That is, the element in theith row andjth column of the product matrixCis obtained
from the ‘scalar’ product of theith row ofAwith thejth column ofB.
As an example:
A=[
a 11 a 12 a 13
a 21 a 22 a 23]
( 2 × 3 )B=[b
11 b 12
b 21 b 22
b 31 b 32]
( 3 × 2 )C=AB=[
a 11 a 12 a 13
a 21 a 22 a 23][b 11 b 12
b 21 b 22
b 31 b 32]=[
a 11 b 11 +a 12 b 21 +a 13 b 31 a 11 b 12 +a 12 b 22 +a 13 b 32
a 21 b 11 +a 22 b 21 +a 23 b 31 a 21 b 12 +a 22 b 22 +a 23 b 32]Note:
(i) ABandBAneed not be the same – in fact, they are only both defined
ifAism×nandBisn×m(thenABism×mandBAn×n)
(ii) AB= 0 is possible even if bothAandBhave non zero elements.Problem 13.5
Evaluate all possible products of the following matrices, excluding powersA=[
1 − 2
31]
B=[
21 0
10 − 1]
C=[ 0 − 12
− 10 − 4
312]Looking at the numbers of rows and columns we see that we can only form the following
products:
AB=[
1 − 2
31][
21 0
10 − 1]
=[
01 2
73 − 1]BC=[
21 0
10 − 1][ 0 − 12
− 10 − 4
312]
=[
− 1 − 20
− 3 − 20]No higher order products are possible without resulting in powers ofA,BorC.
Problem 13.6
Evaluate
[− 123
011
− 102][ 2 − 4 − 1
− 111
1 − 2 − 1]and interpret the result.