## Maximising Revenue With Matrices

Suppose a fruit grower wants to find where to sell his produce to maximise his profit. He has 900, 700 and 400 boxes of oranges, grapefruit and tangerines respectively. We can represent this by a vector\[\begin{pmatrix}900\\700\\400\end{pmatrix}\]

.The table shows the price of a box of each type of fruit in nations of the UK..

Nation\Fruit | Orange | Grapefruit | Tangerine |

England | 4 | 2 | 3 |

Northern Ireland | 5 | 1 | 2 |

Scotland | 4 | 3 | 2 |

Wales | 3 | 2 | 5 |

\[\left( \begin{array}{cccc} 4 & 2 & 3 \\ 5 & 1 & 2 \\ 4 & 3 & 2 \\ 3 & 2 & 5 \end{array} \right)\]

.We can find a vector representing the total revenue from selling all the produce in each country by multiplying the above matrix by the above vector.

\[\left( \begin{array}{cccc} 4 & 2 & 3 \\ 5 & 1 & 2 \\ 4 & 3 & 2 \\ 3 & 2 & 5 \end{array} \right) \begin{pmatrix}900\\700\\400\end{pmatrix}=\begin{pmatrix}4 \times 900+2 \times 700 +3 \times 400\\5 \times 900+1 \times 700+_2 \times 400\\4 \times 900 + 3 \times 700 + 2 \times 400 \\ 3 \times 900 + 2 \times 700 + 5 \times 400 \end{pmatrix}= \begin{pmatrix}6200\\6000\\6500\\6100\end{pmatrix}\]

.The fruit grower show sell his fruit in Scotland.