Page 31 - A Course in Linear Algebra with Applications

1.2: Operations with Matrices 15

w X Y Z
p 2000 3000 1500 4000
Q 1000 500 500 1000
2000 2000 2500 2500

The problem is to find the total monthly costs of material,
labor and overheads at each factory.
Let C be the "cost" matrix formed by the first set of
data and let N be the matrix formed by the second set of
data. Thus

/ l 2 1\ /2000 3000 1500 4000 \
C = 3 2 2 andJV= 1000 500 500 1000 .
\ 2 1 2 / \2000 2000 2500 2500/

The total costs per month at factory W are clearly

material : 1 x 2000 + 2 x 1000 + 1 x 2000 = 6000
labor : 3 x 2000 + 2 x 1000 + 2 x 2000 = 12000
overheads : 2 x 2000 + 1 x 1000 + 2 x 2000 = 9000

Now these amounts arise by multiplying rows 1, 2 and 3 of
matrix C times column 1 of matrix JV, that is, as the (1, 1),
(2, 1), and (3, 1) entries of matrix product CN. Similarly the
costs at the other locations are given by entries in the other
columns of the matrix CN. Thus the complete answer can be
read off from the matrix product

/ 6000 6000 5000 8500 \
CN = I 12000 14000 10500 19000 I .
\ 9000 10500 8500 14000/

Here of course the rows of CN correspond to material, la-
bor and overheads, while the columns correspond to the four
plants W, X, Y, Z.