A market research study of shoppers showed that the buying preferences for the three olive oils, Carmani (), Linelli () and Ohana (), change from month to month according to the transition matrix  below.

The initial state matrix below shows the number of shoppers who bought each brand of olive oil in July 2021.

Let represent the state matrix describing the number of shoppers buying each brand months after July 2021.

1. How many of these 8000 shoppers bought a different brand of olive oil in August 2021 from the brand bought in July 2021?   (1 mark)
   

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2. Using the rule , complete the matrix below.   (1 mark)
   

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3. Consider the shoppers who were expected to buy Carmani olive oil in August 2021.
4. What percentage of these shoppers also bought Carmani olive oil in July 2021?
5. Round your answer to the nearest percentage.   (1 mark)
   

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4. Write a calculation that shows Ohana olive oil is the brand bought by 50% of these shoppers in the long run.   (1 mark)
   

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5. Further research suggests more shoppers will buy olive oil in the coming months.
6. A rule to model this situation is  , where represents the state matrix describing the number of shoppers months after July 2021.

1. represents the extra number of shoppers expected to buy Ohana olive oil each month.
2. If  , what is the value of ?   (1 mark)
   

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The matrix    is determined from the matrix    using the recurrence relation  , where

and  is a column matrix. 

Matrix    is equal to