Question

there are three employees in an office leon, lussy, and lerry. they plan to do a work. leon plans to start work with lerry, lussy would like to work with leon. lerry would like to work with both leon and lussy. organize the given data in a matrix form.

\\(\

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

\\)

\\(\

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

\\)

Explanation:

Define the matrix structure

Let the rows and columns represent Leon, Lussy, and Lerry in order:

• Row 1 / Column 1: Leon
• Row 2 / Column 2: Lussy
• Row 3 / Column 3: Lerry

An entry \(a_{ij} = 1\) represents that employee \(i\) plans or would like to work with employee \(j\), and \(a_{ij} = 0\) otherwise.

Translate the preferences into matrix rows

• Leon (Row 1) plans to work with Lerry:

$$ a_{11} = 0, \quad a_{12} = 0, \quad a_{13} = 1 \implies
LATEXBLOCK0
$$

• Lussy (Row 2) would like to work with Leon:

$$ a_{21} = 1, \quad a_{22} = 0, \quad a_{23} = 0 \implies
LATEXBLOCK1
$$

• Lerry (Row 3) would like to work with both Leon and Lussy:

$$ a_{31} = 1, \quad a_{32} = 1, \quad a_{33} = 0 \implies
LATEXBLOCK2
$$

Construct the final matrix

Combine the rows to form the complete adjacency matrix:

$$
LATEXBLOCK3
$$

This matches the second option.

Answer:

• (A) \(

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

\)

• **(B) \(

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

\) (Correct answer)**