Page 30 - A Course in Linear Algebra with Applications

14 Chapter One: Matrix Algebra

Let d ik be the (i, k) entry of AB ; then d ik = YH=I o-uhk-
d
Thus the (i,j) entry of (AB)C is YX=i ikCkj, that is

p n
J~](yiaubik)ckj.
fc=i 1=1

After a change in the order of summation, this becomes

n p
/^ilj/^lkCkj).
1=1 fc = l

Here it is permissible to change the order of the two summa-
tions since this just corresponds to adding up the numbers
aubikCkj in a different order. Finally, by the same procedure
we recognise the last sum as the (i,j) entry of the matrix
A(BC).
The next two examples illustrate the use of matrices in
real-life situations.

Example 1.2.7

A certain company manufactures three products P, Q, R in
four different plants W, X, Y, Z. The various costs (in whole
dollars) involved in producing a single item of a product are
given in the table

P Q R
material 1 2 1
labor 3 2 2
overheads 2 1 2

The numbers of items produced in one month at the four
locations are as follows: