THE UNIVERSITY OF SYDNEYPURE MATHEMATICS

Linear Mathematics 2012

Tutorial 6 (week 7)Preliminary exercise

1. Let A =(

2 1 21 0 00 1 0

).

a) Find the eigenvalues of A.b) Find the eigenspaces of A.c) Explain why A is diagonalisable.d) Write down matrices P and D such that A = PDP1.e) Find P1, the inverse of the matrix P you found in part d).

Tutorial exercisesYour tutor will award you a participation mark for this weeks tutorial once you have made asignificant attempt on these questions. You will not automatically be awarded a participationmark for attending the tutorial. You are encouraged to work in groups of 45 people.

2. A sequence of real numbers is defined by xn+3 = 2xn+2 + xn+1 2xn, with x0 = 0, x1 = 1and x2 = 1.

a) Let un =(

xn+2xn+1xn

), and un+1 = Aun.

Show that A is the matrix in Question 1.b) Use the formula developed in lectures to find a formula for xn in terms of n.c) Now calculate un using un = Anu0 = PDnP1u0, where P and D are the matrices you

found in Question 1 d). (It will be easiest if you start at the right hand side of PDnP1u0 ie, multiply P1 by u0, then multiply the result by Dn, etc. Note that u0 =

(110

).)

Confirm that this method results in the same formula for xn as in part b).

3. Let M =

54

2 123

0 0

0 12

0

be the Leslie matrix for an animal population divided into 3 age groups.

a) Show that 2 is an eigenvalue of M , and find a corresponding eigenvector.b) If the population distribution is stable, and the total population is 34000, how many ani-

mals are in each age group?

4. At any time t, the populations of two species in symbiotic relationship (each population support-ing the other) are denoted by x(t) and y(t). They are given by the system of linked differentialequations:

x(t) = 3x(t) + 4y(t)

y(t) = 3x(t) + 2y(t)

Math 2061: Tutorial 6 (week 7) A.M. 18/4/2012

Linear Mathematics Tutorial 6 (week 7) Page 2

a) Express the system in the form X(t) = AX(t), where A is a 2 2 matrix andX(t) =

(x(t)y(t)

).

b) Find the eigenvalues and eigenvectors of A and write down the general solution of thesystem of linear differential equations.

c) Find the particular solution of the system for the initial conditions x(0) = 6, y(0) = 1.

5. Suppose that two tanks, each containing 100 litres of salt water of a different concentration, areconnected as shown below. Fresh water is pumped into tank A and then the water in the twotanks mixes as described by the diagram.

12 L/min 16 L/min

4 L/min 12 L/min

Tank A100 Litres

Tank B100 Litres

a) Let x(t) and y(t), respectively, be the number of grams of salt in tank A, tank B, respec-tively, at time t. Find a system of differential equations in x(t) and y(t).

b) Find the general solution of the system of differential equations that you found in part (a).c) Initially, tank A contains 40 grams of salt while tank B contains 20 grams of salt. Deter-

mine the number of grams of salt in each tank at time t.d) Confirm that the concentration of salt in each tank approaches zero over a long period of

time.

6. Define a sequence of numbers { an |n 0 } by a0 = 0, a1 = 1, a2 = 2 and for all n 0,

an+3 = an+2 +14an+1

14an.

a) Express the recurrence relation for an in matrix form.b) Find a formula for an in terms of n.c) Find lim

nan.

7. Let L =

b1 b2 b3s1 0 00 s2 0

be the Leslie matrix for an animal population with 3 age groups.

a) If is an eigenvalue of L show that

3 = b12 + s1b2+ s1s2b3.

b) Verify that u =

1s1

s1s22

is an eigenvector of L corresponding to eigenvalue .

c) Use parts a) and b) to find the unique positive eigenvalue, and a corresponding eigenvector,

for the following Leslie matrix M =

0 2 014

0 0

0 14

0

.

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Linear Mathematics Tutorial 6 (week 7) Page 3

Extra exercises

8. At any time t, the populations of two predator-prey species, x(t) and y(t), are given by thefollowing system of linked differential equations:

x(t) = 4x(t) 2y(t)

y(t) = x(t) + y(t)

Suppose that, initially, x = 300 and y = 200.a) Is x(t) the predator population or the prey population?b) Find formulas for x(t) and y(t) at time t.c) After a long time, what is the proportion of predators to prey?

9. Suppose that two tanks, each containing 200 litres of a mixture, are connected as shown inquestion 5. Fresh water is pumped into tank A at a rate of 15 L/min. Further, the mixture ispumped from tank A to tank B at a rate of 20 L/min, and from tank B to tank A at 5 L/min.Water then flows out of tank B at the rate of 15L/min. Suppose that initially tank A contains 60grams of salt while tank B contains pure water

a) Find a system of differential equations modelling the situation.b) Find the general solution of the system you found in part (a).c) Determine the number of grams of salt in each tank at time t .d) Confirm that the concentration in each tank approaches zero in the long term.

10. The Fibonacci numbers,0, 1, 1, 2, 3, 5, 8, 11, . . .

are defined by Fn+2 = Fn+1 + Fn, with F0 = 0 and F1 = 1.Find a formula for Fn in terms of n.

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