Introduction:
We are given a simple undirected graph with at least 3 vertices and we need to find the number of 3-cycles in the graph using its adjacency matrix.

Explanation:
- Let's assume that the given graph has n vertices. The adjacency matrix A of the graph is an n x n matrix where A[i][j] is 1 if there is an edge between vertices i and j, and 0 otherwise.
- To find the number of 3-cycles in the graph, we need to consider all possible combinations of 3 vertices that form a cycle.
- We can represent a cycle of length 3 as (i, j, k), where i, j, and k are distinct vertices in the graph.
- For a cycle of length 3 to exist, there should be an edge between i and j, j and k, and k and i.
- If we consider the adjacency matrix A, then A[i][j] represents the presence of an edge between vertices i and j.
- To find the number of 3-cycles, we need to consider all possible combinations of i, j, and k such that A[i][j], A[j][k], and A[k][i] are all 1.
- We can achieve this by taking the matrix product of A with itself, i.e., A3 = A * A * A.
- The entry (i, j) in the matrix A3 represents the number of paths of length 3 from vertex i to vertex j in the given graph.
- If we divide A3 by 6, we are essentially counting each cycle 6 times (since a cycle can be traversed in 6 different ways).
- Therefore, the number of 3-cycles in the graph is given by the trace of A3 divided by 6, which is the sum of the diagonal elements of A3 divided by 6.
- Hence, the correct answer is option B, A3/6.

Conclusion:
The number of 3-cycles in a simple undirected graph can be found using its adjacency matrix. By taking the matrix product of the adjacency matrix A with itself three times (A3), we can count the number of paths of length 3 between every pair of vertices. Dividing A3 by 6 gives us the number of 3-cycles in the graph. Therefore, option B, A3/6, is the correct answer.