Page 32 - A Course in Linear Algebra with Applications

16 Chapter One: Matrix Algebra

Example 1.2.8
In a certain city there are 10,000 people of employable age.
At present 7000 are employed and the rest are out of work.
Each year 10% of those employed become unemployed, while
60% of the unemployed find work. Assuming that the total
pool of people remains the same, what will the employment
picture be in three years time?

Let e n and u n denote the numbers of employed and un-
employed persons respectively after n years. The information
given translates into the equations

e n + i = .9e n + .6u n
u n+i = .le n + Au n

These linear equations are converted into a single matrix equa-
tion by introducing matrices

X„ = ( 6n \ and A ( , 9 - 6
u„. I V .1 .4
"n

The equivalent matrix equation is

X n+i = AX n.

Taking n to be 0, 1, 2 successively, we see that X\ = AXo,
3
2
= AXi = A X 0, = = A X Q. In general
X 2 X 3 AX 2
U
= A XQ.
X n
Now we were told that e 0 = 7000 and UQ = 3000, so

Y - f700(A
x
°- ^3oooy •

Thus to find X 3 all that we need to do is to compute the power
3
A . This turns out to be