The Application of Matrices to Business and Economics.

Problem:

Suppose that the economy of a certain region depends on three industries:

service, electricity and oil production. Monitoring the operations of these three industries

over a period of one year, we were able to come up with the following observations:

1- To produce 1 unit worth of service, the service industry must consume 0.3 units of its own production, 0.3 units of electricity and 0.3 units of oil to run its operations.

2-To produce 1 unit of electricity, the power-generating plant must buy 0.4 units of

service, 0.1 units of its own production, and 0.5 units of oil.

3-Finally, the oil production company requires 0.3 units of service, 0.6 units of

electricity and 0.2 units of its own production to produce 1 unit of oil.

Find the production level of each of these industries in order to satisfy the external and the internal demands assuming that the above model is closed, that is, no goods leave or enter the system.

Solution:

Consider the following variables:
1. p1= production level for the service industry
2. p2= production level for the power-generating plant (electricity)
3. p3= production level for the oil production company

Since the model is closed, the total consumption of each industry must equal its total

production. This gives the following linear system:

0.3p1 + 0.3p2 + 0.3p3 = p1

0.4p1 + 0.1p2 + 0.5p3 = p2

0.3p1 + 0.6p2 + 0.2p3 = p3

The input matrix is:

A=

0.3 0.3 0.3

0.4 0.1 0.5

0.3 0.6 0.2

and the above system can be written as (A-I)P=0. Note that this homogeneous system has infinitely many

solutions (and consequently a nontrivial solution) since each column in the coefficient matrix sums to 1. The

augmented matrix of this homogeneous system is :

-0.7 0.3 0.3 0

0.4 -0.9 0.5 0

0.3 0.6 -0.8 0

which can be reduced to :

1 0 -0.82 0

0 1 -0.92 0

0 0 0 0

To solve the system, we let p3 =t (a parameter), then the general solution is :

p1= 0.82t

p2=0.92t

p3=t

The values of the variables in this system must be nonnegative in order for the model to make sense. In other words, t ≥ 0. Taking t=100 for example would give the solution :

p1= 82 units

p2= 92 units

p3= 100 units.

I chose an example of real life problem related to economics that shows how matrices can be effective and helpful when it comes to solving complicated problems. Matrices let us arrange numbers and get solutions to our problem in a quick and easy way !