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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

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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?

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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.

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That's the plus, minus, plus.

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This one is going to
be minus, plus, minus.

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So the minus switches that.

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Plus keeps that the same.

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Minus switches that.

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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.

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So a lot of things that were
negative became positive.

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And I had to change-- maybe
I threw in one negative,

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maybe not.

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But, so all the signs I kept,
this one stayed the same,

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this one stayed the same,
this one stayed the same,

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these two stayed
the same, and then

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these four switched,
because it's

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the plus, minus,
plus sort of grid

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that I have to put
on top of this.

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OK, so that's the
cofactor matrix.

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We're getting closer.

218
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OK, now we need the
transpose of this, right?

219
00:09:04,320 --> 00:09:08,220
So if I look at the
transpose-- actually,

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know what I'm going to do?

221
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Because I'm also just going
to have to take the transpose

222
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and then multiply it by
1 over the determinant,

223
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I'm going to do
that all at once.

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OK.

225
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Because we can do that
all at once, and then

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we don't have to worry about it.

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So A inverse I know is
going to be negative 1/10,

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because the determinant
was minus 10.

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So it's 1 over the determinant
times the transpose

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of this matrix.

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So the transpose
of this matrix--

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remember what I'm going
to do is essentially

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you fix the diagonal and
you're going to flip.

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That's really what, in
the square matrix, that's

235
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how you can think about it.

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But every column is
going to become a row.

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So I'm going to write
this as my first row.

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This first column is going
to become my first row.

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So it's going to be negative
2, 2, 2 as my first row.

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And then the next
column is going

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to be negative 1, negative 4, 1.

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I mean next row.

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I will take a column
and change it to a row.

244
00:10:00,930 --> 00:10:05,220
The next row is going to be
negative 1, negative 4, 1.

245
00:10:05,220 --> 00:10:06,360
And then the last one.

246
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I take this column and
I change it to a row.

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It's going to be 3, 2, 7.

248
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OK.

249
00:10:14,780 --> 00:10:16,446
And because again, I
want to make sure--

250
00:10:16,446 --> 00:10:18,395
this one is really
messy-- I want

251
00:10:18,395 --> 00:10:22,200
to make sure I have something
similar for that, or exactly

252
00:10:22,200 --> 00:10:22,800
that.

253
00:10:22,800 --> 00:10:23,530
OK.

254
00:10:23,530 --> 00:10:25,850
I think I'm still
doing all right.

255
00:10:25,850 --> 00:10:27,587
Now, let's get to
solving the problem.

256
00:10:27,587 --> 00:10:29,920
Because so far, we just were
finding the inverse matrix.

257
00:10:29,920 --> 00:10:31,503
So I'm going to leave
it in this form,

258
00:10:31,503 --> 00:10:33,230
instead of dividing
by 10 in every spot,

259
00:10:33,230 --> 00:10:35,290
because that will be annoying.

260
00:10:35,290 --> 00:10:36,850
So let's think
about how do I want

261
00:10:36,850 --> 00:10:39,870
to solve the system that I had.

262
00:10:39,870 --> 00:10:43,990
I had A*x equals b.

263
00:10:43,990 --> 00:10:46,920
And actually, I
mean, my strategy

264
00:10:46,920 --> 00:10:48,770
is to find the inverse matrix.

265
00:10:48,770 --> 00:10:50,380
I didn't talk to
you about why we

266
00:10:50,380 --> 00:10:53,454
know the inverse
matrix actually exists.

267
00:10:53,454 --> 00:10:55,370
But ultimately, you
haven't even seen this yet

268
00:10:55,370 --> 00:10:56,619
in the lecture videos, really.

269
00:10:56,619 --> 00:11:00,390
Except that you know that the
determinant of A being non-zero

270
00:11:00,390 --> 00:11:01,790
gives you an inverse matrix.

271
00:11:01,790 --> 00:11:03,750
That's all you know, I
think, at this point.

272
00:11:03,750 --> 00:11:06,130
You have the determinant
of A. It's non-zero,

273
00:11:06,130 --> 00:11:08,100
so you can find
an inverse matrix.

274
00:11:08,100 --> 00:11:10,650
Makes sense based on the
formulation you have,

275
00:11:10,650 --> 00:11:14,000
because if the determinant
is 0, then this quantity 1

276
00:11:14,000 --> 00:11:16,010
over the determinant
of A, you run

277
00:11:16,010 --> 00:11:17,590
into quite a bit of trouble.

278
00:11:17,590 --> 00:11:20,050
So that's just as
a little sidebar,

279
00:11:20,050 --> 00:11:22,960
we know the inverse
matrix exists for A.

280
00:11:22,960 --> 00:11:25,470
So what we do-- this
is again the strategy--

281
00:11:25,470 --> 00:11:29,370
you multiply A inverse A
times x on the left side.

282
00:11:29,370 --> 00:11:30,056
Ooh.

283
00:11:30,056 --> 00:11:34,000
Is equal to-- sorry-- that
should be the lowercase b.

284
00:11:34,000 --> 00:11:35,370
Should be a vector there.

285
00:11:35,370 --> 00:11:39,800
It is equal to A inverse
b on the right-hand side.

286
00:11:39,800 --> 00:11:42,100
And you notice,
it's very important,

287
00:11:42,100 --> 00:11:43,830
in the matrix
multiplication video

288
00:11:43,830 --> 00:11:47,490
we saw that it's very
important the order in which

289
00:11:47,490 --> 00:11:48,700
you multiply matrices.

290
00:11:48,700 --> 00:11:51,900
And since I'm putting A
inverse on the far left

291
00:11:51,900 --> 00:11:55,940
of this side of the equality, I
have to put it on the far left

292
00:11:55,940 --> 00:11:58,461
of the right-hand
side of the equality.

293
00:11:58,461 --> 00:11:58,960
Right?

294
00:11:58,960 --> 00:12:00,640
And in fact, you
would run into trouble

295
00:12:00,640 --> 00:12:02,430
if you tried to switch
the order of these.

296
00:12:02,430 --> 00:12:02,929
OK?

297
00:12:02,929 --> 00:12:05,250
We wouldn't be able
to multiply them.

298
00:12:05,250 --> 00:12:06,000
All right?

299
00:12:06,000 --> 00:12:10,000
So A inverse A, we know is
just the identity matrix.

300
00:12:10,000 --> 00:12:13,340
So you get the
identity matrix times x

301
00:12:13,340 --> 00:12:16,430
is equal to A inverse b.

302
00:12:16,430 --> 00:12:22,640
So you can find x by
finding A inverse times b.

303
00:12:22,640 --> 00:12:23,470
Right?

304
00:12:23,470 --> 00:12:25,140
And so now we have A inverse.

305
00:12:25,140 --> 00:12:26,860
Let's see if we can
solve the problem.

306
00:12:26,860 --> 00:12:30,080
One point I want to make is that
now that you have A inverse--

307
00:12:30,080 --> 00:12:32,850
I've tried to ask you to solve
the problem for two different

308
00:12:32,850 --> 00:12:35,320
b's-- you don't have to go and
find A inverse again, right?

309
00:12:35,320 --> 00:12:36,570
You're done finding A inverse.

310
00:12:36,570 --> 00:12:39,130
You just now have to
do the multiplication.

311
00:12:39,130 --> 00:12:41,997
So now for number
1, we had b was

312
00:12:41,997 --> 00:12:43,580
equal to-- I'm going
to write it here,

313
00:12:43,580 --> 00:12:49,120
so I don't have to keep looking
over-- 1, 2, negative 3.

314
00:12:49,120 --> 00:12:54,120
So A inverse b is going
to be equal to-- well

315
00:12:54,120 --> 00:12:57,680
I should get another
vector, so I should just

316
00:12:57,680 --> 00:12:59,610
have three components here.

317
00:12:59,610 --> 00:13:02,200
And I'm probably going to
have to write out what I get,

318
00:13:02,200 --> 00:13:03,359
because it might be long.

319
00:13:03,359 --> 00:13:05,150
But let's see-- actually,
you know what I'm

320
00:13:05,150 --> 00:13:06,787
going to do to make it easier?

321
00:13:06,787 --> 00:13:08,620
Because there's a lot
of junk going on here.

322
00:13:08,620 --> 00:13:10,328
So what I'm going to
do to make it easier

323
00:13:10,328 --> 00:13:13,272
is put the negative
1/10 in front to start.

324
00:13:13,272 --> 00:13:14,730
Because that negative
1/10 is going

325
00:13:14,730 --> 00:13:16,742
to come along with
every term, so I'm

326
00:13:16,742 --> 00:13:18,950
just going to put the negative
1/10 in front and deal

327
00:13:18,950 --> 00:13:20,290
with it at the end.

328
00:13:20,290 --> 00:13:21,516
OK?

329
00:13:21,516 --> 00:13:22,890
So now I'm just
going to multiply

330
00:13:22,890 --> 00:13:27,310
b-- which is this 1, 2, negative
3-- by this big matrix here

331
00:13:27,310 --> 00:13:29,480
without the negative
1/10 in front.

332
00:13:29,480 --> 00:13:30,404
OK?

333
00:13:30,404 --> 00:13:31,320
So let's look at that.

334
00:13:31,320 --> 00:13:35,050
We're just going to have
first row times the column,

335
00:13:35,050 --> 00:13:37,020
and that's going to give
me the first position.

336
00:13:37,020 --> 00:13:40,115
So negative 2 times
1 is negative 2.

337
00:13:40,115 --> 00:13:42,030
I'm going to write
them all down.

338
00:13:42,030 --> 00:13:45,350
Plus 2 times 2 is 4.

339
00:13:45,350 --> 00:13:49,300
Plus 2 times negative
3 is negative 6.

340
00:13:49,300 --> 00:13:50,730
So that's the first position.

341
00:13:50,730 --> 00:13:52,920
We'll simplify in a moment.

342
00:13:52,920 --> 00:13:55,180
So the next one, I get
negative 1 times 1.

343
00:13:55,180 --> 00:13:56,490
That's negative 1.

344
00:13:56,490 --> 00:13:58,450
Then I get negative 4 times 2.

345
00:13:58,450 --> 00:13:59,550
That's negative 8.

346
00:13:59,550 --> 00:14:01,190
So minus 8.

347
00:14:01,190 --> 00:14:05,080
And then I get 1 times
negative 3, so minus 3.

348
00:14:05,080 --> 00:14:07,750
So we've got two
of the rows done.

349
00:14:07,750 --> 00:14:09,940
We just have to simplify
them in a moment.

350
00:14:09,940 --> 00:14:12,360
And now we just do
this third component.

351
00:14:12,360 --> 00:14:17,000
So it's the third row of A
inverse without that scalar

352
00:14:17,000 --> 00:14:19,890
in front, times the
only column of b

353
00:14:19,890 --> 00:14:21,810
to give me the last position.

354
00:14:21,810 --> 00:14:22,450
Right?

355
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,

356
00:14:27,250 --> 00:14:31,250
and then 7 times
negative 3 is minus 21.

357
00:14:31,250 --> 00:14:32,090
OK.

358
00:14:32,090 --> 00:14:33,940
So what do I get when
I write it all out?

359
00:14:33,940 --> 00:14:35,500
I get negative 1/10.

360
00:14:35,500 --> 00:14:40,750
And then-- so negative 8 plus
4, that looks like a minus 4.

361
00:14:40,750 --> 00:14:41,660
Right?

362
00:14:41,660 --> 00:14:44,210
8, 9, 10, 11, 12.

363
00:14:44,210 --> 00:14:46,144
That looks like a negative 12.

364
00:14:46,144 --> 00:14:47,310
It's a lot of adding for me.

365
00:14:47,310 --> 00:14:49,950
I make a lot of adding mistakes,
so we should be careful.

366
00:14:49,950 --> 00:14:52,170
This looks like negative 14.

367
00:14:52,170 --> 00:14:53,030
OK.

368
00:14:53,030 --> 00:14:58,499
So this is a matrix that,
it's just a vector, right?

369
00:14:58,499 --> 00:15:00,040
All the negative
signs will drop out.

370
00:15:00,040 --> 00:15:01,750
I'll get some fractions.

371
00:15:01,750 --> 00:15:05,245
But if it is the correct
answer-- which I'm really

372
00:15:05,245 --> 00:15:07,370
hoping it is, because I
just did this whole problem

373
00:15:07,370 --> 00:15:08,490
and I hope it's the
correct answer--

374
00:15:08,490 --> 00:15:10,640
if it's the correct answer,
then what should it do?

375
00:15:10,640 --> 00:15:13,760
When I take the original A
that I had and I multiply it

376
00:15:13,760 --> 00:15:16,020
by this, I should get b.

377
00:15:16,020 --> 00:15:17,870
I should get 1, 2, negative 3.

378
00:15:17,870 --> 00:15:20,910
So you can check your work
very easily to see if it works.

379
00:15:20,910 --> 00:15:25,350
You can take A times this,
and see if you get b.

380
00:15:25,350 --> 00:15:26,340
Right?

381
00:15:26,340 --> 00:15:30,740
And then you'll know if this
is the x we were looking for.

382
00:15:30,740 --> 00:15:32,090
OK?

383
00:15:32,090 --> 00:15:34,550
And then let's
look at number two.

384
00:15:34,550 --> 00:15:38,260
I just said that b
equals [0, 0,  0].

385
00:15:38,260 --> 00:15:39,960
And the point I
want to make there

386
00:15:39,960 --> 00:15:42,660
is that since this
has an inverse,

387
00:15:42,660 --> 00:15:44,940
A inverse-- since A has
an inverse, A inverse b

388
00:15:44,940 --> 00:15:50,270
is going to be-- in this case--
A inverse times [0, 0, 0],

389
00:15:50,270 --> 00:15:53,860
which is going to
give you [0,  0, 0].

390
00:15:53,860 --> 00:15:57,420
So the only solution we have in
this case-- because A inverse,

391
00:15:57,420 --> 00:15:59,897
if I look and I try and multiply
every row by this column,

392
00:15:59,897 --> 00:16:02,530
right, I'm going to get
0 in the first spot,

393
00:16:02,530 --> 00:16:05,060
0 in the second spot,
and 0 in the third spot--

394
00:16:05,060 --> 00:16:10,570
so the solution I get-- the
x I'm looking for so that Ax

395
00:16:10,570 --> 00:16:13,760
equals [0, 0, 0]-- is [0, 0, 0].

396
00:16:13,760 --> 00:16:17,000
And what I just want
to mention to you,

397
00:16:17,000 --> 00:16:20,390
is that that is true
because A is invertible.

398
00:16:20,390 --> 00:16:23,230
If A were not invertible, you
could get other solutions.

399
00:16:23,230 --> 00:16:24,230
Other things might work.

400
00:16:24,230 --> 00:16:26,990
And that's also true,
actually, in this case as well,

401
00:16:26,990 --> 00:16:29,860
but it's a little harder to
see that it's-- that could be

402
00:16:29,860 --> 00:16:32,480
potentially a weird thing.

403
00:16:32,480 --> 00:16:39,090
To solve A*x equals
[0, 0, 0], it's sort of like,

404
00:16:39,090 --> 00:16:41,380
naturally we see
[0, 0, 0] is a solution.

405
00:16:41,380 --> 00:16:43,884
Right away you can see that,
and that's one that we get.

406
00:16:43,884 --> 00:16:46,050
The point I want to make
is because A is invertible,

407
00:16:46,050 --> 00:16:47,620
that's the only solution.

408
00:16:47,620 --> 00:16:49,330
And if A were not
invertible, you

409
00:16:49,330 --> 00:16:51,260
could get other
solutions to that.

410
00:16:51,260 --> 00:16:53,570
So that's something that
we haven't seen yet--

411
00:16:53,570 --> 00:16:55,680
we haven't dealt
with yet-- but that

412
00:16:55,680 --> 00:16:57,140
is something that can happen.

413
00:16:57,140 --> 00:17:00,160
So I just want to point out that
there could be an oddity if A

414
00:17:00,160 --> 00:17:01,490
were not invertible.

415
00:17:01,490 --> 00:17:03,740
But since A is invertible,
we get just one solution

416
00:17:03,740 --> 00:17:05,470
for both of these things.

417
00:17:05,470 --> 00:17:06,740
OK.

418
00:17:06,740 --> 00:17:08,815
So I'm going to go back
and just remind you

419
00:17:08,815 --> 00:17:11,470
of a few things of how we
found the inverse matrix,

420
00:17:11,470 --> 00:17:13,210
and then I will stop.

421
00:17:13,210 --> 00:17:15,690
So we were given a
matrix A. And to go

422
00:17:15,690 --> 00:17:18,570
through the steps of finding the
inverse matrix, what did we do?

423
00:17:18,570 --> 00:17:21,900
The first thing we did was
we found the determinant.

424
00:17:21,900 --> 00:17:23,690
Then we found the
matrix of minors.

425
00:17:23,690 --> 00:17:26,150
And then I just took
that matrix of minors,

426
00:17:26,150 --> 00:17:28,410
put the plus-minus
grid on top of it

427
00:17:28,410 --> 00:17:31,150
so that I got the
cofactor matrix.

428
00:17:31,150 --> 00:17:31,650
Right?

429
00:17:31,650 --> 00:17:33,360
And then once I had
the cofactor matrix,

430
00:17:33,360 --> 00:17:35,517
you just have to transpose it.

431
00:17:35,517 --> 00:17:36,350
So I came over here.

432
00:17:36,350 --> 00:17:39,740
I transposed that, and I put
1 over the determinant of A

433
00:17:39,740 --> 00:17:40,260
in front.

434
00:17:40,260 --> 00:17:42,700
So the scalar is 1 over
the determinant of A,

435
00:17:42,700 --> 00:17:45,190
times the transpose of
the cofactor matrix.

436
00:17:45,190 --> 00:17:46,971
And that's what
gives me A inverse.

437
00:17:46,971 --> 00:17:48,470
So there are a fair
number of steps,

438
00:17:48,470 --> 00:17:50,710
but you can do them
very systematically,

439
00:17:50,710 --> 00:17:53,300
and then you have the inverse
matrix that you're looking for.

440
00:17:53,300 --> 00:17:58,220
And then you can solve for x,
when you're looking for A*x

441
00:17:58,220 --> 00:18:00,490
equals b, and you
know b and you know A.

442
00:18:00,490 --> 00:18:03,720
And you do this same process
we just outlined here again,

443
00:18:03,720 --> 00:18:05,590
and that gives it to you.

444
00:18:05,590 --> 00:18:07,456
OK, I think I'll stop there.