| Consumer\Producer | Farmer | Builder | Tailor |
| Farmer | 7/16 | 1/2 | 3/16 |
| Builder | 5/16 | 1/6 | 5/16 |
| Tailor | 1/4 | 1/3 | 1/2 |
Let
\[I_1, \: I_2, \: I_3\]
be the income of the farmer, builder and tailor respectively. From the table above, the farmers, builders and tailors expenditures are: \[I_1=7/16I_1+1/2I_2+3/16I_3\]
\[I_2=5/16I_1+1/6I_2+5/16I_3\]
\[I_3=1/4I_1+1/3I_2+1/2I_3\]
We can write this in matrix form as
\[\left( \begin{array}{ccc} 7/16 & 1/2 & 3/16 \\ 5/16 & 1/6 & 5/16 \\ 1/4 & 1/3 & 1/2 \end{array} \right) \begin{pmatrix}I_1\\I_2\\I_3\end{pmatrix}=\begin{pmatrix}I_1\\I_2\\I_3\end{pmatrix}\]
\[(\left( \begin{array}{ccc} 7/16 & 1/2 & 3/16 \\ 5/16 & 1/6 & 5/16 \\ 1/4 & 1/3 & 1/2 \end{array} \right)- \left( \begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array} \right) ) \begin{pmatrix}I_1\\I_2\\I_3\end{pmatrix}=\begin{pmatrix}0\\0\\0\end{pmatrix}\]
\[\left( \begin{array}{ccc} -9/16 & 1/2 & 3/16 \\ 5/16 & -5/6 & 5/16 \\ 1/4 & 1/3 & -1/2 \end{array} \right) \begin{pmatrix}I_1\\I_2\\I_3\end{pmatrix}=\begin{pmatrix}0\\0\\0\end{pmatrix}\]
We can reduce the matrix above to
\[\left( \begin{array}{ccc} 1 & 0 & -1 \\ 0 & 1 & -3/4 \\ 0 & 0 & 0 \end{array} \right)\]
Now we can take
\[I_2=3/4I_2, \: I_1=I_3\]