Linear Algebra and Economics 
Steve Levandusky 
Input Coefficient Matrix 
We start by defining an input coefficient matrix
The elements, a_{ij}, indicate the amount needed of the good i to produce the good j. The sum of the elements in the j^{th} column represents the cost of producing one unit of good j, and is assumed to be less than 1 because when we combine all the inputs there is only one product. Since each good can be an input for the other goods, including itself, A is considered to be a square matrix. Now, let a_{ij}x_{j} be the portion demanded of good i for the production of good j. Also there exists a demand for the final good i by the consumer denoted d_{i}. For the market to be in equilibrium, the total supply x_{i} must be equal to the total demand for good i,
for i = 1, 2, …, n. And expressed as a matrix:
Or x = Ax + d, which can be rewritten as (I  A)x = d. The matrix (IA) is called the technology matrix and if nonsingular there exists a solution for the total output x needed for the final demand d, x = (I  A)^{1}d. [1]

To contact: 
Email: steve.levandusky@bucknell.edu 