# Application involving matrices!!!

• May 22nd 2007, 12:00 AM
Skelly
Application involving matrices!!!
So I was going through the old exam papers and found this question. I tried to look for a something a bit similar to this but found nothing. It was hinted that a similar question might appear in this year's exam so if anybody can help me then it'd be greatly appreciated.:) The question is as follows:

A study of a colony of a rare sea bird, the fairy tern, indicates that the population changes on a yearly basis as a discrete dynamic system. Dividing the population into 2 classes, juveniles and adults, there are initially 40 juvenile chicks and 20 breeding adults. The system is defined by: (1) an initial state vector x0 = (40 j
20) a

and (2) a transition matrix
A = (0 1.1
0.5 0.6)

and (3) the relationship x(n+1) = A*x(n)
(a) Calculate the state vector x1.
(b) Which entry in the transition matrix gives the annual birthrate of chicks per adult?
(c) Which entry of A tells us the probability a chick will survive to become a breeding adult?
(d) Find the eigenvalues n' corresponding eigenvectors for matrix A.
• May 23rd 2007, 12:39 AM
JakeD
Quote:

Originally Posted by Skelly
So I was going through the old exam papers and found this question. I tried to look for a something a bit similar to this but found nothing. It was hinted that a similar question might appear in this year's exam so if anybody can help me then it'd be greatly appreciated.:) The question is as follows:

A study of a colony of a rare sea bird, the fairy tern, indicates that the population changes on a yearly basis as a discrete dynamic system. Dividing the population into 2 classes, juveniles and adults, there are initially 40 juvenile chicks and 20 breeding adults. The system is defined by: (1) an initial state vector x0 = (40 j
20) a

and (2) a transition matrix
A = (0 1.1
0.5 0.6)

and (3) the relationship x(n+1) = A*x(n)
(a) Calculate the state vector x1.

Multiply x1 = A x0.

Quote:

(b) Which entry in the transition matrix gives the annual birthrate of chicks per adult?
(c) Which entry of A tells us the probability a chick will survive to become a breeding adult?
Using x1 = A x0, interpret x1 = [chicks, adults] output and x0 = [chicks,adults] input.

Quote:

(d) Find the eigenvalues n' corresponding eigenvectors for matrix A.
The eigenvalues are the roots of characteristic polynomial and the eigenvectors are the corresponding solutions to the characteristic equation.