Modelling a Simple Community

Suppose we are to model a simple closed society with a with three individuals: farmer, a tailor and a builder. During the year the proportion of each commodity produced by each individual is given in the table.
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
One of each commodity is produced during the year.
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 
\[I_2=3/4I_2, \: I_1=I_3\]

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