Matrix multiplication (part 1)

Multiplying two 2x2 matrices.

Well, after the last video,
hopefully, we're a little

familiar with how you
add matrices.

So now let's learn how
to multiply matrices.

And keep in mind, these are
human-created definitions for

matrix multiplication.

We could have come up with
completely different ways to

multiply it.

But I encourage you to learn
this way because it'll help

you in math class.

And we will see later that
there's actually a lot of

applications that come out
of this type of matrix

multiplication.

So let me think of
two matrices.

I will do two 2 by 2 matrices,
and let's multiply them.

Let's say-- let me pick some
random numbers: 2,

minus 3, 7, and 5.

And I'm going to multiply that
matrix, or that table of

numbers, times 10, minus 8--
let me pick a good number

here-- 12, and then minus 2.

So now there might be a strong
temptation-- and you know in

some ways it's not even an
illegitimate temptation-- to

do the same thing with
multiplication that we did

with addition, to just multiply
the corresponding

terms. So you might be tempted
to say, well, the first term

right here, the 1, 1 term, or
in the first row and first

column, is going to
be 2 times 10.

And this term is going
to be minus 3 times

minus 8 and so forth.

And that's how we added matrices
so maybe it's a

natural extension to multiply
matrices the same way.

And that is legitimate.

One could define it that way,
but that's not the way it is

in the real world.

And the way in the real world,

unfortunately, is more complex.

But if you look at a
bunch of examples I

think you'll get it.

And you'll learn that
it's actually fairly

straightforward.

So how do we do it?

So this first term that's in
the first row and its first

column, is equal to essentially
this first row's

vector-- no, this first
row vector--

times this column vector.

Now what do I mean
by that, right?

So it's getting it's row
information from the first

matrix's row, and it's getting
it's column information from

the second matrix's column.

So how do I do that?

If you're familiar with dot
product, it's essentially the

dot product of these
two matrices.

Or without saying it that fancy,
it's just this: it's 2

times 10, so 2-- I'm going to
write small-- times 10, plus

minus 3 times 12.

I'm going to run out of space.

And so what's this second
term over here?

Well, we're still on the first
row of the product vector but

now we're on the
second column.

We get our column information
from here.

So let's pick a good color--
this is a slightly different

shade of purple.

So now this is going to be--
I'll do that in another

color-- 2 times minus 8-- let me
just write out the number--

2 times minus 8 is minus 16,
plus minus 3 times minus 2--

what's minus 3 times minus 2?

That is plus 6, right?

So that's in row 1 column 2.

It's minus 16 plus 6.

And then let's come down here.

So now we're in the
second row.

So now we're going to use--
we're getting our row

information from the first
matrix-- I know this is

confusing and I feel bad for you
right now, but we're going

to a bunch of examples and
I think it'll make sense.

So this term-- the bottom left
term-- is going to be this row

times this column.

So it's going to be 7 times
10, so 70, plus 7 times 10

plus 5 times 12, plus 60.

And then the bottom right term
is going to be 7 times minus

8, which is minus 56 plus
5 times minus 2.

So that's minus 10.

So the final product is going to
be 2 times 10 is 20, minus

36, so that's minus 16
plus 6, that's 10.

90-- was that what I said?

No, it was-- 70, plus
60, that's 130.

And then minus 56 minus
10, so minus 66.

So there you have it.

We just multiplied this matrix
times this matrix.

Let me do another example.

And I think I'll actually
squeeze it on this side so

that we can write this side out
a little bit more neatly.

So let's take the matrix and
now 1, 2, 3, 4, times the

matrix 5, 6, 7, 8.

Now we have much more space to
work with so this should come

out neater.

OK, but I'm going to do the
same thing, so to get this

term right here-- the top left
term-- we're going to take--

or the one that has row 1 column
1-- we're going to take

the row 1 information from
here, and the column 1

information from here.

So you can view it as
this row vector

times this column vector.

So it results, 1 times
5 plus 2 times 7.

Right?

There you go.

And so this term, it'll be this
row vector times this

column vector-- let me do that
in a different color-- will be

1 times 6 plus 2 times 8.

Let me write that down.

So it's 1 times 6
plus 2 times 8.

Now we go down to
the second row.

And we get our row information
from the first vector-- let me

circle it with this color--
and it is 3 times 5

plus 4 times 7.

And then we are in the bottom
right, so we're in the bottom

row and second column.

So we get our row information
from here and our column

information from here.

So it's 3 times 6
plus 4 times 8.

And if we simplify,
this is 5 plus--

Well actually, let me just
remind you where all the

numbers came from.

So we have that green
color, right?

This 1 and this 2, that's
this 1 and this 2,

this 1 and this 2.

Right?

And notice, these were in the
first row and they're in the

first row here.

And this 5 and this 7?

Well, that's this 5 and this
7, and this 5 and this 7.

So, interesting.

This was in column 1 of the
second matrix and this is in

column 1 in the product
matrix.

And similarly, the
6 and the 8.

That's this 6, this 8, and then
it's used here, this 6

and this 8.

And then finally this 3 and the
4 in the brown, so that's

this 3, this 4, and
this 3 and this 4.

And we could of course
simplify all of it.

This was 1 times 5 plus 2 times
7, so that's 5 plus 14,

so this is 19.

This is 1 times 6 plus 2
times 8, so it's 6 plus

16, so that's 22.

This is 3 times 5
plus 4 times 7.

So 15 plus 28, 38, 43-- if my
math is correct-- and then we

have 3 times 6 plus 4 times 8.

So that's 18 plus
32, that's 50.

So now let me ask you-- just so
you know that the product

matrix-- just write
it neatly-- is

19, 22, 43, and 50.

So now let me ask
you a question.

When we did matrix addition we
learned that if I had two

matrices-- it didn't matter what
order we added them in.

So if we said, A plus B-- and
these are matrices; that's why

I'm making them all bold-- we
said this is the same thing as

B plus A, based on how
we define matrix

addition, B plus A.

So now let me ask
you a question.

Is multiplying two matrices,
is AB-- that's just means

we're multiplying A and B-- is
that the same thing as BA?

Does it matter?

Does the order of the matrix
multiplication matter?

And so, I'll tell you right
now, it actually matters a

tremendous amount.

And actually there are certain
matrices that you can add in

one direction that you can't
add in the other-- oh, that

you can multiply in one way but
you can't multiply in the

other order.

And well, I'll show you that
in an example-- but just to

show that this isn't even equal
for most matrices, I

encourage you to multiply these
two matrices in the

other order.

Actually let me do that.

Let me do that really
fast just to prove

the point to you.

So let me delete all
this top part.

Let me delete all of it, and
actually I can delete to this.

So hopefully, you know that when
I multiply this matrix

times this matrix, I got this.

So let me switch the order--
and I'll do it fairly fast

just so as to not bore you-- so
let me switch the order of

the matrix multiplication.

This is good as this is another
example-- so I'm going

to multiply this matrix: 5, 6,
7, 8, times this matrix-- and

I just switched the order; and
we're testing to see whether

order matters-- 1, 2, 3, 4.

Let's do it-- and I won't do all
the colors and everything,

I'll just do it systematically.

I think you just have to see a
lot of examples here-- So this

first term gets its row
information from the first

matrix, column information
from the second matrix.

So it's 5 times 1 plus 6 times
3, so it's 5 times 1--

Let me just write,
actually edit.

I'm going to skip a step here--
OK so it's 5 times 1

plus 6 times 3, plus 18.

What's the second term here?

It's going to be 5 times
2 plus 6 times 4.

So 5 times 2 is 10, plus
6 times 4 is 24.

Right, now we just took
this row times this

column right here.

OK now we're down here for the
set-- so then we're doing this

row, this element right here at
the bottom left is going to

use this row, and this column.

So it's 7 times 1
plus 8 times 3.

8 times 3 is 24.

And then finally, to get this
element we're essentially

multiplying this row times this
column, so it's 7 times 2

is 14, plus 8 times
4, plus 32.

So this is equal to 5
plus 18 is 23, 34.

What's 7 plus 24?

That's 31, 46.

So notice, if we called
this matrix A and this

is matrix B, right?

In the last example, we showed
that A times B is equal to 19,

22, 43, 50.

And we just showed that, well,
if you reverse the order, B

times A is actually this
completely different matrix.

So the order in which
you multiply

matrices completely matters.

So I'm actually running
out of time.

In the next video I going talk
a little bit more about the

types of matrix-- well, one, we
know that order matters--

and in the next video I'll
show that what type of

matrices can be multiplied
by each other.

When we added or subtracted
matrices, we just said, well

they have to have the same
dimensions because you're

adding or subtracting
corresponding terms. But

you'll see with multiplication
it's a little bit different.

And we'll do that in
the next video.

See you soon.