A Markov matrix is a transition matrix that models a sequence of events whose changes are dependent only on their

most recent state. The entries of a Markov matrix are all probabilities between $0$ and $1$ inclusive, and in particular the

$(i,j)-$th entry is the probability that an event in the $j-$th state changes to the $i-$th state in the next step.

Writing the recorded values for each state at time $t$ in a vector $x_t,$ the Markov matrix $M$ governs the relation

$x_{t+1}=M{x}_t.$

A certain store sells three different kinds of cereal created by three different companies: brands $A,B$ and $C.$ Anyone

who visits the store does so regularly, and purchases one box of cereal each week.

It is observed that someone who buys brand $A\text{'}$s product is $77\%$ likely to buy it again on their next visit, $7\%$ likely to

buy brand $B$ on their next visit, and $16\%$ likely to buy brand $C$ on their next visit.

Someone who buys brand $B^{\text{'}}$ s product is $71\%$ likely to buy it again on their next visit, $7\%$ likely to buy brand $A$ on

their next visit, and $22\%$ likely to buy brand $C$ on their next visit.

Someone who buys brand $C\text{'}$s product is $70\%$ likely to buy it again on their next visit, $9\%$ likely to buy brand $A$ on

their next visit, and $21\%$ likely to buy brand $B$ on their next visit.

Assigning brands $A,B$ and $C$ to row$/$column numbers $1,2$ and $3$ respectively, what is the Markov matrix that

represents the above information?

Note: The Matlab syntax for the matrix $([1,2,3],[4,5,6],[7,8,9])$ is

$[[1,2,3]$;$[4,5,6]$;$[7,8,9]].$

With this in mind, we can answer the following questions:

In one particular week, a survey reveals that out of all the customers who purchased cereal, $46\%$ bought brand

$A,22\%$ bought brand $B,$ and $32\%$ bought brand $C.$ In the following week, what percentage of customers

purchasing cereal will buy brand $A?$

$\%($answer correct to $2$ decimal places$)$

What is the expected change in percentage of customers who prefer brand $B$ five weeks after the above

survey? $($Give your answer as a negative number if the share decreases$).$