0:00:05.130,0:00:08.300
One of the most important[br]operations we can do with

0:00:08.300,0:00:11.470
matrices is to learn how to[br]multiply them together. That's

0:00:11.470,0:00:14.957
what we're going to do now. And[br]when we multiply matrices

0:00:14.957,0:00:18.444
together, we find that we[br]combine the elements in the two

0:00:18.444,0:00:21.931
matrices in rather a strange[br]way, and the easiest way to

0:00:21.931,0:00:25.101
explain that is by example, so[br]let's have a look.

0:00:26.040,0:00:29.793
Suppose we've got a[br]row of a matrix 37.

0:00:31.650,0:00:36.057
And we want to combine it or[br]multiply it with a column of

0:00:36.057,0:00:37.074
another matrix 29.

0:00:38.790,0:00:42.222
And what we do is we combine[br]these numbers in a rather

0:00:42.222,0:00:46.226
strange way. What we do is we[br]pair off the elements in the row

0:00:46.226,0:00:49.658
of the first matrix with the[br]column with the column in the

0:00:49.658,0:00:52.804
Second matrix, and we pair them[br]off and we multiply the

0:00:52.804,0:00:55.664
corresponding elements together.[br]So we pair off the three with

0:00:55.664,0:00:58.460
the two. The seven with the 9.

0:00:59.250,0:01:02.310
And we multiply the paired[br]elements together so we

0:01:02.310,0:01:03.670
have 3 * 2.

0:01:06.290,0:01:08.546
I mean multiply the[br]seven with the 9.

0:01:10.750,0:01:12.388
And we add the results together.

0:01:13.390,0:01:18.542
So we have 3 * 2 which is[br]6 and 7 * 9, which is 63

0:01:18.542,0:01:22.084
and we add them together[br]and we get the answer 69.

0:01:25.560,0:01:29.004
So we have this rather strange[br]way in which we have combined

0:01:29.004,0:01:32.448
the elements in the row of the[br]first matrix with the column

0:01:32.448,0:01:33.596
in the Second matrix.

0:01:34.620,0:01:38.652
Let's have a look at another[br]example. Suppose we have a row

0:01:38.652,0:01:39.660
which is 425.

0:01:41.020,0:01:45.947
And we're going to learn how to[br]multiply it with a column 368.

0:01:48.670,0:01:53.416
And again, what we do is we pair[br]the elements off elements in the

0:01:53.416,0:01:57.823
row of the first matrix with the[br]column of the Second matrix. We

0:01:57.823,0:01:59.179
have 4 * 3.

0:02:03.200,0:02:05.060
2 * 6.

0:02:07.370,0:02:09.230
5 * 8.

0:02:12.050,0:02:17.003
And we add these products[br]together, so with 4 * 3 which is

0:02:17.003,0:02:23.861
12 two times 6 which is 12 and 5[br]* 8 which is 40. And if we add

0:02:23.861,0:02:26.909
these up will get 12 and 12 is

0:02:26.909,0:02:29.970
24. And 40 which is 64.

0:02:34.190,0:02:37.742
So this is a rather strange way[br]in which we've combined the

0:02:37.742,0:02:40.998
elements in the first matrix[br]with the elements in the second

0:02:40.998,0:02:43.958
matrix, but it's the basis of[br]matrix multiplication, as we'll

0:02:43.958,0:02:47.214
see shortly Now, suppose we have[br]to general matrices A&B, say.

0:02:48.900,0:02:52.050
And we want to find the[br]product of these two

0:02:52.050,0:02:54.885
matrices. In other words, we[br]want to multiply A&B

0:02:54.885,0:02:55.200
together.

0:02:56.470,0:03:00.694
Now suppose that this matrix[br]A. The first matrix has P,

0:03:00.694,0:03:04.534
rossion, Q columns, so it's[br]a P by Q matrix.

0:03:06.270,0:03:09.663
And the second matrix be.[br]Let's suppose that Scott

0:03:09.663,0:03:13.810
are rows and S columns,[br]so it's an arby S matrix.

0:03:15.820,0:03:19.746
Now it turns out that we can[br]only form this product. We can

0:03:19.746,0:03:21.256
only multiply the two matrices

0:03:21.256,0:03:24.692
together. If the number of[br]columns in the first matrix,

0:03:24.692,0:03:28.304
which is Q is the same as the[br]number of rows in the second

0:03:28.304,0:03:31.658
matrix, these two numbers have[br]got to be the same. Q Must equal

0:03:31.658,0:03:35.012
R and the reason for that will[br]become apparent when we start to

0:03:35.012,0:03:38.366
do the calculation. But you've[br]got to be able to pair up the

0:03:38.366,0:03:41.204
elements in the first matrix[br]with the elements in the Second

0:03:41.204,0:03:44.816
matrix and will only be able to[br]do that if the number of columns

0:03:44.816,0:03:48.428
in the first is the same as the[br]number of rows in the second.

0:03:49.960,0:03:54.100
When that's the case, we can[br]actually find the product AB and

0:03:54.100,0:03:57.895
the answer is another matrix.[br]And let's suppose this answer is

0:03:57.895,0:04:02.380
matrix C and the size of matrix.[br]See, we can determine in advance

0:04:02.380,0:04:07.210
from the sizes of matrix A&B,[br]the size of matrix C will be P

0:04:07.210,0:04:11.967
by S. So it's got the same[br]number of rows as the first

0:04:11.967,0:04:16.088
matrix and columns as the second[br]matrix, so this will be an R.

0:04:16.088,0:04:18.624
This will be a P by S matrix.

0:04:20.690,0:04:25.622
Let's have a look at a specific[br]example. Suppose we want to

0:04:25.622,0:04:27.266
multiply the matrix 37.

0:04:27.990,0:04:29.670
45

0:04:32.040,0:04:34.508
by the Matrix 29.

0:04:35.920,0:04:39.077
And the first question we should[br]ask ourselves is, do these

0:04:39.077,0:04:42.234
matrices have the right size so[br]that we can actually multiply

0:04:42.234,0:04:47.470
them together? Well, this matrix[br]is a two row two column matrix.

0:04:50.350,0:04:53.374
And the second matrix[br]is 2 rows one column.

0:04:55.960,0:04:59.951
And we note that the number of[br]columns here in the first matrix

0:04:59.951,0:05:03.942
is the same as the number of[br]rows in the second matrix. So

0:05:03.942,0:05:07.626
these two numbers are the same,[br]so we can do this multiplication

0:05:07.626,0:05:11.924
and the size of the answer. The[br]size of the result that will get

0:05:11.924,0:05:15.915
is obtained by the number of[br]rows in the 1st and columns in

0:05:15.915,0:05:19.906
the second. So the size of the[br]answer that we're looking for is

0:05:19.906,0:05:24.204
a 2 by 1 matrix. So you see[br]right at the beginning we can

0:05:24.204,0:05:27.046
tell how many. Elements are[br]going to be in our answer.

0:05:27.046,0:05:29.770
There's going to be a number[br]there, and a number there so

0:05:29.770,0:05:31.586
that we have a 2 by 1 matrix.

0:05:32.920,0:05:36.169
Now to determine these[br]numbers, we use the same

0:05:36.169,0:05:39.779
operations as we've just[br]seen. We take the first row

0:05:39.779,0:05:43.750
here and we pair the elements[br]with those in the first

0:05:43.750,0:05:44.111
column.

0:05:45.380,0:05:50.308
We multiply the paired elements[br]together and add the result. So

0:05:50.308,0:05:53.892
we want 3 * 2, which is 6.

0:05:55.540,0:06:02.125
We want 7 * 9 which is 63 and[br]we add the results together. 6

0:06:02.125,0:06:07.832
and 63 is 69, so the element[br]that goes in the first position

0:06:07.832,0:06:13.539
in our answer is 69. That's 3 *[br]2 at 7 * 9.

0:06:15.380,0:06:19.043
The element that's going in this[br]position here is obtained by

0:06:19.043,0:06:22.373
working with the 2nd row and[br]this first column here.

0:06:23.120,0:06:25.871
Again, we pair the elements up 4

0:06:25.871,0:06:28.670
* 2. Which is 8th.

0:06:29.950,0:06:36.541
5 * 9 which is 45, and[br]we add the results together. 45

0:06:36.541,0:06:38.062
+ 8 is.

0:06:39.600,0:06:40.610
53

0:06:43.270,0:06:46.810
So the result is 6953, so the[br]result of multiplying these two

0:06:46.810,0:06:50.350
matrices together is another[br]matrix which is a 2 by 1 matrix

0:06:50.350,0:06:53.890
and the elements are obtained in[br]the way I've just shown you.

0:06:54.490,0:06:56.116
Let's have a look at another

0:06:56.116,0:07:01.054
example. This time I'm going to[br]try to multiply together the two

0:07:01.054,0:07:06.010
matrices A&B where a is this two[br]by two Matrix 2453 and B. Is

0:07:06.010,0:07:09.550
this two by two Matrix three 6[br]-- 1 nine?

0:07:10.380,0:07:13.339
And again, the first question we[br]should ask ourselves is, do

0:07:13.339,0:07:16.298
these matrices have the right[br]size so that we can actually

0:07:16.298,0:07:21.893
multiply them together? Well,[br]matrix a this matrix A is a two

0:07:21.893,0:07:25.916
row two column matrix. So that's[br]two by two.

0:07:27.550,0:07:30.960
Matrix B is 2 rows and two[br]columns, so that's also

0:07:30.960,0:07:31.890
two by two.

0:07:33.290,0:07:36.631
And you can see that these two[br]numbers are the same. That is,

0:07:36.631,0:07:39.715
the number of columns in the[br]first is the same as the

0:07:39.715,0:07:42.799
number of rows in the second.[br]So we can perform the matrix

0:07:42.799,0:07:43.056
multiplication.

0:07:44.280,0:07:47.604
The size of the answer we can[br]determine right at the start.

0:07:47.604,0:07:51.205
The size of the matrix that we[br]get is determined by the number

0:07:51.205,0:07:54.806
here and the number there two by[br]two. So what we can decide

0:07:54.806,0:07:58.130
before we do any calculation at[br]all is that this answer matrix

0:07:58.130,0:07:59.792
is a two by two matrix.

0:08:00.470,0:08:03.198
Be 2 rows and two columns,[br]so we're looking for four

0:08:03.198,0:08:04.438
numbers to pop in there.

0:08:05.970,0:08:09.389
Let's try and figure out how we[br]work out, what the answer is.

0:08:10.940,0:08:15.269
When we want to find the element[br]that goes in here, observe that

0:08:15.269,0:08:18.599
this is the first row first[br]column of the answer.

0:08:20.140,0:08:23.781
And the number in the first[br]row first column comes from

0:08:23.781,0:08:26.760
looking at the first row and[br]1st Column here.

0:08:28.370,0:08:32.491
If we pair off the elements in[br]the first row and 1st Column

0:08:32.491,0:08:35.027
will have 2 * 3 which is 6.

0:08:36.410,0:08:40.960
4 * -- 1, which is minus four,[br]and we add them together.

0:08:44.870,0:08:49.430
When we come to this element[br]here, this element is in the

0:08:49.430,0:08:50.950
first row, second column.

0:08:51.610,0:08:55.558
So we use the first row, second[br]column in the original matrices.

0:08:56.270,0:09:03.155
2 * 6 which is 12 and 4[br]* 9 Four nines of 36. And

0:09:03.155,0:09:05.909
we add those paired[br]products together.

0:09:08.610,0:09:13.303
When we want the element that's[br]in the 2nd row first column, we

0:09:13.303,0:09:17.996
use the 2nd row in the first[br]matrix and 1st column in the

0:09:17.996,0:09:22.328
Second Matrix. Again, pairing[br]the elements off 5 * 3 is 15.

0:09:23.790,0:09:28.122
3 * -- 1 is minus three.[br]When we add the paired

0:09:28.122,0:09:28.844
elements together.

0:09:30.730,0:09:35.570
Finally. The element that's in[br]the 2nd row, second column of

0:09:35.570,0:09:39.860
the answer is obtained by using[br]the elements in the 2nd row of

0:09:39.860,0:09:42.830
the first matrix, second column[br]of the Second matrix.

0:09:43.810,0:09:50.984
5 * 6 which is 30 and 3 *[br]9, which is 27, and we add the

0:09:50.984,0:09:52.250
paired elements together.

0:09:53.480,0:09:57.596
So finally, just to tidy it[br]up, we've got 6 subtract 4

0:09:57.596,0:09:58.625
which is 2.

0:10:00.540,0:10:03.606
12 + 36, which is 48.

0:10:05.750,0:10:12.533
15 subtract 3 which is 12 and 30[br]+ 27 which is 57, so we can find

0:10:12.533,0:10:18.119
the matrix product AB in this[br]case and the result is a two by

0:10:18.119,0:10:21.874
two matrix. Let's have a look[br]at another example. Suppose

0:10:21.874,0:10:24.654
we're asked to find the[br]product of these two matrices.

0:10:25.990,0:10:29.270
And again, we should ask all[br]these matrices of the

0:10:29.270,0:10:32.550
appropriate size. The first[br]matrix here has two rows, one

0:10:32.550,0:10:34.846
column, it's a 2 by 1 matrix.

0:10:35.760,0:10:40.154
And the second matrix has two[br]rows and two columns, so it's a

0:10:40.154,0:10:44.210
two by two matrix. Now in this[br]case, you'll see that the

0:10:44.210,0:10:48.604
number of columns in the first[br]matrix is not the same as the

0:10:48.604,0:10:52.322
number of rows in the second[br]matrix. Those two numbers are

0:10:52.322,0:10:55.026
not equal, so we cannot[br]multiply these matrices

0:10:55.026,0:10:58.406
together. We say the product of[br]these matrices doesn't exist,

0:10:58.406,0:11:01.110
so we stop there. We can't[br]calculate that.

0:11:02.540,0:11:07.950
Let's have another example.[br]Suppose we want to try to

0:11:07.950,0:11:11.737
find the product of the[br]matrices 3214.

0:11:14.180,0:11:17.700
With the matrix XY. Now this[br]is the first example we've

0:11:17.700,0:11:20.900
looked at where we've had[br]symbols rather than numbers in

0:11:20.900,0:11:23.780
our matrix, but the operation[br]the process is, the

0:11:23.780,0:11:25.380
calculations are just the[br]same.

0:11:26.430,0:11:29.466
First of all, we should ask can[br]we multiply these together? Are

0:11:29.466,0:11:30.731
they of the right size?

0:11:31.660,0:11:34.188
This is a two row two[br]column matrix.

0:11:35.970,0:11:38.562
And this is a two row one[br]column matrix.

0:11:40.030,0:11:43.969
And these numbers are the same[br]in here the number of columns in

0:11:43.969,0:11:48.211
the first is the same as the[br]number of rows in the second. So

0:11:48.211,0:11:50.938
we can actually perform the[br]matrix multiplication and the

0:11:50.938,0:11:55.786
answer we get will be a 2 by 1[br]matrix, so we know the shape of

0:11:55.786,0:11:59.725
the answer. It's a two row one[br]column matrix, so it looks the

0:11:59.725,0:12:01.846
same shape as this one. This one

0:12:01.846,0:12:04.348
here. Two rows, one column.

0:12:06.340,0:12:09.750
Let's actually work out what the[br]elements in the answers are.

0:12:10.600,0:12:15.176
As before, we take the first row[br]and pair the elements with the

0:12:15.176,0:12:21.620
first column. So it's 3[br]multiplied by X 2 * y and we add

0:12:21.620,0:12:26.130
the resulting products, so we[br]get three X + 2 Y.

0:12:33.190,0:12:37.138
The element that's down here,[br]which is in the 2nd row, first

0:12:37.138,0:12:41.086
column of our answer, is[br]obtained by using the 2nd row in

0:12:41.086,0:12:43.060
the first matrix and the first

0:12:43.060,0:12:47.612
column here. Multiply the pad[br]elements together and add so

0:12:47.612,0:12:54.344
it's 1 * X which is X 4 * y,[br]which is 4 Y and we add the

0:12:54.344,0:13:00.328
products together and we get X +[br]4 Y. So the result we found is a

0:13:00.328,0:13:02.198
two row one column matrix.

0:13:03.770,0:13:06.914
In this case, the answers got[br]symbols in as well, but that's

0:13:06.914,0:13:09.534
the result of finding the[br]product of these two matrices.

0:13:09.534,0:13:12.940
Now we can go on and look at[br]more examples and trying to

0:13:12.940,0:13:15.298
find products of matrices of[br]different sizes and shapes,

0:13:15.298,0:13:18.180
and we'll do some more of that[br]in the next video.