# Incidence Matrix [A_{a}]

The incidence matrix symbolically describes a network. It also facilitates the testing and identification of the independent variables. Incidence matrix is a matrix which represents a graph **uniquely**.

For a given graph with

*n*nodes and

*b*branches, the complete incidence matrix

*A*is a rectangular matrix of order

_{a}*n*Ã—

*b*, whose elements have the following values.

Number of columns in [

*A*] = Number of branches =_{a}*b*Number of rows in [

*A*] = Number of nodes =_{a}*n**A*= 1, if branch

_{ij}*j*is associated with node

*i*and oriented away from node

*i*.

= â€“1, if branch
= 0, if branch

*j*is associated with node*i*and oriented towards node*i*.*j*is not associated with node*i*.

# Properties of Complete Incidence Matrix

- The sum of the entries in any column is zero.
- The determinant of the incidence matrix of a closed loop is zero.
- The rank of incidence matrix of a connected graph is (
*n*â€“1).