1. **Problem Statement:** Given an adjacency matrix of a graph, our tasks are: a) Verify the given parts about the graph. b) For each pair of vertices, determine how many paths of length 3 join them. c) List all the 3-cycles in the graph, considering different permutations of the cycle as distinct. 2. **Understanding the adjacency matrix:** The adjacency matrix $A$ of a graph with $n$ vertices is an $n \times n$ matrix where the element $A_{ij}$ is 1 if there is an edge from vertex $i$ to vertex $j$, and 0 otherwise. 3. **Paths of length 3 between vertices:** The number of paths of length 3 from vertex $i$ to vertex $j$ is given by the entry $(i,j)$ in the matrix $A^3$, where $A^3 = A \times A \times A$. To find $A^3$, multiply the adjacency matrix $A$ by itself three times. 4. **Finding 3-cycles:** A 3-cycle is a cycle of length 3, i.e., vertices $a$, $b$, and $c$ where $a$ is connected to $b$, $b$ to $c$, and $c$ back to $a$. Use the adjacency matrix to check for these triples: for vertices $i$, $j$, $k$, if $A_{ij} = 1$, $A_{jk} = 1$, and $A_{ki} = 1$, then $(i,j,k)$ forms a 3-cycle. Since different permutations count separately, count all ordered triples that satisfy this. 5. **Summary:** - Calculate $A^3$ to find the number of paths length 3 between all pairs. - Identify all ordered triples $(i,j,k)$ where the above holds to enumerate different 3-cycles. *Note:* Without the explicit adjacency matrix, numerical answers cannot be computed here. Use the above method with the given matrices to proceed.