Page 24 - A Course in Linear Algebra with Applications

8 Chapter One: Matrix Algebra

This has coefficient matrix

aii&n + ai2&2i 011612 + 012622^
«21&11 + C122&21 ^21^12 + ^22^22 /
and represents a change of variables from zi, z 2 to xi, x 2 which
may be thought of as the composite of the original changes of
variables.
At first sight this new matrix looks formidable. However
it is in fact obtained from A and B in quite a simple fashion,
namely by the row-times-column rule. For example, the (1,2)
entry arises from multiplying corresponding entries of row 1 of
A and column 2 of B, and then adding the resulting numbers;
thus

&12
(an a 12) Qll^l2 + Oi2^22-
^22
Other entries arise in a similar fashion from a row of A and a
column of B.
Having made this observation, we are now ready to define
the product AB where A is an m x n matrix and B i s a n n x p
matrix. The rule is that the (i,j) entry of AB is obtained by
multiplying corresponding entries of row i of A and column j
of B, and then adding up the resulting products. This is the
row-times-column rule. Now row i of A and column j of B are

/ bij \
->2j
an a%2 a in) and
\ b nj /

Hence the (i,j) entry of AB is

-
Uilblj + CLi202j + • • + O-inbnj,