Matrices have various engineering applications, as we shall see. For instance, they can be used to characterize connections in electrical networks, in nets of roads, in production processes, etc., as follows.

The network in Fig. 155 consists of six branches (connections) and four nodes (points where two or more branches come together). One node is the reference node (grounded node, whose voltage is zero). We number the other nodes and number and direct the branches. This we do arbitrarily. The network can now be described by a matrix A = [a_jk], where

A is called the nodal incidence matrix of the network. Show that for the network in Fig. 155 the matrix A has the given form.

(a) Find the nodal incidence matrices of the networks in Fig. 156.

(b) Sketch the three networks corresponding to the nodal incidence matrices

+1 if branch k leaves node -1 if branch k enters node O if branch k does not touch node ajk 2) (3) 2 4 (Reference node) Branch 4 6. Node (1) -1 -1 Node (2) Node (3) -1 -1 2.