Introduction - Environmental Modeling - page 24

Exercises.
1. Find an example to show that the commutation-property A B ¼ B A,whichis
valid for single numbers, is not valid for matrix multiplication! Take care that A
and B are square matrices of the same size. Is it valid for pointwise multiplication?
2. Confirm by random matrices the validity of the following identities:

ð A þ B Þ C ¼ A C þ B C

C ð A þ B Þ¼ C A þ C B

ð A B Þ C ¼ A ð B C Þ

Þ 1 B 1 A 1 ð A B ¼

A 1 1

¼ A

A ðÞ¼c A c B ð B Þ¼c A ðÞ B

3. Find the inverse of the following matrices:

12
48

13
45

4. Confirm by random matrices and some integer values of p and q the validity of
the following identity:

A pþq A p A q

¼

1.3 A Simple Environmental Model

A simple example may illustrate the methodology used in this topic. As an
introductory simple situation consider the population of a biological species,
which is denoted by c . As a first approach it seems reasonable that the number of
children increases with the population. The number of children stands for
the reproduction rate of the species, denoted as

@c=@t , the temporal change of the
population at each time instant. For the sake of simplicity one may consider that the
reproduction rate is proportional to c :

@ c
@t / c

(1.11)