Question

use a matrix to represent the given network.

\\(\

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

\\)

\\(\

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

\\)

Explanation:

Identify the vertices and directed edges

Using the Directed Graph knowledge point, we identify the vertices and directed edges from the network diagram.
The vertices are \(X\), \(Y\), and \(Z\). We order them alphabetically: row/column 1 corresponds to \(X\), row/column 2 to \(Y\), and row/column 3 to \(Z\).
The directed edges (arrows) in the network are:

• From \(Z\) to \(X\)
• From \(X\) to \(Y\)
• From \(Z\) to \(Y\)

Construct the adjacency matrix

Using the Adjacency Matrix Construction knowledge point, we construct a \(3 \times 3\) matrix where the element in row \(i\) and column \(j\) is \(1\) if there is a directed edge from vertex \(i\) to vertex \(j\), and \(0\) otherwise.

• Row 1 (\(X\)):
• Edge from \(X\) to \(X\): None (\(0\))
• Edge from \(X\) to \(Y\): Yes (\(1\))
• Edge from \(X\) to \(Z\): None (\(0\))
• Row 1 is \(

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

\).

• Row 2 (\(Y\)):
• Edge from \(Y\) to \(X\): None (\(0\))
• Edge from \(Y\) to \(Y\): None (\(0\))
• Edge from \(Y\) to \(Z\): None (\(0\))
• Row 2 is \(

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

\).

• Row 3 (\(Z\)):
• Edge from \(Z\) to \(X\): Yes (\(1\))
• Edge from \(Z\) to \(Y\): Yes (\(1\))
• Edge from \(Z\) to \(Z\): None (\(0\))
• Row 3 is \(

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

\).

Combine into the final matrix

Combining the rows, we get the complete adjacency matrix:

$$
LATEXBLOCK3
$$

This matches the first option.

Answer:

• **\(

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

\) (Correct answer)**

• \(

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

\)