# Input-Output Matrix

## Definition

The Industry Transaction Matrix (or Transactions Table) is the fundamental quantitative information used in Input-Output Analysis. It concerns the flow of products from each industrial sector (considered as a producer) to each of the sectors, itself and others (considered as consumers).[1]

## Formula

### Single Region Case

Usually denoted as Z, if there are n sectors in an economy the matrix reads:

\begin{align} Z & = \begin{pmatrix} Z_{11} & Z_{12} & \cdots & Z_{1n} \\ Z_{21} & Z_{22} & \cdots & Z_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ Z_{n1} & Z_{n2} & \cdots & Z_{nn} \end{pmatrix} \end{align}
• The entries of the matrix may denote either monetary values (in some defined currency) or physical (activity) values, e.g. volumes.
• The matrix is a flow matrix, hence values refer to a particular time period.

### Multi-Regional Case

In the case of a Multiregional Input-Output Model where industrial sectors operate within distinct "regions" (e.g., countries) the transactions matrix is most convinently expressed as a Tensor $Z^{pq}_{ij}$ that captures exchanges between sector i and j located in regions p and q respectively. In practice this tensor is represented as a "rolled-out", Partitioned Matrix.

## Usage

This basic information from which an input-output model is developed is contained in an interindustry transactions table. The rows of such a table describe the distribution of a producer’s output throughout the economy. The columns describe the composition of inputs required by a particular industry to produce its output.

The Transaction Matrix is of fundamental importance and may underpin alternative possible input-output models.