1. **State the problem:** We need to complete the adjacency matrix for a graph with vertices P, Q, R, S, and T. 2. **Recall the adjacency matrix definition:** An adjacency matrix $A$ for a graph with vertices $V = \{P, Q, R, S, T\}$ is a $5 \times 5$ matrix where the entry $A_{ij}$ is 1 if there is an edge between vertex $i$ and vertex $j$, and 0 otherwise. 3. **List the edges given:** The edges are $P-R$, $P-Q$, $P-T$, $Q-T$, $R-S$, and $S-T$. 4. **Fill the matrix:** Since the graph is undirected, the matrix is symmetric. For each edge, set $A_{ij} = A_{ji} = 1$. - $P-Q$: $A_{P,Q} = A_{Q,P} = 1$ - $P-R$: $A_{P,R} = A_{R,P} = 1$ - $P-T$: $A_{P,T} = A_{T,P} = 1$ - $Q-T$: $A_{Q,T} = A_{T,Q} = 1$ - $R-S$: $A_{R,S} = A_{S,R} = 1$ - $S-T$: $A_{S,T} = A_{T,S} = 1$ 5. **No self-loops:** Diagonal entries $A_{ii} = 0$. 6. **Final adjacency matrix:** $$ A = \begin{bmatrix} 0 & 1 & 1 & 0 & 1 \\ 1 & 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 1 \\ 1 & 1 & 0 & 1 & 0 \end{bmatrix} $$ This matrix correctly represents all edges in the graph.