Question

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

\\( a = \

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

\\)

\\( a = \

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

\\)

Explanation:

Identify vertices and directed edges

The network has three vertices: \(P\), \(Q\), and \(R\).
The directed edges (arrows) are:

• From \(P\) to \(Q\)
• From \(P\) to \(R\)
• From \(R\) to \(Q\)

Set up the adjacency matrix structure

Let the rows and columns represent the vertices in alphabetical order: \(P\), \(Q\), \(R\).
An entry \(a_{ij} = 1\) if there is a directed edge from vertex \(i\) to vertex \(j\), and \(0\) otherwise.

Determine matrix elements

• Row 1 (\(P\)):
• To \(P\): \(0\)
• To \(Q\): \(1\)
• To \(R\): \(1\)
• Row 1 is \(

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

\)

• Row 2 (\(Q\)):
• To \(P\): \(0\)
• To \(Q\): \(0\)
• To \(R\): \(0\)
• Row 2 is \(

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

\)

• Row 3 (\(R\)):
• To \(P\): \(0\)
• To \(Q\): \(1\)
• To \(R\): \(0\)
• Row 3 is \(

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

\)

Combining these rows gives:

$$ A =
LATEXBLOCK3
$$

Answer:

• **(A) \(A =

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

\) (Correct answer)**

• (B) \(A =

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

\)