WEBVTT

00:00:05.130 --> 00:00:08.300
One of the most important
operations we can do with

00:00:08.300 --> 00:00:11.470
matrices is to learn how to
multiply them together. That's

00:00:11.470 --> 00:00:14.957
what we're going to do now. And
when we multiply matrices

00:00:14.957 --> 00:00:18.444
together, we find that we
combine the elements in the two

00:00:18.444 --> 00:00:21.931
matrices in rather a strange
way, and the easiest way to

00:00:21.931 --> 00:00:25.101
explain that is by example, so
let's have a look.

00:00:26.040 --> 00:00:29.793
Suppose we've got a
row of a matrix 37.

00:00:31.650 --> 00:00:36.057
And we want to combine it or
multiply it with a column of

00:00:36.057 --> 00:00:37.074
another matrix 29.

00:00:38.790 --> 00:00:42.222
And what we do is we combine
these numbers in a rather

00:00:42.222 --> 00:00:46.226
strange way. What we do is we
pair off the elements in the row

00:00:46.226 --> 00:00:49.658
of the first matrix with the
column with the column in the

00:00:49.658 --> 00:00:52.804
Second matrix, and we pair them
off and we multiply the

00:00:52.804 --> 00:00:55.664
corresponding elements together.
So we pair off the three with

00:00:55.664 --> 00:00:58.460
the two. The seven with the 9.

00:00:59.250 --> 00:01:02.310
And we multiply the paired
elements together so we

00:01:02.310 --> 00:01:03.670
have 3 * 2.

00:01:06.290 --> 00:01:08.546
I mean multiply the
seven with the 9.

00:01:10.750 --> 00:01:12.388
And we add the results together.

00:01:13.390 --> 00:01:18.542
So we have 3 * 2 which is
6 and 7 * 9, which is 63

00:01:18.542 --> 00:01:22.084
and we add them together
and we get the answer 69.

00:01:25.560 --> 00:01:29.004
So we have this rather strange
way in which we have combined

00:01:29.004 --> 00:01:32.448
the elements in the row of the
first matrix with the column

00:01:32.448 --> 00:01:33.596
in the Second matrix.

00:01:34.620 --> 00:01:38.652
Let's have a look at another
example. Suppose we have a row

00:01:38.652 --> 00:01:39.660
which is 425.

00:01:41.020 --> 00:01:45.947
And we're going to learn how to
multiply it with a column 368.

00:01:48.670 --> 00:01:53.416
And again, what we do is we pair
the elements off elements in the

00:01:53.416 --> 00:01:57.823
row of the first matrix with the
column of the Second matrix. We

00:01:57.823 --> 00:01:59.179
have 4 * 3.

00:02:03.200 --> 00:02:05.060
2 * 6.

00:02:07.370 --> 00:02:09.230
5 * 8.

00:02:12.050 --> 00:02:17.003
And we add these products
together, so with 4 * 3 which is

00:02:17.003 --> 00:02:23.861
12 two times 6 which is 12 and 5
* 8 which is 40. And if we add

00:02:23.861 --> 00:02:26.909
these up will get 12 and 12 is

00:02:26.909 --> 00:02:29.970
24. And 40 which is 64.

00:02:34.190 --> 00:02:37.742
So this is a rather strange way
in which we've combined the

00:02:37.742 --> 00:02:40.998
elements in the first matrix
with the elements in the second

00:02:40.998 --> 00:02:43.958
matrix, but it's the basis of
matrix multiplication, as we'll

00:02:43.958 --> 00:02:47.214
see shortly Now, suppose we have
to general matrices A&B, say.

00:02:48.900 --> 00:02:52.050
And we want to find the
product of these two

00:02:52.050 --> 00:02:54.885
matrices. In other words, we
want to multiply A&B

00:02:54.885 --> 00:02:55.200
together.

00:02:56.470 --> 00:03:00.694
Now suppose that this matrix
A. The first matrix has P,

00:03:00.694 --> 00:03:04.534
rossion, Q columns, so it's
a P by Q matrix.

00:03:06.270 --> 00:03:09.663
And the second matrix be.
Let's suppose that Scott

00:03:09.663 --> 00:03:13.810
are rows and S columns,
so it's an arby S matrix.

00:03:15.820 --> 00:03:19.746
Now it turns out that we can
only form this product. We can

00:03:19.746 --> 00:03:21.256
only multiply the two matrices

00:03:21.256 --> 00:03:24.692
together. If the number of
columns in the first matrix,

00:03:24.692 --> 00:03:28.304
which is Q is the same as the
number of rows in the second

00:03:28.304 --> 00:03:31.658
matrix, these two numbers have
got to be the same. Q Must equal

00:03:31.658 --> 00:03:35.012
R and the reason for that will
become apparent when we start to

00:03:35.012 --> 00:03:38.366
do the calculation. But you've
got to be able to pair up the

00:03:38.366 --> 00:03:41.204
elements in the first matrix
with the elements in the Second

00:03:41.204 --> 00:03:44.816
matrix and will only be able to
do that if the number of columns

00:03:44.816 --> 00:03:48.428
in the first is the same as the
number of rows in the second.

00:03:49.960 --> 00:03:54.100
When that's the case, we can
actually find the product AB and

00:03:54.100 --> 00:03:57.895
the answer is another matrix.
And let's suppose this answer is

00:03:57.895 --> 00:04:02.380
matrix C and the size of matrix.
See, we can determine in advance

00:04:02.380 --> 00:04:07.210
from the sizes of matrix A&B,
the size of matrix C will be P

00:04:07.210 --> 00:04:11.967
by S. So it's got the same
number of rows as the first

00:04:11.967 --> 00:04:16.088
matrix and columns as the second
matrix, so this will be an R.

00:04:16.088 --> 00:04:18.624
This will be a P by S matrix.

00:04:20.690 --> 00:04:25.622
Let's have a look at a specific
example. Suppose we want to

00:04:25.622 --> 00:04:27.266
multiply the matrix 37.

00:04:27.990 --> 00:04:29.670
45

00:04:32.040 --> 00:04:34.508
by the Matrix 29.

00:04:35.920 --> 00:04:39.077
And the first question we should
ask ourselves is, do these

00:04:39.077 --> 00:04:42.234
matrices have the right size so
that we can actually multiply

00:04:42.234 --> 00:04:47.470
them together? Well, this matrix
is a two row two column matrix.

00:04:50.350 --> 00:04:53.374
And the second matrix
is 2 rows one column.

00:04:55.960 --> 00:04:59.951
And we note that the number of
columns here in the first matrix

00:04:59.951 --> 00:05:03.942
is the same as the number of
rows in the second matrix. So

00:05:03.942 --> 00:05:07.626
these two numbers are the same,
so we can do this multiplication

00:05:07.626 --> 00:05:11.924
and the size of the answer. The
size of the result that will get

00:05:11.924 --> 00:05:15.915
is obtained by the number of
rows in the 1st and columns in

00:05:15.915 --> 00:05:19.906
the second. So the size of the
answer that we're looking for is

00:05:19.906 --> 00:05:24.204
a 2 by 1 matrix. So you see
right at the beginning we can

00:05:24.204 --> 00:05:27.046
tell how many. Elements are
going to be in our answer.

00:05:27.046 --> 00:05:29.770
There's going to be a number
there, and a number there so

00:05:29.770 --> 00:05:31.586
that we have a 2 by 1 matrix.

00:05:32.920 --> 00:05:36.169
Now to determine these
numbers, we use the same

00:05:36.169 --> 00:05:39.779
operations as we've just
seen. We take the first row

00:05:39.779 --> 00:05:43.750
here and we pair the elements
with those in the first

00:05:43.750 --> 00:05:44.111
column.

00:05:45.380 --> 00:05:50.308
We multiply the paired elements
together and add the result. So

00:05:50.308 --> 00:05:53.892
we want 3 * 2, which is 6.

00:05:55.540 --> 00:06:02.125
We want 7 * 9 which is 63 and
we add the results together. 6

00:06:02.125 --> 00:06:07.832
and 63 is 69, so the element
that goes in the first position

00:06:07.832 --> 00:06:13.539
in our answer is 69. That's 3 *
2 at 7 * 9.

00:06:15.380 --> 00:06:19.043
The element that's going in this
position here is obtained by

00:06:19.043 --> 00:06:22.373
working with the 2nd row and
this first column here.

00:06:23.120 --> 00:06:25.871
Again, we pair the elements up 4

00:06:25.871 --> 00:06:28.670
* 2. Which is 8th.

00:06:29.950 --> 00:06:36.541
5 * 9 which is 45, and
we add the results together. 45

00:06:36.541 --> 00:06:38.062
+ 8 is.

00:06:39.600 --> 00:06:40.610
53

00:06:43.270 --> 00:06:46.810
So the result is 6953, so the
result of multiplying these two

00:06:46.810 --> 00:06:50.350
matrices together is another
matrix which is a 2 by 1 matrix

00:06:50.350 --> 00:06:53.890
and the elements are obtained in
the way I've just shown you.

00:06:54.490 --> 00:06:56.116
Let's have a look at another

00:06:56.116 --> 00:07:01.054
example. This time I'm going to
try to multiply together the two

00:07:01.054 --> 00:07:06.010
matrices A&B where a is this two
by two Matrix 2453 and B. Is

00:07:06.010 --> 00:07:09.550
this two by two Matrix three 6
-- 1 nine?

00:07:10.380 --> 00:07:13.339
And again, the first question we
should ask ourselves is, do

00:07:13.339 --> 00:07:16.298
these matrices have the right
size so that we can actually

00:07:16.298 --> 00:07:21.893
multiply them together? Well,
matrix a this matrix A is a two

00:07:21.893 --> 00:07:25.916
row two column matrix. So that's
two by two.

00:07:27.550 --> 00:07:30.960
Matrix B is 2 rows and two
columns, so that's also

00:07:30.960 --> 00:07:31.890
two by two.

00:07:33.290 --> 00:07:36.631
And you can see that these two
numbers are the same. That is,

00:07:36.631 --> 00:07:39.715
the number of columns in the
first is the same as the

00:07:39.715 --> 00:07:42.799
number of rows in the second.
So we can perform the matrix

00:07:42.799 --> 00:07:43.056
multiplication.

00:07:44.280 --> 00:07:47.604
The size of the answer we can
determine right at the start.

00:07:47.604 --> 00:07:51.205
The size of the matrix that we
get is determined by the number

00:07:51.205 --> 00:07:54.806
here and the number there two by
two. So what we can decide

00:07:54.806 --> 00:07:58.130
before we do any calculation at
all is that this answer matrix

00:07:58.130 --> 00:07:59.792
is a two by two matrix.

00:08:00.470 --> 00:08:03.198
Be 2 rows and two columns,
so we're looking for four

00:08:03.198 --> 00:08:04.438
numbers to pop in there.

00:08:05.970 --> 00:08:09.389
Let's try and figure out how we
work out, what the answer is.

00:08:10.940 --> 00:08:15.269
When we want to find the element
that goes in here, observe that

00:08:15.269 --> 00:08:18.599
this is the first row first
column of the answer.

00:08:20.140 --> 00:08:23.781
And the number in the first
row first column comes from

00:08:23.781 --> 00:08:26.760
looking at the first row and
1st Column here.

00:08:28.370 --> 00:08:32.491
If we pair off the elements in
the first row and 1st Column

00:08:32.491 --> 00:08:35.027
will have 2 * 3 which is 6.

00:08:36.410 --> 00:08:40.960
4 * -- 1, which is minus four,
and we add them together.

00:08:44.870 --> 00:08:49.430
When we come to this element
here, this element is in the

00:08:49.430 --> 00:08:50.950
first row, second column.

00:08:51.610 --> 00:08:55.558
So we use the first row, second
column in the original matrices.

00:08:56.270 --> 00:09:03.155
2 * 6 which is 12 and 4
* 9 Four nines of 36. And

00:09:03.155 --> 00:09:05.909
we add those paired
products together.

00:09:08.610 --> 00:09:13.303
When we want the element that's
in the 2nd row first column, we

00:09:13.303 --> 00:09:17.996
use the 2nd row in the first
matrix and 1st column in the

00:09:17.996 --> 00:09:22.328
Second Matrix. Again, pairing
the elements off 5 * 3 is 15.

00:09:23.790 --> 00:09:28.122
3 * -- 1 is minus three.
When we add the paired

00:09:28.122 --> 00:09:28.844
elements together.

00:09:30.730 --> 00:09:35.570
Finally. The element that's in
the 2nd row, second column of

00:09:35.570 --> 00:09:39.860
the answer is obtained by using
the elements in the 2nd row of

00:09:39.860 --> 00:09:42.830
the first matrix, second column
of the Second matrix.

00:09:43.810 --> 00:09:50.984
5 * 6 which is 30 and 3 *
9, which is 27, and we add the

00:09:50.984 --> 00:09:52.250
paired elements together.

00:09:53.480 --> 00:09:57.596
So finally, just to tidy it
up, we've got 6 subtract 4

00:09:57.596 --> 00:09:58.625
which is 2.

00:10:00.540 --> 00:10:03.606
12 + 36, which is 48.

00:10:05.750 --> 00:10:12.533
15 subtract 3 which is 12 and 30
+ 27 which is 57, so we can find

00:10:12.533 --> 00:10:18.119
the matrix product AB in this
case and the result is a two by

00:10:18.119 --> 00:10:21.874
two matrix. Let's have a look
at another example. Suppose

00:10:21.874 --> 00:10:24.654
we're asked to find the
product of these two matrices.

00:10:25.990 --> 00:10:29.270
And again, we should ask all
these matrices of the

00:10:29.270 --> 00:10:32.550
appropriate size. The first
matrix here has two rows, one

00:10:32.550 --> 00:10:34.846
column, it's a 2 by 1 matrix.

00:10:35.760 --> 00:10:40.154
And the second matrix has two
rows and two columns, so it's a

00:10:40.154 --> 00:10:44.210
two by two matrix. Now in this
case, you'll see that the

00:10:44.210 --> 00:10:48.604
number of columns in the first
matrix is not the same as the

00:10:48.604 --> 00:10:52.322
number of rows in the second
matrix. Those two numbers are

00:10:52.322 --> 00:10:55.026
not equal, so we cannot
multiply these matrices

00:10:55.026 --> 00:10:58.406
together. We say the product of
these matrices doesn't exist,

00:10:58.406 --> 00:11:01.110
so we stop there. We can't
calculate that.

00:11:02.540 --> 00:11:07.950
Let's have another example.
Suppose we want to try to

00:11:07.950 --> 00:11:11.737
find the product of the
matrices 3214.

00:11:14.180 --> 00:11:17.700
With the matrix XY. Now this
is the first example we've

00:11:17.700 --> 00:11:20.900
looked at where we've had
symbols rather than numbers in

00:11:20.900 --> 00:11:23.780
our matrix, but the operation
the process is, the

00:11:23.780 --> 00:11:25.380
calculations are just the
same.

00:11:26.430 --> 00:11:29.466
First of all, we should ask can
we multiply these together? Are

00:11:29.466 --> 00:11:30.731
they of the right size?

00:11:31.660 --> 00:11:34.188
This is a two row two
column matrix.

00:11:35.970 --> 00:11:38.562
And this is a two row one
column matrix.

00:11:40.030 --> 00:11:43.969
And these numbers are the same
in here the number of columns in

00:11:43.969 --> 00:11:48.211
the first is the same as the
number of rows in the second. So

00:11:48.211 --> 00:11:50.938
we can actually perform the
matrix multiplication and the

00:11:50.938 --> 00:11:55.786
answer we get will be a 2 by 1
matrix, so we know the shape of

00:11:55.786 --> 00:11:59.725
the answer. It's a two row one
column matrix, so it looks the

00:11:59.725 --> 00:12:01.846
same shape as this one. This one

00:12:01.846 --> 00:12:04.348
here. Two rows, one column.

00:12:06.340 --> 00:12:09.750
Let's actually work out what the
elements in the answers are.

00:12:10.600 --> 00:12:15.176
As before, we take the first row
and pair the elements with the

00:12:15.176 --> 00:12:21.620
first column. So it's 3
multiplied by X 2 * y and we add

00:12:21.620 --> 00:12:26.130
the resulting products, so we
get three X + 2 Y.

00:12:33.190 --> 00:12:37.138
The element that's down here,
which is in the 2nd row, first

00:12:37.138 --> 00:12:41.086
column of our answer, is
obtained by using the 2nd row in

00:12:41.086 --> 00:12:43.060
the first matrix and the first

00:12:43.060 --> 00:12:47.612
column here. Multiply the pad
elements together and add so

00:12:47.612 --> 00:12:54.344
it's 1 * X which is X 4 * y,
which is 4 Y and we add the

00:12:54.344 --> 00:13:00.328
products together and we get X +
4 Y. So the result we found is a

00:13:00.328 --> 00:13:02.198
two row one column matrix.

00:13:03.770 --> 00:13:06.914
In this case, the answers got
symbols in as well, but that's

00:13:06.914 --> 00:13:09.534
the result of finding the
product of these two matrices.

00:13:09.534 --> 00:13:12.940
Now we can go on and look at
more examples and trying to

00:13:12.940 --> 00:13:15.298
find products of matrices of
different sizes and shapes,

00:13:15.298 --> 00:13:18.180
and we'll do some more of that
in the next video.