Chapter 12.2, Problem 1E

Solution

To write: The matrix from the given information.

Required matrix is $A=\left[\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 1& 1\\ 0& 1& 0& 1\\ 1& 1& 1& 0\end{array}\right]$ .

Given information: Graph is,

  

Formula used:

If there is a path from $N_i$ to $N_j$ then $a_{ij}=1$ otherwise $a_{ij}=0$ .

Here, $i\ne j$ .

If $i=j$ , then $a_{ij}=0$ .

Explanation:

Consider, the graph as,

  

Let, the matrix is $A=\left[\begin{array}{cccc}{a}_{11}& a_{12}& a_{13}& a_{14}\\ a_{21}& a_{22}& a_{23}& a_{24}\\ a_{31}& a_{32}& a_{33}& a_{34}\\ a_{41}& a_{42}& a_{43}& a_{44}\end{array}\right]$ .

Since, there is no path between $N_1$ and $N_3$ .

Hence, $a_{13}=0$ and $a_{31}=0$ .

Also, all the diagonal elements are $0$ .

Then, $A=\left[\begin{array}{cccc}0& 1& 0& 1\\ 1& 0& 1& 1\\ 0& 1& 0& 1\\ 1& 1& 1& 0\end{array}\right]$ .