 --> 00:09:16.980
OK.
00:09:16.980 --> 00:09:18.940
Because we can do that
all at once, and then
00:09:18.940 --> 00:09:21.750
we don't have to worry about it.
00:09:21.750 --> 00:09:25.230
So A inverse I know is
going to be negative 1/10,
00:09:25.230 --> 00:09:27.720
because the determinant
was minus 10.
00:09:27.720 --> 00:09:31.680
So it's 1 over the determinant
times the transpose
00:09:31.680 --> 00:09:32.790
of this matrix.
00:09:32.790 --> 00:09:34.417
So the transpose
of this matrix--
00:09:34.417 --> 00:09:36.250
remember what I'm going
to do is essentially
00:09:36.250 --> 00:09:39.000
you fix the diagonal and
you're going to flip.
00:09:39.000 --> 00:09:40.050
That's really what, in
the square matrix, that's
00:09:40.050 --> 00:09:41.230
how you can think about it.
00:09:41.230 --> 00:09:43.310
But every column is
going to become a row.
00:09:43.310 --> 00:09:45.417
So I'm going to write
this as my first row.
00:09:45.417 --> 00:09:47.500
This first column is going
to become my first row.
00:09:47.500 --> 00:09:52.390
So it's going to be negative
2, 2, 2 as my first row.
00:09:52.390 --> 00:09:53.765
And then the next
column is going
00:09:53.765 --> 00:09:57.670
to be negative 1, negative 4, 1.
00:09:57.670 --> 00:09:59.010
I mean next row.
00:09:59.010 --> 00:10:00.930
I will take a column
and change it to a row.
00:10:00.930 --> 00:10:05.220
The next row is going to be
negative 1, negative 4, 1.
00:10:05.220 --> 00:10:06.360
And then the last one.
00:10:06.360 --> 00:10:08.620
I take this column and
I change it to a row.
00:10:08.620 --> 00:10:09.970
It's going to be 3, 2, 7.
00:10:14.280 --> 00:10:14.780
OK.
00:10:14.780 --> 00:10:16.446
And because again, I
want to make sure--
00:10:16.446 --> 00:10:18.395
this one is really
messy-- I want
00:10:18.395 --> 00:10:22.200
to make sure I have something
similar for that, or exactly
00:10:22.200 --> 00:10:22.800
that.
00:10:22.800 --> 00:10:23.530
OK.
00:10:23.530 --> 00:10:25.850
I think I'm still
doing all right.
00:10:25.850 --> 00:10:27.587
Now, let's get to
solving the problem.
00:10:27.587 --> 00:10:29.920
Because so far, we just were
finding the inverse matrix.
00:10:29.920 --> 00:10:31.503
So I'm going to leave
it in this form,
00:10:31.503 --> 00:10:33.230
instead of dividing
by 10 in every spot,
00:10:33.230 --> 00:10:35.290
because that will be annoying.
00:10:35.290 --> 00:10:36.850
So let's think
about how do I want
00:10:36.850 --> 00:10:39.870
to solve the system that I had.
00:10:39.870 --> 00:10:43.990
I had A*x equals b.
00:10:43.990 --> 00:10:46.920
And actually, I
mean, my strategy
00:10:46.920 --> 00:10:48.770
is to find the inverse matrix.
00:10:48.770 --> 00:10:50.380
I didn't talk to
you about why we
00:10:50.380 --> 00:10:53.454
know the inverse
matrix actually exists.
00:10:53.454 --> 00:10:55.370
But ultimately, you
haven't even seen this yet
00:10:55.370 --> 00:10:56.619
in the lecture videos, really.
00:10:56.619 --> 00:11:00.390
Except that you know that the
determinant of A being non-zero
00:11:00.390 --> 00:11:01.790
gives you an inverse matrix.
00:11:01.790 --> 00:11:03.750
That's all you know, I
think, at this point.
00:11:03.750 --> 00:11:06.130
You have the determinant
of A. It's non-zero,
00:11:06.130 --> 00:11:08.100
so you can find
an inverse matrix.
00:11:08.100 --> 00:11:10.650
Makes sense based on the
formulation you have,
00:11:10.650 --> 00:11:14.000
because if the determinant
is 0, then this quantity 1
00:11:14.000 --> 00:11:16.010
over the determinant
of A, you run
00:11:16.010 --> 00:11:17.590
into quite a bit of trouble.
00:11:17.590 --> 00:11:20.050
So that's just as
a little sidebar,
00:11:20.050 --> 00:11:22.960
we know the inverse
matrix exists for A.
00:11:22.960 --> 00:11:25.470
So what we do-- this
is again the strategy--
00:11:25.470 --> 00:11:29.370
you multiply A inverse A
times x on the left side.
00:11:29.370 --> 00:11:30.056
Ooh.
00:11:30.056 --> 00:11:34.000
Is equal to-- sorry-- that
should be the lowercase b.
00:11:34.000 --> 00:11:35.370
Should be a vector there.
00:11:35.370 --> 00:11:39.800
It is equal to A inverse
b on the right-hand side.
00:11:39.800 --> 00:11:42.100
And you notice,
it's very important,
00:11:42.100 --> 00:11:43.830
in the matrix
multiplication video
00:11:43.830 --> 00:11:47.490
we saw that it's very
important the order in which
00:11:47.490 --> 00:11:48.700
you multiply matrices.
00:11:48.700 --> 00:11:51.900
And since I'm putting A
inverse on the far left
00:11:51.900 --> 00:11:55.940
of this side of the equality, I
have to put it on the far left
00:11:55.940 --> 00:11:58.461
of the right-hand
side of the equality.
00:11:58.461 --> 00:11:58.960
Right?
00:11:58.960 --> 00:12:00.640
And in fact, you
would run into trouble
00:12:00.640 --> 00:12:02.430
if you tried to switch
the order of these.
00:12:02.430 --> 00:12:02.929
OK?
00:12:02.929 --> 00:12:05.250
We wouldn't be able
to multiply them.
00:12:05.250 --> 00:12:06.000
All right?
00:12:06.000 --> 00:12:10.000
So A inverse A, we know is
just the identity matrix.
00:12:10.000 --> 00:12:13.340
So you get the
identity matrix times x
00:12:13.340 --> 00:12:16.430
is equal to A inverse b.
00:12:16.430 --> 00:12:22.640
So you can find x by
finding A inverse times b.
00:12:22.640 --> 00:12:23.470
Right?
00:12:23.470 --> 00:12:25.140
And so now we have A inverse.
00:12:25.140 --> 00:12:26.860
Let's see if we can
solve the problem.
00:12:26.860 --> 00:12:30.080
One point I want to make is that
now that you have A inverse--
00:12:30.080 --> 00:12:32.850
I've tried to ask you to solve
the problem for two different
00:12:32.850 --> 00:12:35.320
b's-- you don't have to go and
find A inverse again, right?
00:12:35.320 --> 00:12:36.570
You're done finding A inverse.
00:12:36.570 --> 00:12:39.130
You just now have to
do the multiplication.
00:12:39.130 --> 00:12:41.997
So now for number
1, we had b was
00:12:41.997 --> 00:12:43.580
equal to-- I'm going
to write it here,
00:12:43.580 --> 00:12:49.120
so I don't have to keep looking
over-- 1, 2, negative 3.
00:12:49.120 --> 00:12:54.120
So A inverse b is going
to be equal to-- well
00:12:54.120 --> 00:12:57.680
I should get another
vector, so I should just
00:12:57.680 --> 00:12:59.610
have three components here.
00:12:59.610 --> 00:13:02.200
And I'm probably going to
have to write out what I get,
00:13:02.200 --> 00:13:03.359
because it might be long.
00:13:03.359 --> 00:13:05.150
But let's see-- actually,
you know what I'm
00:13:05.150 --> 00:13:06.787
going to do to make it easier?
00:13:06.787 --> 00:13:08.620
Because there's a lot
of junk going on here.
00:13:08.620 --> 00:13:10.328
So what I'm going to
do to make it easier
00:13:10.328 --> 00:13:13.272
is put the negative
1/10 in front to start.
00:13:13.272 --> 00:13:14.730
Because that negative
1/10 is going
00:13:14.730 --> 00:13:16.742
to come along with
every term, so I'm
00:13:16.742 --> 00:13:18.950
just going to put the negative
1/10 in front and deal
00:13:18.950 --> 00:13:20.290
with it at the end.
00:13:20.290 --> 00:13:21.516
OK?
00:13:21.516 --> 00:13:22.890
So now I'm just
going to multiply
00:13:22.890 --> 00:13:27.310
b-- which is this 1, 2, negative
3-- by this big matrix here
00:13:27.310 --> 00:13:29.480
without the negative
1/10 in front.
00:13:29.480 --> 00:13:30.404
OK?
00:13:30.404 --> 00:13:31.320
So let's look at that.
00:13:31.320 --> 00:13:35.050
We're just going to have
first row times the column,
00:13:35.050 --> 00:13:37.020
and that's going to give
me the first position.
00:13:37.020 --> 00:13:40.115
So negative 2 times
1 is negative 2.
00:13:40.115 --> 00:13:42.030
I'm going to write
them all down.
00:13:42.030 --> 00:13:45.350
Plus 2 times 2 is 4.
00:13:45.350 --> 00:13:49.300
Plus 2 times negative
3 is negative 6.
00:13:49.300 --> 00:13:50.730
So that's the first position.
00:13:50.730 --> 00:13:52.920
We'll simplify in a moment.
00:13:52.920 --> 00:13:55.180
So the next one, I get
negative 1 times 1.
00:13:55.180 --> 00:13:56.490
That's negative 1.
00:13:56.490 --> 00:13:58.450
Then I get negative 4 times 2.
00:13:58.450 --> 00:13:59.550
That's negative 8.
00:13:59.550 --> 00:14:01.190
So minus 8.
00:14:01.190 --> 00:14:05.080
And then I get 1 times
negative 3, so minus 3.
00:14:05.080 --> 00:14:07.750
So we've got two
of the rows done.
00:14:07.750 --> 00:14:09.940
We just have to simplify
them in a moment.
00:14:09.940 --> 00:14:12.360
And now we just do
this third component.
00:14:12.360 --> 00:14:17.000
So it's the third row of A
inverse without that scalar
00:14:17.000 --> 00:14:19.890
in front, times the
only column of b
00:14:19.890 --> 00:14:21.810
to give me the last position.
00:14:21.810 --> 00:14:22.450
Right?
00:14:22.450 --> 00:14:27.250
So 3 times 1 is 3, plus 2 times
2 is 4, so I get 3 plus 4,
00:14:27.250 --> 00:14:31.250
and then 7 times
negative 3 is minus 21.
00:14:31.250 --> 00:14:32.090
OK.
00:14:32.090 --> 00:14:33.940
So what do I get when
I write it all out?
00:14:33.940 --> 00:14:35.500
I get negative 1/10.
00:14:35.500 --> 00:14:40.750
And then-- so negative 8 plus
4, that looks like a minus 4.
00:14:40.750 --> 00:14:41.660
Right?
00:14:41.660 --> 00:14:44.210
8, 9, 10, 11, 12.
00:14:44.210 --> 00:14:46.144
That looks like a negative 12.
00:14:46.144 --> 00:14:47.310
It's a lot of adding for me.
00:14:47.310 --> 00:14:49.950
I make a lot of adding mistakes,
so we should be careful.
00:14:49.950 --> 00:14:52.170
This looks like negative 14.
00:14:52.170 --> 00:14:53.030
OK.
00:14:53.030 --> 00:14:58.499
So this is a matrix that,
it's just a vector, right?
00:14:58.499 --> 00:15:00.040
All the negative
signs will drop out.
00:15:00.040 --> 00:15:01.750
I'll get some fractions.
00:15:01.750 --> 00:15:05.245
But if it is the correct
answer-- which I'm really
00:15:05.245 --> 00:15:07.370
hoping it is, because I
just did this whole problem
00:15:07.370 --> 00:15:08.490
and I hope it's the
correct answer--
00:15:08.490 --> 00:15:10.640
if it's the correct answer,
then what should it do?
00:15:10.640 --> 00:15:13.760
When I take the original A
that I had and I multiply it
00:15:13.760 --> 00:15:16.020
by this, I should get b.
00:15:16.020 --> 00:15:17.870
I should get 1, 2, negative 3.
00:15:17.870 --> 00:15:20.910
So you can check your work
very easily to see if it works.
00:15:20.910 --> 00:15:25.350
You can take A times this,
and see if you get b.
00:15:25.350 --> 00:15:26.340
Right?
00:15:26.340 --> 00:15:30.740
And then you'll know if this
is the x we were looking for.
00:15:30.740 --> 00:15:32.090
OK?
00:15:32.090 --> 00:15:34.550
And then let's
look at number two.
00:15:34.550 --> 00:15:38.260
I just said that b
equals [0, 0, 0].
00:15:38.260 --> 00:15:39.960
And the point I
want to make there
00:15:39.960 --> 00:15:42.660
is that since this
has an inverse,
00:15:42.660 --> 00:15:44.940
A inverse-- since A has
an inverse, A inverse b
00:15:44.940 --> 00:15:50.270
is going to be-- in this case--
A inverse times [0, 0, 0],
00:15:50.270 --> 00:15:53.860
which is going to
give you [0, 0, 0].
00:15:53.860 --> 00:15:57.420
So the only solution we have in
this case-- because A inverse,
00:15:57.420 --> 00:15:59.897
if I look and I try and multiply
every row by this column,
00:15:59.897 --> 00:16:02.530
right, I'm going to get
0 in the first spot,
00:16:02.530 --> 00:16:05.060
0 in the second spot,
and 0 in the third spot--
00:16:05.060 --> 00:16:10.570
so the solution I get-- the
x I'm looking for so that Ax
00:16:10.570 --> 00:16:13.760
equals [0, 0, 0]-- is [0, 0, 0].
00:16:13.760 --> 00:16:17.000
And what I just want
to mention to you,
00:16:17.000 --> 00:16:20.390
is that that is true
because A is invertible.
00:16:20.390 --> 00:16:23.230
If A were not invertible, you
could get other solutions.
00:16:23.230 --> 00:16:24.230
Other things might work.
00:16:24.230 --> 00:16:26.990
And that's also true,
actually, in this case as well,
00:16:26.990 --> 00:16:29.860
but it's a little harder to
see that it's-- that could be
00:16:29.860 --> 00:16:32.480
potentially a weird thing.
00:16:32.480 --> 00:16:39.090
To solve A*x equals
[0, 0, 0], it's sort of like,
00:16:39.090 --> 00:16:41.380
naturally we see
[0, 0, 0] is a solution.
00:16:41.380 --> 00:16:43.884
Right away you can see that,
and that's one that we get.
00:16:43.884 --> 00:16:46.050
The point I want to make
is because A is invertible,
00:16:46.050 --> 00:16:47.620
that's the only solution.
00:16:47.620 --> 00:16:49.330
And if A were not
invertible, you
00:16:49.330 --> 00:16:51.260
could get other
solutions to that.
00:16:51.260 --> 00:16:53.570
So that's something that
we haven't seen yet--
00:16:53.570 --> 00:16:55.680
we haven't dealt
with yet-- but that
00:16:55.680 --> 00:16:57.140
is something that can happen.
00:16:57.140 --> 00:17:00.160
So I just want to point out that
there could be an oddity if A
00:17:00.160 --> 00:17:01.490
were not invertible.
00:17:01.490 --> 00:17:03.740
But since A is invertible,
we get just one solution
00:17:03.740 --> 00:17:05.470
for both of these things.
00:17:05.470 --> 00:17:06.740
OK.
00:17:06.740 --> 00:17:08.815
So I'm going to go back
and just remind you
00:17:08.815 --> 00:17:11.470
of a few things of how we
found the inverse matrix,
00:17:11.470 --> 00:17:13.210
and then I will stop.
00:17:13.210 --> 00:17:15.690
So we were given a
matrix A. And to go
00:17:15.690 --> 00:17:18.570
through the steps of finding the
inverse matrix, what did we do?
00:17:18.570 --> 00:17:21.900
The first thing we did was
we found the determinant.
00:17:21.900 --> 00:17:23.690
Then we found the
matrix of minors.
00:17:23.690 --> 00:17:26.150
And then I just took
that matrix of minors,
00:17:26.150 --> 00:17:28.410
put the plus-minus
grid on top of it
00:17:28.410 --> 00:17:31.150
so that I got the
cofactor matrix.
00:17:31.150 --> 00:17:31.650
Right?
00:17:31.650 --> 00:17:33.360
And then once I had
the cofactor matrix,
00:17:33.360 --> 00:17:35.517
you just have to transpose it.
00:17:35.517 --> 00:17:36.350
So I came over here.
00:17:36.350 --> 00:17:39.740
I transposed that, and I put
1 over the determinant of A
00:17:39.740 --> 00:17:40.260
in front.
00:17:40.260 --> 00:17:42.700
So the scalar is 1 over
the determinant of A,
00:17:42.700 --> 00:17:45.190
times the transpose of
the cofactor matrix.
00:17:45.190 --> 00:17:46.971
And that's what
gives me A inverse.
00:17:46.971 --> 00:17:48.470
So there are a fair
number of steps,
00:17:48.470 --> 00:17:50.710
but you can do them
very systematically,
00:17:50.710 --> 00:17:53.300
and then you have the inverse
matrix that you're looking for.
00:17:53.300 --> 00:17:58.220
And then you can solve for x,
when you're looking for A*x
00:17:58.220 --> 00:18:00.490
equals b, and you
know b and you know A.
00:18:00.490 --> 00:18:03.720
And you do this same process
we just outlined here again,
00:18:03.720 --> 00:18:05.590
and that gives it to you.
00:18:05.590 --> 00:18:07.456
OK, I think I'll stop there.