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Matrix Multiplication:
3 x 3 by 3 x 3
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Find the product
AB = C
, where
A
=
and
B
=
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 the same numbers of rows and columns. Can they really be multiplied?
Yes they can.
What is the requirement on the number of rows and columns of
A
and
B
?
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?
Three.
How many columns will C have?
It will have the same number of columns as the second matrix has.
How many is that here?
Three.
Use C
11
, C
12
, C
13
, C
21
, C
22
, C
23
, C
31
, C
32
, and C
33
to represent the elements of
C
. Show how
C
will look.
Now we can set up our problem:
The bold type indicates how we obtain C
11
.
Show the details with the numerical values.
We get C
11
= 4*1 + 1*3 + 5*(–2) = –3
Which row of
A
is involved in C
12
?
The first.
Which column of
B
is involved in C
12
?
The second.
Indicate this with bolding on the matrices.
Show the details with the numerical values.
We get C
12
= 4*(–4) + 1*2 + 5*(–3) = –29
Which row of
A
is involved in C
13
?
The first.
Which column of
B
is involved in C
13
?
The third.
Indicate this with bolding on the matrices.
Show the details with the numerical values.
We get C
13
= 4*5 + 1*0 + 5*2 = 30
Which row of
A
is involved in C
21
?
The second.
Which column of
B
is involved in C
21
?
The first.
Indicate this on the matrices.
Show the details with the numerical values.
We get C
21
= 3*1 + (–2)*3 + 1*(–2) = –5
Which row of
A
is involved in C
22
?
The second.
Which column of
B
is involved in C
22
?
The second.
Indicate this on the matrices.
Show the details with the numerical values.
We get C
22
= 3*(–4) + (–2)*2 + 1*(–3) = –19
Which row of
A
is involved in C
23
?
The second.
Which column of
B
is involved in C
23
?
The third.
Indicate this on the matrices.
Show the details with the numerical values.
We get C
23
= 3*5 + (–2)*0 + 1*2 = 17
Which row of
A
is involved in C
31
?
The third.
Which column of
B
is involved in C
31
?
The first.
Indicate this on the matrices.
Show the details with the numerical values.
We get C
31
= (–4)*1 + 3*3 + (–1)*(–2) = 7
Which row of
A
is involved in C
32
?
The third.
Which column of
B
is involved in C
32
?
The second.
Indicate this on the matrices.
Show the details with the numerical values.
We get C
32
= (–4)*(–4) + 3*2 + (–1)*(–3) = 25
Which row of
A
is involved in C
33
?
The third.
Which column of
B
is involved in C
33
?
The third.
Indicate this on the matrices.
Show the details with the numerical values.
We get C
33
= –4*5 + 3*0 + (–1)*2 = –22
Combine the results to show the numerical elements of
C
.
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