Solved on Nov 12, 2023
Find the most accurate statement about the given matrix , which could be the adjacency matrix of a graph or a digraph.
STEP 1
Assumptions1. The given matrix is a6x6 matrix.
. The entries of the matrix are either0 or1.
3. The matrix represents an adjacency matrix of a graph or a digraph.
4. In an adjacency matrix for a graph, the matrix is symmetric because the edges have no direction. If there is an edge from vertex i to vertex j, there is also an edge from vertex j to vertex i.
5. In an adjacency matrix for a digraph, the matrix does not have to be symmetric because the edges have direction. If there is an edge from vertex i to vertex j, there is not necessarily an edge from vertex j to vertex i.
STEP 2
First, we need to check if the given matrix is symmetric. A matrix is symmetric if it is equal to its transpose.
STEP 3
The transpose of a matrix is obtained by interchanging its rows and columns. Let's find the transpose of the given matrix.
STEP 4
The transpose of the matrix A is
STEP 5
Now, we compare the given matrix A with its transpose A^. If they are equal, then the matrix is symmetric and it represents a graph. If they are not equal, then it represents a digraph.
STEP 6
By comparing the matrix A and its transpose A^, we can see that they are not equal. Therefore, the matrix A is not symmetric.
STEP 7
Since the matrix A is not symmetric, it cannot represent a graph. Therefore, it must represent a digraph.
The statement "This must be the adjacency matrix for a digraph (it can't be a regular graph)" is most true.
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