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CHRISTINE BREINER: Welcome
back to recitation.

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In this video, I really just
want to practice matrix

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multiplication, which is
potentially something new for

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some of you, and maybe some of
you have been doing it for a

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while and are very good at it.

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But I want to make sure that
everyone is feeling confident

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in their ability to
multiply matrices.

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So we have three
matrices here.

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We have A, B, and C. And what I
want you to do is I want you

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to compute what makes
sense below.

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I have four products
of matrices below.

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a is A times B, b is B times A,
c is B times C, and d is A

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times C. So I want you to
multiply the matrices that

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make sense to multiply, and then
the ones that don't, make

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sure you understand why.

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Give yourself a brief
explanation of why you can't

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multiply them.

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So why don't you work on that,
pause the video, and when you

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feel confident in your answers,
bring the video back

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up, and you can check them
against my work.

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OK, welcome back.

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Well, we wanted to make sure
we felt comfortable

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multiplying matrices.

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So what we're going to do is
look at the four products I

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mentioned below, and we're
going to see how they do,

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whether or not we can actually
compute them.

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So let's first look at a, which
was A times B. So before

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I write it down again,
A times B is--

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notice A is a 2 by 2 matrix,
and B is a 2

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by 3 matrix, right?

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And so if I write letter a,
we know we're taking a 2

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by 2 by a 2 by 3.

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And so the fact that the
interior dimensions agree,

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that the number of columns of
A is equal to the number of

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rows of B, means that
I can multiply them.

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So I can multiply them, and my
result I expect to get is, of

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course, the dimensions we
have on the outside.

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So I expect to get
a 2 by 3 matrix.

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So I'm going to rewrite A and
B here, so that I don't have

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to keep walking back and forth,
and then we'll do the

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multiplication.

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So I have 6, 5, 1, 2, times
2, minus 1, 3, 1, 0, 4.

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OK.

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So I want to perform this
multiplication.

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Now remember that when you are
looking for a value in your

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resulting matrix, which I know
is 2 by 3, so I can even make

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a little space for myself.

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I know it's 2 by 3, so I know
I'm going to have to fill in

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these spots.

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When I look at this position,
it's row 1, column 1.

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That means I take row 1 of
the first matrix, and I'm

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essentially just dotting
it with column 1

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of the second matrix.

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So I'm taking row 1 times column
1 in the way it was

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described, which is I take
6 times 2, and I add

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it to 5 times 1.

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So row 1, column 1 gives me 6
times 2 is 12, plus 5 times 1

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which is 5, so I get 17.

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And then if I come in to the
next spot, what is this?

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And the resulting matrix's
position is row 1, column 2.

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So now I take row 1 of the first
matrix and column 2 of

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the second matrix, and I get 6
times negative 1 is negative

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6, plus 0 times 5, so I
get a negative 6 here.

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Negative 6 times 0.

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Maybe I should show
you this way.

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Negative 6 times 0.

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OK.

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And then here I am now in the
third spot of the first row,

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so I'm in a row 1, column 3.

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So that's again, row 1 of the
first, column 3 of the second.

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So you see a pattern here about
where we're getting our

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things from that we're
multiplying.

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For row 1, column 3 of the
resulting, I take row 1 of the

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first and column 3
of the second.

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So 6 times 3 is 18, plus--

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5 times 4 is--

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20.

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So 20 plus 18 is 38.

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OK.

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Now I have to do the same
thing on the bottom.

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Right?

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So I have now here, the row,
notice I'm in row 2, so I'm

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always going to use row
2 of ths first matrix.

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And then what we saw last time
is I used column 1 in the

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first spot, column 2 in the
second spot, column 3 in the

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third spot.

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Right?

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That's what happens over
and over again.

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So what do I do?

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I take 1, 2, and I multiply
it by 2, 1.

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So I take 1 times 2,
plus 2 times 1.

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So I get 2 there and 2
there, so I get a 4.

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And then the next column:
row 2, column 2.

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1 times negative 1
is negative 1--

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2 times 0 is 0--

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so I get a negative 1.

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And then the last column.

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1 times 3 is 3.

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2 times 4 is 8.

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So I get 3 plus 8,
so I get 11.

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Hopefully I didn't make any
stupid summing mistakes there,

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but if I did, you probably
caught it.

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Because I was trying to say
what I did as we went.

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So that is the answer to a.

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OK.

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So now let's think
about what is b.

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b was take B times A, which is
just to switch the order.

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So let's look at the
dimension matchup.

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Now we have a 2 by 3 matrix, and
I'm trying to multiply it

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by a 2 by 2 matrix.

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Well, I don't have to
do any more work,

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because I can't do it.

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Because the dimensions
of the insides here--

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three columns for B,
two rows for A--

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means that I can't actually
multiply them.

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OK?

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So this isn't even defined.

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OK, so that was easy.

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That was b.

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All right.

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Letter c--

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I'll give myself a lot
of room to do that--

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letter c was B times C. And so
I'm going to write down the

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dimensions to see if
I even need to

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write down the matrices.

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B was two rows by three columns,
and C was three rows

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by two columns.

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So if I look at the dimensions,
the 3 and the 3

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match up, so I am going
to be able to multiply

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them, and my result--

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as I mentioned before--
should be a 2 by 2.

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So let me write down B and C
here, so we don't have to keep

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going to the side.

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OK.

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And then C is 1, 2, negative
1, negative 1, 3, 2.

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All right, let me just make
sure I didn't transcribe

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anything incorrectly.

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I think it looks good.

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OK.

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So row 1 of B. Row 2 of B.
Column 1 of C. Column 2 of C.

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We're going to be dealing with
those, specifically.

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So we want to multiply these.

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We said our resulting matrix
is going to be 2 by 2.

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OK.

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Because I'm going to have a lot
of terms, I might write

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them down on this one,
and then simplify.

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Because I may make a mistake,
so to be more careful, I'll

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write down all the pieces.

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So here I am in row 1, column
1 of the resulting.

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So I take row 1 of the first,
column 1 of the second, and

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what do I get when I multiply?

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I get 2 times 1-- that's 2.

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Plus negative 1 times 2--
that's negative 2.

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Plus 3 times negative 1--
that's negative 3.

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Right?

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That's all we have to do
for the first position.

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Then I do, for the second one,
it's row 1, column 2.

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So I do row 1, column 2.

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So I'll try to keep my
head out of the way.

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I realize I keep stepping
in front.

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So it's 2 times negative 1.

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I get negative 2.

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Negative 1 times 3, so
I get negative 3.

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And 3 times 2 gives me 6.

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OK.

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And then the bottom two.

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I get row 2, column 1 over
here, and then row 2,

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column 2 over here.

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So row 2, column 1 is going
to be 1 times 1.

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Plus 0 times 2.

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Plus 4 times negative 1,
so I get negative 4.

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And then here.

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Row 2, column 2.

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I get 1 times negative 1,
so I have negative 1.

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Plus 0 times 3--

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plus 0.

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And then 4 times 2 is 8.

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So if I simplify these, it looks
like in the first spot I

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should get a negative 3.

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And the second spot,
I should get a 1.

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This is just for you to
check your answer.

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And the third spot, I
get a negative 3.

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And then the fourth
spot, I get a 7.

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So hopefully I added correctly
all throughout.

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I think I did, so I think
we're good there.

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So that is the answer to C.

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And again, the reason we can
multiply those, was that the

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dimensions--

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when you wrote them down in
order-- the dimensions to the

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inside agreed, and then the
outside gives us the size of

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the resulting matrix.

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So there was one more problem,
and that was d.

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And I wanted you to take A times
C. And A was a 2 by 2.

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And C was a 3 by 2.

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And so again, we see we can't
do it, because the two

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interior dimensions here--

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when I write them
in that order--

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don't agree.

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OK.

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So d is not defined.

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All right, so the basic idea of
this whole video is just to

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make sure we felt comfortable
multiplying matrices.

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We're trying to use some
simple examples

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to understand that.

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Understand how we can recognize
from the dimensions

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whether or not multiplication is
even defined, and then what

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size the resulting
matrix will be.

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I do want to point
out one thing.

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And I want to point out that if
we come over to our example

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back in the beginning.

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We had AB as our first
example and then BA

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as our second example.

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And AB-- well, I think,
I got to remember

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what they were, yeah--

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AB you could multiply,
but BA you could not.

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So I think it has been stressed
before, but I think I

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should stress it again,
that order matters in

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multiplication.

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OK?

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You can't commute
these things.

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You can't switch the order
and get the same result.

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So matrix multiplication, you
have to be very careful about

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keeping things in the same order
as you're multiplying.

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OK, I think that is
where I will stop.

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