40S Applied Math
Mr. Knight – Killarney School
Unit: Matrices
Lesson: MAT-5 Transitions
Transition Problems
MAT-L5 Objectives:
To solve transition problems using matrix operations.
Learning Outcome L-4
Slide 1
40S Applied Math
Mr. Knight – Killarney School
Unit: Matrices
Lesson: MAT-5 Transitions
Changes occur around us all the time. Some of these
changes may be random events, but some changes
follow predictable patterns over an extended period of
time. For example, the type and amount of recorded
music bought by teenagers probably follows
predictable patterns, and the music industry needs to
adjust to the buying patterns of
teenagers. After the patterns have
been identified, matrix operations
can be used to predict future events.
Theory – Transitions & Predictions
Slide 2
40S Applied Math
Mr. Knight – Killarney School
Unit: Matrices
Lesson: MAT-5 Transitions
Before we can predict future events, we must write an initial population
matrix. This is a row matrix that identifies the initial population before
changes occur.
Consider the following example:
A small community has a workforce of 1800 people, with 1680 employed
and 120 unemployed. During the course of one year, 10% of the
employed workers will lose their jobs and 60% of the unemployed will find
jobs.
We will label the initial population matrix A.
Now:
Therefore, the initial population matrix is:
Theory – Initial Population Matrix
Slide 3
40S Applied Math
Mr. Knight – Killarney School
Unit: Matrices
Lesson: MAT-5 Transitions
The previous example is repeated below:
A small community has a workforce of 1800 people, with 1680 employed
and 120 unemployed. During the course of one year, 10% of the
employed workers will lose their jobs and 60% of the unemployed will find
jobs.
The changes in employment are illustrated in the following transition
diagram.
This shows that 0.90 or 90% of the employed remain employed, and 10%
become unemployed. Likewise, 0.40 or 40% of the unemployed remain
unemployed, and 60% become employed.
Theory – Transition Diagram
Slide 4
40S Applied Math
Mr. Knight – Killarney School
Unit: Matrices
Lesson: MAT-5 Transitions
A transition matrix is a square matrix that shows the probability of
the population remaining in the same state or changing to another
state. For example, there is a 100% chance that an employed person
will either remain employed or become unemployed. Likewise, there is
a 100% chance that an unemployed person will become employed or
remain unemployed.
The previous example is repeated again:
A small community has a workforce of 1800 people, with 1680
employed and 120 unemployed. During the course of one year, 10%
of the employed workers will lose their jobs and 60% of the
unemployed will find jobs.
Let B represent the transition matrix.
Then
We now write the elements in the transition matrix in decimal form.
Note that each row must add to 1.0, which indicates
that 100% of the population is accounted for.
Theory – Transition Matrix
Slide 5
40S Applied Math
Mr. Knight – Killarney School
Unit: Matrices
Lesson: MAT-5 Transitions
From the previous example, the initial population matrix is
and the transition matrix is:
To predict the number of people employed and unemployed after one year,
you multiply the population matrix and the transition matrix.
Let matrix C represent the number employed and unemployed after one
year.
Therefore, after one year, the number employed is 1584 and the number
unemployed is 216.
Theory – After One Year
Slide 6
40S Applied Math
Mr. Knight – Killarney School
Unit: Matrices
Lesson: MAT-5 Transitions
To predict the number of people employed and unemployed after two years,
you can multiply population matrix C with transition matrix B, or multiply
population matrix A by the square of transition matrix B. Let matrix D
represent the populations after two years.
We now have
Therefore, after two years, the number employed is 1555 and the number
unemployed is 245.
Likewise, after five years the number employed and unemployed could be
represented by matrix E, and calculated as:
Therefore, after five years, the number employed is 1543 and the number
unemployed is 257. (These numbers represent rounded values.)
Theory – After Two (+) Years
Slide 7
40S Applied Math
Mr. Knight – Killarney School
Unit: Matrices
Lesson: MAT-5 Transitions
The Paper Street Soap Company sells laundry detergent in two-litre and
five-litre packages. Their research shows that 34% of the people buying
the small package will switch to the large package for their next
purchase, and 12% of the buyers of the large package will switch to the
small package for their next purchase. The original market share was 55%
for the small package and 45% for the large package.
Test Yourself – Paper Street
Soap
Slide 8
40S Applied Math
Mr. Knight – Killarney School
Unit: Matrices
Lesson: MAT-5 Transitions
A small store in a remote community sells three brands of soft drinks.
The three brands are Popsie, Sparkle, and Fizz. The current market share
is 60% for Popsie, 31% for
Sparkle, 9% for Fizz.
Since the store is located in a
remote community, the store manager
needs to place orders 12 months in
advance. (The winter roads to the
community are operational for a few
weeks of the year, and airfare for soft
drinks is too expensive.)
Test Yourself – Popsie
Slide 9