Matrices review.

1. Matrices
Workshop ...
…he talks in maths.

6. HOMEWORK
Suppose that the population of a small island is classified into three distinct
groups, children, teenagers and adults, and that each year:
• children are born at a proportion of • 1 % of the children die
6% of the adult population
• 10 % of children become teenagers • 5 % of the teenagers die
• 14% of teenagers become adults • 7 % of the adults die
This year, in Youngville, a city with a population of 25 000, there are 5 000
children, 18 000 teenagers and 2 000 adults.
(a) Write a row matrix that represents the current population of
Youngville.
(b) Write a 3 × 3 transition matrix that shows how the population is
changing over time.

7. HOMEWORK
Suppose that the population of a small island is classified into three distinct
groups, children, teenagers and adults, and that each year:
• children are born at a proportion of • 1 % of the children die
6% of the adult population
• 10 % of children become teenagers • 5 % of the teenagers die
• 14% of teenagers become adults • 7 % of the adults die
This year, in Youngville, a city with a population of 25 000, there are 5 000
children, 18 000 teenagers and 2 000 adults.
(c) Find the population of each group in 10 years from now.

8. HOMEWORK
If a train is late on one day, there is a 90% probability that the same train will
be on time the next day, while if the train is on time, there is a 20% chance it
will be late the next day. If in a given week it arrives on time on Monday,
compute the probabilities that it will be on time or late for each of the
subsequent days of the week. What would the corresponding probabilities have
been if the train had been late on the Monday?

9. I have two coins in a bag, a fair coin and a biased (double headed) coin. The fair
coin is tossed first. If the result of the previous toss is a head, then the biased
coin is tossed next. If the result of the previous toss is a tail, then the fair coin is
tossed next. Use transition matrices to describe the sequence of likely outcomes.
What happens if a different biased coin with both head and tail sides, but biased
60:40 in favour of tails is used instead?

10. A small community has Thrifty's Co-opers Food Mart
three grocery stores. The Bread (1loaf) $1.49 $1.59 $1.35
table shows each store's Milk (4 litre) $3.25 $3.75 $3.50
prices for a grocery order. Sugar (5 kg) $5.25 $4.95 $5.75
Eggs (1 doz) $1.35 $1.35 $1.20
Cereal (750 g) $3.75 $3.35 $3.95
(a) Use matrix operations to determine which store has the lowest
overall cost if the order consists of five loaves of bread, two 4-litre jugs
of milk, one 5-kg package of sugar, three dozen eggs, and one box of
cereal. Write your answer as matrix C.
(b) The community has organized a quot;Midnight Madnessquot; sale, and the
three stores are reducing all their prices as follows: Thrifty's by 15%;
Co-opers by 12%; and Food Mart by 18%. Write matrix E that shows all
the prices (rounded to the nearest cent) for each item during the sale.

11. Write a square network matrix where each element represents the number of
direct paths between points or towns. List the matrix entries in alphabetical
form.