Page 104 - Matrix Analysis & Applied Linear Algebra

98 Chapter 3 Matrix Algebra

There are various ways to express the individual rows and columns of a
matrix product. For example, the i th row of AB is

[AB] i∗ = A i∗ B ∗1 | A i∗ B ∗2 |· · · | A i∗ B ∗n = A i∗ B
 
B 1∗
 B 2∗ 
··· a ip )   = a i1 B 1∗ + a i2 B 2∗ + ··· + a ip B p∗ .
=( a i1 a i2 .
.
 . 
B p∗
As shown below, there are similar representations for the individual columns.
Rows and Columns of a Product

Suppose that A =[a ij ]is m × p and B =[b ij ]is p × n.
th th
• [AB] i∗ = A i∗ B ( i row of AB )=( i row of A ) ×B . (3.5.4)

th th
• [AB] ∗j = AB ∗j ( j col of AB )= A× ( j col of B ) . (3.5.5)
p
• [AB] i∗ = a i1 B 1∗ + a i2 B 2∗ + ··· + a ip B p∗ = a ik B k∗ . (3.5.6)
k=1
p
• [AB] ∗j = A ∗1 b 1j + A ∗2 b 2j + ··· + A ∗p b pj = A ∗k b kj . (3.5.7)
k=1
These last two equations show that rows of AB are combinations of
rows of B, while columns of AB are combinations of columns of A.

 
3 −51
1 −20
For example, if A = and B =  2 −72  , then the
3 −45
1 −20
second row of AB is
 
3 −51
[AB] 2∗ = A 2∗ B =( 3 −45 )  2 −72  =( 6 3 −5) ,
1 −20
and the second column of AB is
 
−5
1 −20 9
[AB] ∗2 = AB ∗2 =  −7  = .
3 −45 3
−2
This example makes the point that it is wasted effort to compute the entire
product if only one row or column is called for. Although it’s not necessary to
compute the complete product, you may wish to verify that
 
3 −51
1 −20 −19 −3
AB =  2 −72  = .
3 −45 63 −5
1 −20