Math / Discrete

QuestionConsider the following matrix:
Which of the following is most true? This is not a valid adjacency matrix - that is, there is no graph or digraph that has this for its adjacency matrix This could be the adjacency matrix for either a graph or a digraph This must be the adjacency matrix for a graph (it can't be a digraph) This must be the adjacency matrix for a digraph (it can't be a regular graph)

Studdy Solution

STEP 1

1. The matrix $A$ is a square matrix.
2. The matrix $A$ is being evaluated to determine if it can represent an adjacency matrix for a graph or digraph.
3. An adjacency matrix for a graph is symmetric, while for a digraph it is not necessarily symmetric.

STEP 2

1. Define the properties of adjacency matrices for graphs and digraphs.
2. Analyze the given matrix to determine its symmetry.
3. Determine the most accurate statement about the matrix.

STEP 3

An adjacency matrix for an undirected graph is symmetric, meaning $A[i][j] = A[j][i]$ for all $i, j$ .
An adjacency matrix for a directed graph (digraph) does not require symmetry.

STEP 4

Examine the given matrix $A$ to check for symmetry:
$\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ \end{bmatrix}$
Check if $A[i][j] = A[j][i]$ for all $i, j$ .

STEP 5

Compare elements: - $A[1][2] = 1$ and $A[2][1] = 0$ , not equal. - $A[1][3] = 0$ and $A[3][1] = 1$ , not equal. - $A[2][3] = 1$ and $A[3][2] = 0$ , not equal.
The matrix is not symmetric.

STEP 6

Since the matrix is not symmetric, it cannot be the adjacency matrix for an undirected graph.
However, it can be the adjacency matrix for a directed graph (digraph) since symmetry is not required.
The most accurate statement about the matrix is:
$\boxed{\text{This must be the adjacency matrix for a digraph (it can't be a regular graph)}}$

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Studdy Solution

STEP 1

1. The matrix $A$ is a square matrix.
2. The matrix $A$ is being evaluated to determine if it can represent an adjacency matrix for a graph or digraph.
3. An adjacency matrix for a graph is symmetric, while for a digraph it is not necessarily symmetric.

STEP 2

1. Define the properties of adjacency matrices for graphs and digraphs.
2. Analyze the given matrix to determine its symmetry.
3. Determine the most accurate statement about the matrix.

STEP 3

An adjacency matrix for an undirected graph is symmetric, meaning $A[i][j] = A[j][i]$ for all $i, j$ .
An adjacency matrix for a directed graph (digraph) does not require symmetry.

STEP 4

Examine the given matrix $A$ to check for symmetry:
$\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ \end{bmatrix}$
Check if $A[i][j] = A[j][i]$ for all $i, j$ .

STEP 5

Compare elements: - $A[1][2] = 1$ and $A[2][1] = 0$ , not equal. - $A[1][3] = 0$ and $A[3][1] = 1$ , not equal. - $A[2][3] = 1$ and $A[3][2] = 0$ , not equal.
The matrix is not symmetric.

STEP 6

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Studdy Solution

STEP 1

1. The matrix A A A is a square matrix.2. The matrix A A A is being evaluated to determine if it can represent an adjacency matrix for a graph or digraph.3. An adjacency matrix for a graph is symmetric, while for a digraph it is not necessarily symmetric.

STEP 2

1. Define the properties of adjacency matrices for graphs and digraphs.2. Analyze the given matrix to determine its symmetry.3. Determine the most accurate statement about the matrix.

STEP 3

An adjacency matrix for an undirected graph is symmetric, meaning A[i][j]=A[j][i] A[i][j] = A[j][i] A[i][j]=A[j][i] for all i,j i, j i,j.An adjacency matrix for a directed graph (digraph) does not require symmetry.

STEP 4

Examine the given matrix A A A to check for symmetry:[0100001010010100]\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ \end{bmatrix}⎣⎡​0010​1001​0100​0010​⎦⎤​Check if A[i][j]=A[j][i] A[i][j] = A[j][i] A[i][j]=A[j][i] for all i,j i, j i,j.

STEP 5

Compare elements: - A[1][2]=1 A[1][2] = 1 A[1][2]=1 and A[2][1]=0 A[2][1] = 0 A[2][1]=0, not equal. - A[1][3]=0 A[1][3] = 0 A[1][3]=0 and A[3][1]=1 A[3][1] = 1 A[3][1]=1, not equal. - A[2][3]=1 A[2][3] = 1 A[2][3]=1 and A[3][2]=0 A[3][2] = 0 A[3][2]=0, not equal.The matrix is not symmetric.

STEP 6

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1. The matrix $A$ is a square matrix.
2. The matrix $A$ is being evaluated to determine if it can represent an adjacency matrix for a graph or digraph.
3. An adjacency matrix for a graph is symmetric, while for a digraph it is not necessarily symmetric.

1. Define the properties of adjacency matrices for graphs and digraphs.
2. Analyze the given matrix to determine its symmetry.
3. Determine the most accurate statement about the matrix.

An adjacency matrix for an undirected graph is symmetric, meaning $A[i][j] = A[j][i]$ for all $i, j$ .
An adjacency matrix for a directed graph (digraph) does not require symmetry.

Examine the given matrix $A$ to check for symmetry:
$\begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ \end{bmatrix}$
Check if $A[i][j] = A[j][i]$ for all $i, j$ .

Compare elements: - $A[1][2] = 1$ and $A[2][1] = 0$ , not equal. - $A[1][3] = 0$ and $A[3][1] = 1$ , not equal. - $A[2][3] = 1$ and $A[3][2] = 0$ , not equal.
The matrix is not symmetric.