Example 1.  Consider the closed three sector economy   consisting of say:  Energy, Manufacturing, and Services where the input-output matrix is given by

.   Find the production vector  .   

Solution 1.

Enter the matrix  .  

If   is an eigenvalue of    and    an eigen-pair, then the solution to   will be a multiple of the eigenvector  .
Let us verify that    is an eigenvalue of  .  

The eigen-pair     is easily computed using Mathematica's subroutines  Eigenvalues and  Eigenvectors.  

Since    is a solution for any constant  c, we are permitted to choose any multiple of   we desire for the solution.  For illustration:

Aside.  Any value of  c  is permitted, it is your choice.  When we get

We are done.

How did we get that "nice" vector ?
By using exact arithmetic and the following calculations.

Form the homogeneous system .   
and the augmented matrix     and find its reduced row echelon form.

The step used in finding the row reduced echelon form are familiar row operations.

The equations for this augmented matrix are  

      

There is one free variables which we choose to be  .   It is used in computing    and   .  
Use and solve the previous equations for    and   

      

Get    
    

The solution vector    is

We are done.

Aside.  We can verify that this is the solution by direct multiplication A X.  This is just for fun !

We are really done.

Aside.  Iteration can be used to solve for the solution of the closed Leontief model.  If the sum of the production levels is known and a starting vector is given   then the simple iteration    will converge to  .  Because the largest eigenvalue is  ,  this is a simplified version of the "power method" for  finding the dominant eigen-pair.  For the example above   and the following iteration will converge.

(c) John H. Mathews 2004