MAT 2401 Handout 3.4
Population Modeling
Suppose we are interested in the population of a certain type of bird in a forest area.
We can divide the population in two age groups – hatchlings (age<1) and adults.
Suppose we can estimate the following parameters:
Birth rate from hatchlings Bh
Birth rate from adults Ba
Survival rate of hatchlings S h
Survival rate of adults S a
We can model the population from year to year by the matrix equation
x1  Ax0
 hn 1   Bh
a    S
 n 1   h
xn 1  Axn
x2  Ax1
Ba   hn 
Sa   an 
x3  Ax2
xn 1  Axn
Stable proportion of population in the age groups
xn 1  Axn
xn 1   xn
Eigenvalues and Eigenvectors
Let A be a n  n matrix,  a scalar, and x a non-zero n1 column vector.
 and x are called an eigenvalue and eigenvector of A respectively if
Ax   x .
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1 4 
Example 1 Let A  
 , find the eigenvalue of A if the eigenvector is
 2 3
1
(a) x1   
1
2
(b) x2   
 1
Theorem (3.3)
A square matrix A is invertible if and only if det  A  0 .
A square matrix A is singular if and only if det  A  0 .
Now,
 x  Ax
2
1 4 
Example 2 Find the eigenvalues and eigenvectors of A  
.
 2 3
I  A 
I  A 
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 1 2 2 
Example 3 Find the eigenvalues and eigenvectors of A   1 2 1  .
 1 1 0 
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Remarks
1. det   I  A is called the ____________________________ of A .
2. It is a polynomial equation of degree ___.
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MAT 2401 Homework 3.4 Name:________________________________
WARNING: Make sure your solutions are logically presented.
3 1
Let A  
.
5 3
(a) Find the characteristic equation of A .
(b) Find the eigenvalues of A .
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(c) Find the corresponding eigenvectors of A .
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