equal the row dimension of B, then the matrices Aand Bcannot be multiplied. The
matrix product of two matrices Aand B, having the proper dimensions, is written
C=AB and then one can say either
(a) Apremultiplies B
(b) or Bpostmultiplies A
If the row dimension of Bequals the column dimension of Aone can write p=n, then
the two matrices Aand Bcan be multiplied and the resulting matrix product Chas
dimension m×q. This is sometimes expressed in the form
where attention is drawn to the fact that the matrices satisfy the compatibility
condition for matrix multiplication. Expressing the matrices A,Band Cin expanded
form one can write
The element cij belong to the matrix product Cis calculated using the elements
from the ith row vector of Aand the elements from the jth column vector of Bto
represent cij as a dot or inner product. The ith row vector of Ais dotted with the j
column vector from Band the resulting single number is called cij. This inner or dot
product is defined as above, but now a double subscript notation is in use so that
one obtains
cij = (ai 1 ai 2... ain )
b 1 j
b 2 j..
.
bnj
=ai 1 b 1 j+ai 2 b 2 j+···+ain bnj =∑nk=1aik bkjPerforming all possible inner products of the ith row vector with the jth column