How many units of each are needed from crude oil to satisfy the orders?
We can represent the conversion from one fuel to another in a table.
| Consumed\Product | Petrol | Oil | Gas |
| Petrol | 0 | 0 | 1/5 |
| Oil | 1 | 1/5 | 2/5 |
| Gas | 1 | 2/5 | 1/5 |
\[C =\left( \begin{array}{ccc} 0 & 0 & 1/5 \\ 1 & 1/5 & 2/5 \\ 1 & 2/5 & 1/5 \end{array} \right)\]
.Let
\[P, \: O, \: G\]
represent the production of petrol, oil and gas respectively. We can represent conversion between fuels by \[\left( \begin{array}{ccc} 0 & 0 & 1/5 \\ 1 & 1/5 & 2/5 \\ 1 & 2/5 & 1/5 \end{array} \right) \mathbf{Y}=\begin{pmatrix}P\\O\\G\end{pmatrix}\]
.Let
\[\mathbf{D}=\begin{pmatrix}100\\100\\100\end{pmatrix}\]
be the demand vector, then \[\left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right) \begin{pmatrix}P\\O\\G\end{pmatrix} - \left( \begin{array}{ccc} 0 & 0 & 1/5 \\ 1 & 1/5 & 2/5 \\ 1 & 2/5 & 1/5 \end{array} \right) \begin{pmatrix}P\\O\\G\end{pmatrix}=\begin{pmatrix}100\\100\\100\end{pmatrix}\]
\[\left( \begin{array}{ccc} 1 & 0 & -1/5 \\ -1 & 4/5 & 3/5 \\ -1 & -2/5 & 4/5 \end{array} \right) \begin{pmatrix}P\\O\\G\end{pmatrix}=\begin{pmatrix}100\\100\\100\end{pmatrix}\]
\[\begin{pmatrix}P\\O\\G\end{pmatrix}={\left( \begin{array}{ccc} 1 & 0 & -1/5 \\ -1 & 4/5 & 3/5 \\ -1 & -2/5 & 4/5 \end{array} \right)}^{-1}\begin{pmatrix}100\\100\\100\end{pmatrix}=\left( \begin{array}{ccc} 2 & 1/3 & 2/3 \\ 5 & 5/2 & 5/2 \\ 5 & 5/3 & 10/3 \end{array} \right) \begin{pmatrix}100\\100\\100\end{pmatrix}=\begin{pmatrix}300\\1000\\1000\end{pmatrix}\]