1
00:00:05,130 --> 00:00:08,300
One of the most important
operations we can do with

2
00:00:08,300 --> 00:00:11,470
matrices is to learn how to
multiply them together. That's

3
00:00:11,470 --> 00:00:14,957
what we're going to do now. And
when we multiply matrices

4
00:00:14,957 --> 00:00:18,444
together, we find that we
combine the elements in the two

5
00:00:18,444 --> 00:00:21,931
matrices in rather a strange
way, and the easiest way to

6
00:00:21,931 --> 00:00:25,101
explain that is by example, so
let's have a look.

7
00:00:26,040 --> 00:00:29,793
Suppose we've got a
row of a matrix 37.

8
00:00:31,650 --> 00:00:36,057
And we want to combine it or
multiply it with a column of

9
00:00:36,057 --> 00:00:37,074
another matrix 29.

10
00:00:38,790 --> 00:00:42,222
And what we do is we combine
these numbers in a rather

11
00:00:42,222 --> 00:00:46,226
strange way. What we do is we
pair off the elements in the row

12
00:00:46,226 --> 00:00:49,658
of the first matrix with the
column with the column in the

13
00:00:49,658 --> 00:00:52,804
Second matrix, and we pair them
off and we multiply the

14
00:00:52,804 --> 00:00:55,664
corresponding elements together.
So we pair off the three with

15
00:00:55,664 --> 00:00:58,460
the two. The seven with the 9.

16
00:00:59,250 --> 00:01:02,310
And we multiply the paired
elements together so we

17
00:01:02,310 --> 00:01:03,670
have 3 * 2.

18
00:01:06,290 --> 00:01:08,546
I mean multiply the
seven with the 9.

19
00:01:10,750 --> 00:01:12,388
And we add the results together.

20
00:01:13,390 --> 00:01:18,542
So we have 3 * 2 which is
6 and 7 * 9, which is 63

21
00:01:18,542 --> 00:01:22,084
and we add them together
and we get the answer 69.

22
00:01:25,560 --> 00:01:29,004
So we have this rather strange
way in which we have combined

23
00:01:29,004 --> 00:01:32,448
the elements in the row of the
first matrix with the column

24
00:01:32,448 --> 00:01:33,596
in the Second matrix.

25
00:01:34,620 --> 00:01:38,652
Let's have a look at another
example. Suppose we have a row

26
00:01:38,652 --> 00:01:39,660
which is 425.

27
00:01:41,020 --> 00:01:45,947
And we're going to learn how to
multiply it with a column 368.

28
00:01:48,670 --> 00:01:53,416
And again, what we do is we pair
the elements off elements in the

29
00:01:53,416 --> 00:01:57,823
row of the first matrix with the
column of the Second matrix. We

30
00:01:57,823 --> 00:01:59,179
have 4 * 3.

31
00:02:03,200 --> 00:02:05,060
2 * 6.

32
00:02:07,370 --> 00:02:09,230
5 * 8.

33
00:02:12,050 --> 00:02:17,003
And we add these products
together, so with 4 * 3 which is

34
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

35
00:02:23,861 --> 00:02:26,909
these up will get 12 and 12 is

36
00:02:26,909 --> 00:02:29,970
24. And 40 which is 64.

37
00:02:34,190 --> 00:02:37,742
So this is a rather strange way
in which we've combined the

38
00:02:37,742 --> 00:02:40,998
elements in the first matrix
with the elements in the second

39
00:02:40,998 --> 00:02:43,958
matrix, but it's the basis of
matrix multiplication, as we'll

40
00:02:43,958 --> 00:02:47,214
see shortly Now, suppose we have
to general matrices A&B, say.

41
00:02:48,900 --> 00:02:52,050
And we want to find the
product of these two

42
00:02:52,050 --> 00:02:54,885
matrices. In other words, we
want to multiply A&B

43
00:02:54,885 --> 00:02:55,200
together.

44
00:02:56,470 --> 00:03:00,694
Now suppose that this matrix
A. The first matrix has P,

45
00:03:00,694 --> 00:03:04,534
rossion, Q columns, so it's
a P by Q matrix.

46
00:03:06,270 --> 00:03:09,663
And the second matrix be.
Let's suppose that Scott

47
00:03:09,663 --> 00:03:13,810
are rows and S columns,
so it's an arby S matrix.

48
00:03:15,820 --> 00:03:19,746
Now it turns out that we can
only form this product. We can

49
00:03:19,746 --> 00:03:21,256
only multiply the two matrices

50
00:03:21,256 --> 00:03:24,692
together. If the number of
columns in the first matrix,

51
00:03:24,692 --> 00:03:28,304
which is Q is the same as the
number of rows in the second

52
00:03:28,304 --> 00:03:31,658
matrix, these two numbers have
got to be the same. Q Must equal

53
00:03:31,658 --> 00:03:35,012
R and the reason for that will
become apparent when we start to

54
00:03:35,012 --> 00:03:38,366
do the calculation. But you've
got to be able to pair up the

55
00:03:38,366 --> 00:03:41,204
elements in the first matrix
with the elements in the Second

56
00:03:41,204 --> 00:03:44,816
matrix and will only be able to
do that if the number of columns

57
00:03:44,816 --> 00:03:48,428
in the first is the same as the
number of rows in the second.

58
00:03:49,960 --> 00:03:54,100
When that's the case, we can
actually find the product AB and

59
00:03:54,100 --> 00:03:57,895
the answer is another matrix.
And let's suppose this answer is

60
00:03:57,895 --> 00:04:02,380
matrix C and the size of matrix.
See, we can determine in advance

61
00:04:02,380 --> 00:04:07,210
from the sizes of matrix A&B,
the size of matrix C will be P

62
00:04:07,210 --> 00:04:11,967
by S. So it's got the same
number of rows as the first

63
00:04:11,967 --> 00:04:16,088
matrix and columns as the second
matrix, so this will be an R.

64
00:04:16,088 --> 00:04:18,624
This will be a P by S matrix.

65
00:04:20,690 --> 00:04:25,622
Let's have a look at a specific
example. Suppose we want to

66
00:04:25,622 --> 00:04:27,266
multiply the matrix 37.

67
00:04:27,990 --> 00:04:29,670
45

68
00:04:32,040 --> 00:04:34,508
by the Matrix 29.

69
00:04:35,920 --> 00:04:39,077
And the first question we should
ask ourselves is, do these

70
00:04:39,077 --> 00:04:42,234
matrices have the right size so
that we can actually multiply

71
00:04:42,234 --> 00:04:47,470
them together? Well, this matrix
is a two row two column matrix.

72
00:04:50,350 --> 00:04:53,374
And the second matrix
is 2 rows one column.

73
00:04:55,960 --> 00:04:59,951
And we note that the number of
columns here in the first matrix

74
00:04:59,951 --> 00:05:03,942
is the same as the number of
rows in the second matrix. So

75
00:05:03,942 --> 00:05:07,626
these two numbers are the same,
so we can do this multiplication

76
00:05:07,626 --> 00:05:11,924
and the size of the answer. The
size of the result that will get

77
00:05:11,924 --> 00:05:15,915
is obtained by the number of
rows in the 1st and columns in

78
00:05:15,915 --> 00:05:19,906
the second. So the size of the
answer that we're looking for is

79
00:05:19,906 --> 00:05:24,204
a 2 by 1 matrix. So you see
right at the beginning we can

80
00:05:24,204 --> 00:05:27,046
tell how many. Elements are
going to be in our answer.

81
00:05:27,046 --> 00:05:29,770
There's going to be a number
there, and a number there so

82
00:05:29,770 --> 00:05:31,586
that we have a 2 by 1 matrix.

83
00:05:32,920 --> 00:05:36,169
Now to determine these
numbers, we use the same

84
00:05:36,169 --> 00:05:39,779
operations as we've just
seen. We take the first row

85
00:05:39,779 --> 00:05:43,750
here and we pair the elements
with those in the first

86
00:05:43,750 --> 00:05:44,111
column.

87
00:05:45,380 --> 00:05:50,308
We multiply the paired elements
together and add the result. So

88
00:05:50,308 --> 00:05:53,892
we want 3 * 2, which is 6.

89
00:05:55,540 --> 00:06:02,125
We want 7 * 9 which is 63 and
we add the results together. 6

90
00:06:02,125 --> 00:06:07,832
and 63 is 69, so the element
that goes in the first position

91
00:06:07,832 --> 00:06:13,539
in our answer is 69. That's 3 *
2 at 7 * 9.

92
00:06:15,380 --> 00:06:19,043
The element that's going in this
position here is obtained by

93
00:06:19,043 --> 00:06:22,373
working with the 2nd row and
this first column here.

94
00:06:23,120 --> 00:06:25,871
Again, we pair the elements up 4

95
00:06:25,871 --> 00:06:28,670
* 2. Which is 8th.

96
00:06:29,950 --> 00:06:36,541
5 * 9 which is 45, and
we add the results together. 45

97
00:06:36,541 --> 00:06:38,062
+ 8 is.

98
00:06:39,600 --> 00:06:40,610
53

99
00:06:43,270 --> 00:06:46,810
So the result is 6953, so the
result of multiplying these two

100
00:06:46,810 --> 00:06:50,350
matrices together is another
matrix which is a 2 by 1 matrix

101
00:06:50,350 --> 00:06:53,890
and the elements are obtained in
the way I've just shown you.

102
00:06:54,490 --> 00:06:56,116
Let's have a look at another

103
00:06:56,116 --> 00:07:01,054
example. This time I'm going to
try to multiply together the two

104
00:07:01,054 --> 00:07:06,010
matrices A&B where a is this two
by two Matrix 2453 and B. Is

105
00:07:06,010 --> 00:07:09,550
this two by two Matrix three 6
-- 1 nine?

106
00:07:10,380 --> 00:07:13,339
And again, the first question we
should ask ourselves is, do

107
00:07:13,339 --> 00:07:16,298
these matrices have the right
size so that we can actually

108
00:07:16,298 --> 00:07:21,893
multiply them together? Well,
matrix a this matrix A is a two

109
00:07:21,893 --> 00:07:25,916
row two column matrix. So that's
two by two.

110
00:07:27,550 --> 00:07:30,960
Matrix B is 2 rows and two
columns, so that's also

111
00:07:30,960 --> 00:07:31,890
two by two.

112
00:07:33,290 --> 00:07:36,631
And you can see that these two
numbers are the same. That is,

113
00:07:36,631 --> 00:07:39,715
the number of columns in the
first is the same as the

114
00:07:39,715 --> 00:07:42,799
number of rows in the second.
So we can perform the matrix

115
00:07:42,799 --> 00:07:43,056
multiplication.

116
00:07:44,280 --> 00:07:47,604
The size of the answer we can
determine right at the start.

117
00:07:47,604 --> 00:07:51,205
The size of the matrix that we
get is determined by the number

118
00:07:51,205 --> 00:07:54,806
here and the number there two by
two. So what we can decide

119
00:07:54,806 --> 00:07:58,130
before we do any calculation at
all is that this answer matrix

120
00:07:58,130 --> 00:07:59,792
is a two by two matrix.

121
00:08:00,470 --> 00:08:03,198
Be 2 rows and two columns,
so we're looking for four

122
00:08:03,198 --> 00:08:04,438
numbers to pop in there.

123
00:08:05,970 --> 00:08:09,389
Let's try and figure out how we
work out, what the answer is.

124
00:08:10,940 --> 00:08:15,269
When we want to find the element
that goes in here, observe that

125
00:08:15,269 --> 00:08:18,599
this is the first row first
column of the answer.

126
00:08:20,140 --> 00:08:23,781
And the number in the first
row first column comes from

127
00:08:23,781 --> 00:08:26,760
looking at the first row and
1st Column here.

128
00:08:28,370 --> 00:08:32,491
If we pair off the elements in
the first row and 1st Column

129
00:08:32,491 --> 00:08:35,027
will have 2 * 3 which is 6.

130
00:08:36,410 --> 00:08:40,960
4 * -- 1, which is minus four,
and we add them together.

131
00:08:44,870 --> 00:08:49,430
When we come to this element
here, this element is in the

132
00:08:49,430 --> 00:08:50,950
first row, second column.

133
00:08:51,610 --> 00:08:55,558
So we use the first row, second
column in the original matrices.

134
00:08:56,270 --> 00:09:03,155
2 * 6 which is 12 and 4
* 9 Four nines of 36. And

135
00:09:03,155 --> 00:09:05,909
we add those paired
products together.

136
00:09:08,610 --> 00:09:13,303
When we want the element that's
in the 2nd row first column, we

137
00:09:13,303 --> 00:09:17,996
use the 2nd row in the first
matrix and 1st column in the

138
00:09:17,996 --> 00:09:22,328
Second Matrix. Again, pairing
the elements off 5 * 3 is 15.

139
00:09:23,790 --> 00:09:28,122
3 * -- 1 is minus three.
When we add the paired

140
00:09:28,122 --> 00:09:28,844
elements together.

141
00:09:30,730 --> 00:09:35,570
Finally. The element that's in
the 2nd row, second column of

142
00:09:35,570 --> 00:09:39,860
the answer is obtained by using
the elements in the 2nd row of

143
00:09:39,860 --> 00:09:42,830
the first matrix, second column
of the Second matrix.

144
00:09:43,810 --> 00:09:50,984
5 * 6 which is 30 and 3 *
9, which is 27, and we add the

145
00:09:50,984 --> 00:09:52,250
paired elements together.

146
00:09:53,480 --> 00:09:57,596
So finally, just to tidy it
up, we've got 6 subtract 4

147
00:09:57,596 --> 00:09:58,625
which is 2.

148
00:10:00,540 --> 00:10:03,606
12 + 36, which is 48.

149
00:10:05,750 --> 00:10:12,533
15 subtract 3 which is 12 and 30
+ 27 which is 57, so we can find

150
00:10:12,533 --> 00:10:18,119
the matrix product AB in this
case and the result is a two by

151
00:10:18,119 --> 00:10:21,874
two matrix. Let's have a look
at another example. Suppose

152
00:10:21,874 --> 00:10:24,654
we're asked to find the
product of these two matrices.

153
00:10:25,990 --> 00:10:29,270
And again, we should ask all
these matrices of the

154
00:10:29,270 --> 00:10:32,550
appropriate size. The first
matrix here has two rows, one

155
00:10:32,550 --> 00:10:34,846
column, it's a 2 by 1 matrix.

156
00:10:35,760 --> 00:10:40,154
And the second matrix has two
rows and two columns, so it's a

157
00:10:40,154 --> 00:10:44,210
two by two matrix. Now in this
case, you'll see that the

158
00:10:44,210 --> 00:10:48,604
number of columns in the first
matrix is not the same as the

159
00:10:48,604 --> 00:10:52,322
number of rows in the second
matrix. Those two numbers are

160
00:10:52,322 --> 00:10:55,026
not equal, so we cannot
multiply these matrices

161
00:10:55,026 --> 00:10:58,406
together. We say the product of
these matrices doesn't exist,

162
00:10:58,406 --> 00:11:01,110
so we stop there. We can't
calculate that.

163
00:11:02,540 --> 00:11:07,950
Let's have another example.
Suppose we want to try to

164
00:11:07,950 --> 00:11:11,737
find the product of the
matrices 3214.

165
00:11:14,180 --> 00:11:17,700
With the matrix XY. Now this
is the first example we've

166
00:11:17,700 --> 00:11:20,900
looked at where we've had
symbols rather than numbers in

167
00:11:20,900 --> 00:11:23,780
our matrix, but the operation
the process is, the

168
00:11:23,780 --> 00:11:25,380
calculations are just the
same.

169
00:11:26,430 --> 00:11:29,466
First of all, we should ask can
we multiply these together? Are

170
00:11:29,466 --> 00:11:30,731
they of the right size?

171
00:11:31,660 --> 00:11:34,188
This is a two row two
column matrix.

172
00:11:35,970 --> 00:11:38,562
And this is a two row one
column matrix.

173
00:11:40,030 --> 00:11:43,969
And these numbers are the same
in here the number of columns in

174
00:11:43,969 --> 00:11:48,211
the first is the same as the
number of rows in the second. So

175
00:11:48,211 --> 00:11:50,938
we can actually perform the
matrix multiplication and the

176
00:11:50,938 --> 00:11:55,786
answer we get will be a 2 by 1
matrix, so we know the shape of

177
00:11:55,786 --> 00:11:59,725
the answer. It's a two row one
column matrix, so it looks the

178
00:11:59,725 --> 00:12:01,846
same shape as this one. This one

179
00:12:01,846 --> 00:12:04,348
here. Two rows, one column.

180
00:12:06,340 --> 00:12:09,750
Let's actually work out what the
elements in the answers are.

181
00:12:10,600 --> 00:12:15,176
As before, we take the first row
and pair the elements with the

182
00:12:15,176 --> 00:12:21,620
first column. So it's 3
multiplied by X 2 * y and we add

183
00:12:21,620 --> 00:12:26,130
the resulting products, so we
get three X + 2 Y.

184
00:12:33,190 --> 00:12:37,138
The element that's down here,
which is in the 2nd row, first

185
00:12:37,138 --> 00:12:41,086
column of our answer, is
obtained by using the 2nd row in

186
00:12:41,086 --> 00:12:43,060
the first matrix and the first

187
00:12:43,060 --> 00:12:47,612
column here. Multiply the pad
elements together and add so

188
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

189
00:12:54,344 --> 00:13:00,328
products together and we get X +
4 Y. So the result we found is a

190
00:13:00,328 --> 00:13:02,198
two row one column matrix.

191
00:13:03,770 --> 00:13:06,914
In this case, the answers got
symbols in as well, but that's

192
00:13:06,914 --> 00:13:09,534
the result of finding the
product of these two matrices.

193
00:13:09,534 --> 00:13:12,940
Now we can go on and look at
more examples and trying to

194
00:13:12,940 --> 00:13:15,298
find products of matrices of
different sizes and shapes,

195
00:13:15,298 --> 00:13:18,180
and we'll do some more of that
in the next video.