Matrix Multiplication: 2x3 times 3x1
General Contents
Detailed Contents
Index
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Find the product
AB
, where
A
=
┌
┐
│
2
1
4
│
│
-1
0
5
│
└
┘
and
B
=
┌
┐
│
-2
│
│
6
│
│
3
│
└
┘
We’ll be talking about rows and columns, so let’s be sure which is which. Is a row left-right or up-and-down?
Left-right.
These matrices have different numbers of rows and columns. Can they really be multiplied?
Yes they can.
What is the requirement on their dimensions?
The only requirement is that the number of columns in the first matrix must be the same as the number of rows in the second.
Will the result of
AB = C
be a number or a matrix?
A matrix.
How many rows will
C
have?
It will have the same number of rows as the first matrix has.
How many is that here?
Two.
How many columns will
C
have?
It will have the same number of columns as the second matrix has.
How many is that here?
One.
Use C
11
and C
21
to represent the elements of
C
. Show how
C
will look.
┌
┐
│
C
11
│
│
C
21
│
└
┘
Now we can set up our problem:
┌
┐
┌
┐
┌
┐
│
2
1
4
│
*
│
-2
│
=
│
C
11
│
│
-1
0
5
│
│
6
│
│
C
21
│
└
┘
│
3
│
└
┘
└
┘
The color indicates how we obtain C
11
. Show the details with the numerical values.
We get C
11
= 2*(-2) + 1*6 + 4*3 = 14
Which row of A is involved in C
21
?
The second.
┌
┐
┌
┐
┌
┐
│
2
1
4
│
*
│
-2
│
=
│
C
11
│
│
-1
0
5
│
│
6
│
│
C
21
│
└
┘
│
3
│
└
┘
└
┘
Show the details.
C
21
= (-1)*(-2) + (0)*6 + 5*3 = 17
Combine the results to show the numerical elements of
C
.
┌
┐
┌
┐
│
C
11
│
=
│
14
│
│
C
21
│
│
17
│
└
┘
└
┘
Will the product always have one column?
No.
What determines the number of columns?
The number of columns in the second matrix determines the number of columns in the product. In this problem, that was matrix
B
with one column. Here is an example in which the product has
two rows and three columns.
The end. If you found this helpful and would recommend that I create more pages like this one, please let me know:
Email to John Taylor
General Contents
Detailed Contents
Index