Question

consider the given network. use a matrix to represent the data.

Explanation:

Identify the network vertices and directed edges

The network has three vertices: \(X\), \(Y\), and \(Z\). We define the adjacency matrix \(A\) where the rows and columns represent \(X\), \(Y\), and \(Z\) in alphabetical order. The element \(a_{ij}\) represents a directed edge from the vertex of row \(i\) to the vertex of column \(j\).

Determine the directed connections

Analyze the directed arrows between vertices:

• From \(X\): Arrows go to \(Y\) and \(Z\). Thus, row 1 (\(X\)) is \(

$$\begin{bmatrix} 0 & 1 & 1 \end{bmatrix}$$

\).

• From \(Y\): Arrows go to \(X\) and \(Z\). Thus, row 2 (\(Y\)) is \(

$$\begin{bmatrix} 1 & 0 & 1 \end{bmatrix}$$

\).

• From \(Z\): Arrows go to \(X\) and \(Y\). Thus, row 3 (\(Z\)) is \(

$$\begin{bmatrix} 1 & 1 & 0 \end{bmatrix}$$

\).

Construct the final adjacency matrix

Combine the rows to form the complete \(3 \times 3\) representation matrix:

$$
LATEXBLOCK3
$$

Answer:

• **(A) \(

$$\begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$$

\) (Correct answer)**

• (B) \(

$$\begin{bmatrix} 0 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 1 & 0 \end{bmatrix}$$

\)