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.