Page 102 - Matrix Analysis & Applied Linear Algebra

96 Chapter 3 Matrix Algebra

Matrix Multiplication

• Matrices A and B are said to be conformable for multiplication
in the order AB whenever A has exactly as many columns as B
has rows—i.e., A is m × p and B is p × n.

• For conformable matrices A m×p =[a ij ] and B p×n =[b ij ], the
matrix product AB is defined to be the m × n matrix whose
(i, j) -entry is the inner product of the i th row of A with the j th
column in B. That is,
p

[AB] = A i∗ B ∗j = a i1 b 1j + a i2 b 2j + ··· + a ip b pj = a ik b kj .
ij
k=1

• In case A and B fail to be conformable—i.e., A is m × p and B
is q × n with p
= q —then no product AB is defined.

For example, if

 
b 11 b 12 b 13 b 14

a 11 a 12 a 13
A = and B =  b 21 b 22 b 23 b 24 
a 21 a 22 a 23
2×3
b 31 b 32 b 33 b 34
3×4

 ↑ inside ones match ↑ 
 
 
shape of the product
then the product AB exists and has shape 2 × 4. Consider a typical entry of
this product, say, the (2,3)-entry. The definition says [AB] 23 is obtained by
forming the inner product of the second row of A with the third column of B
 

a 11 a 12 a 13 b 11 b 12 b 13 b 14
 b 21 b 22 b 23 b 24  ,
a 21 a 22 a 23
b 31 b 32 b 33 b 34
so
3

[AB] = A 2∗ B ∗3 = a 21 b 13 + a 22 b 23 + a 23 b 33 = a 2k b k3 .
23
k=1