
One of the most important
operations we can do with

matrices is to learn how to
multiply them together. That's

what we're going to do now. And
when we multiply matrices

together, we find that we
combine the elements in the two

matrices in rather a strange
way, and the easiest way to

explain that is by example, so
let's have a look.

Suppose we've got a
row of a matrix 37.

And we want to combine it or
multiply it with a column of

another matrix 29.

And what we do is we combine
these numbers in a rather

strange way. What we do is we
pair off the elements in the row

of the first matrix with the
column with the column in the

Second matrix, and we pair them
off and we multiply the

corresponding elements together.
So we pair off the three with

the two. The seven with the 9.

And we multiply the paired
elements together so we

have 3 * 2.

I mean multiply the
seven with the 9.

And we add the results together.

So we have 3 * 2 which is
6 and 7 * 9, which is 63

and we add them together
and we get the answer 69.

So we have this rather strange
way in which we have combined

the elements in the row of the
first matrix with the column

in the Second matrix.

Let's have a look at another
example. Suppose we have a row

which is 425.

And we're going to learn how to
multiply it with a column 368.

And again, what we do is we pair
the elements off elements in the

row of the first matrix with the
column of the Second matrix. We

have 4 * 3.

2 * 6.

5 * 8.

And we add these products
together, so with 4 * 3 which is

12 two times 6 which is 12 and 5
* 8 which is 40. And if we add

these up will get 12 and 12 is

24. And 40 which is 64.

So this is a rather strange way
in which we've combined the

elements in the first matrix
with the elements in the second

matrix, but it's the basis of
matrix multiplication, as we'll

see shortly Now, suppose we have
to general matrices A&B, say.

And we want to find the
product of these two

matrices. In other words, we
want to multiply A&B

together.

Now suppose that this matrix
A. The first matrix has P,

rossion, Q columns, so it's
a P by Q matrix.

And the second matrix be.
Let's suppose that Scott

are rows and S columns,
so it's an arby S matrix.

Now it turns out that we can
only form this product. We can

only multiply the two matrices

together. If the number of
columns in the first matrix,

which is Q is the same as the
number of rows in the second

matrix, these two numbers have
got to be the same. Q Must equal

R and the reason for that will
become apparent when we start to

do the calculation. But you've
got to be able to pair up the

elements in the first matrix
with the elements in the Second

matrix and will only be able to
do that if the number of columns

in the first is the same as the
number of rows in the second.

When that's the case, we can
actually find the product AB and

the answer is another matrix.
And let's suppose this answer is

matrix C and the size of matrix.
See, we can determine in advance

from the sizes of matrix A&B,
the size of matrix C will be P

by S. So it's got the same
number of rows as the first

matrix and columns as the second
matrix, so this will be an R.

This will be a P by S matrix.

Let's have a look at a specific
example. Suppose we want to

multiply the matrix 37.

45

by the Matrix 29.

And the first question we should
ask ourselves is, do these

matrices have the right size so
that we can actually multiply

them together? Well, this matrix
is a two row two column matrix.

And the second matrix
is 2 rows one column.

And we note that the number of
columns here in the first matrix

is the same as the number of
rows in the second matrix. So

these two numbers are the same,
so we can do this multiplication

and the size of the answer. The
size of the result that will get

is obtained by the number of
rows in the 1st and columns in

the second. So the size of the
answer that we're looking for is

a 2 by 1 matrix. So you see
right at the beginning we can

tell how many. Elements are
going to be in our answer.

There's going to be a number
there, and a number there so

that we have a 2 by 1 matrix.

Now to determine these
numbers, we use the same

operations as we've just
seen. We take the first row

here and we pair the elements
with those in the first

column.

We multiply the paired elements
together and add the result. So

we want 3 * 2, which is 6.

We want 7 * 9 which is 63 and
we add the results together. 6

and 63 is 69, so the element
that goes in the first position

in our answer is 69. That's 3 *
2 at 7 * 9.

The element that's going in this
position here is obtained by

working with the 2nd row and
this first column here.

Again, we pair the elements up 4

* 2. Which is 8th.

5 * 9 which is 45, and
we add the results together. 45

+ 8 is.

53

So the result is 6953, so the
result of multiplying these two

matrices together is another
matrix which is a 2 by 1 matrix

and the elements are obtained in
the way I've just shown you.

Let's have a look at another

example. This time I'm going to
try to multiply together the two

matrices A&B where a is this two
by two Matrix 2453 and B. Is

this two by two Matrix three 6
 1 nine?

And again, the first question we
should ask ourselves is, do

these matrices have the right
size so that we can actually

multiply them together? Well,
matrix a this matrix A is a two

row two column matrix. So that's
two by two.

Matrix B is 2 rows and two
columns, so that's also

two by two.

And you can see that these two
numbers are the same. That is,

the number of columns in the
first is the same as the

number of rows in the second.
So we can perform the matrix

multiplication.

The size of the answer we can
determine right at the start.

The size of the matrix that we
get is determined by the number

here and the number there two by
two. So what we can decide

before we do any calculation at
all is that this answer matrix

is a two by two matrix.

Be 2 rows and two columns,
so we're looking for four

numbers to pop in there.

Let's try and figure out how we
work out, what the answer is.

When we want to find the element
that goes in here, observe that

this is the first row first
column of the answer.

And the number in the first
row first column comes from

looking at the first row and
1st Column here.

If we pair off the elements in
the first row and 1st Column

will have 2 * 3 which is 6.

4 *  1, which is minus four,
and we add them together.

When we come to this element
here, this element is in the

first row, second column.

So we use the first row, second
column in the original matrices.

2 * 6 which is 12 and 4
* 9 Four nines of 36. And

we add those paired
products together.

When we want the element that's
in the 2nd row first column, we

use the 2nd row in the first
matrix and 1st column in the

Second Matrix. Again, pairing
the elements off 5 * 3 is 15.

3 *  1 is minus three.
When we add the paired

elements together.

Finally. The element that's in
the 2nd row, second column of

the answer is obtained by using
the elements in the 2nd row of

the first matrix, second column
of the Second matrix.

5 * 6 which is 30 and 3 *
9, which is 27, and we add the

paired elements together.

So finally, just to tidy it
up, we've got 6 subtract 4

which is 2.

12 + 36, which is 48.

15 subtract 3 which is 12 and 30
+ 27 which is 57, so we can find

the matrix product AB in this
case and the result is a two by

two matrix. Let's have a look
at another example. Suppose

we're asked to find the
product of these two matrices.

And again, we should ask all
these matrices of the

appropriate size. The first
matrix here has two rows, one

column, it's a 2 by 1 matrix.

And the second matrix has two
rows and two columns, so it's a

two by two matrix. Now in this
case, you'll see that the

number of columns in the first
matrix is not the same as the

number of rows in the second
matrix. Those two numbers are

not equal, so we cannot
multiply these matrices

together. We say the product of
these matrices doesn't exist,

so we stop there. We can't
calculate that.

Let's have another example.
Suppose we want to try to

find the product of the
matrices 3214.

With the matrix XY. Now this
is the first example we've

looked at where we've had
symbols rather than numbers in

our matrix, but the operation
the process is, the

calculations are just the
same.

First of all, we should ask can
we multiply these together? Are

they of the right size?

This is a two row two
column matrix.

And this is a two row one
column matrix.

And these numbers are the same
in here the number of columns in

the first is the same as the
number of rows in the second. So

we can actually perform the
matrix multiplication and the

answer we get will be a 2 by 1
matrix, so we know the shape of

the answer. It's a two row one
column matrix, so it looks the

same shape as this one. This one

here. Two rows, one column.

Let's actually work out what the
elements in the answers are.

As before, we take the first row
and pair the elements with the

first column. So it's 3
multiplied by X 2 * y and we add

the resulting products, so we
get three X + 2 Y.

The element that's down here,
which is in the 2nd row, first

column of our answer, is
obtained by using the 2nd row in

the first matrix and the first

column here. Multiply the pad
elements together and add so

it's 1 * X which is X 4 * y,
which is 4 Y and we add the

products together and we get X +
4 Y. So the result we found is a

two row one column matrix.

In this case, the answers got
symbols in as well, but that's

the result of finding the
product of these two matrices.

Now we can go on and look at
more examples and trying to

find products of matrices of
different sizes and shapes,

and we'll do some more of that
in the next video.