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3.5 Matrix Multiplication 95
3.5 MATRIX MULTIPLICATION

The purpose of this section is to further develop the concept of matrix multipli-
cation as introduced in the previous section. In order to do this, it is helpful to
begin by composing a single row with a single column. If

 
c 1
 c 2 
··· r n ) and C =   ,
R =( r 1 r 2 .
.
 . 
c n
the standard inner product of R with C is defined to be the scalar
n

RC = r 1 c 1 + r 2 c 2 + ··· + r n c n = r i c i .
i=1
For example,

 
1
2
(2 4 −2)   = (2)(1) + (4)(2) + (−2)(3) = 4.
3

Recall from (3.4.1) that the product of two 2 × 2 matrices

a b A B
F = and G =
c d C D
was defined naturally by writing

a b A B aA + bC aB + bD
FG = = = H.
c d C D cA + dC cB + dD
Notice that the (i, j) -entry in the product H can be described as the inner
product of the i th row of F with the j th column in G. That is,

A B
h 11 = F 1∗ G ∗1 =( ab ) , h 12 = F 1∗ G ∗2 =( ab ) ,
C D

A B
h 21 = F 2∗ G ∗1 =( cd ) , h 22 = F 2∗ G ∗2 =( cd ) .
C D
This is exactly the way that the general definition of matrix multiplication is
formulated.