Question: 1. Let G be a directed graph, possibly with parallel arcs, and let A be its adjacency matrix. Then A may not be a Boolean matrix. Prove that the i, j entry of matrix a2 is the number of paths of length 2 from node i to node j.
2. Let A be the adjacency matrix of a directed graph G, possibly with parallel arcs. Prove that the i, j entry of matrix an gives the number of paths of length n from node i to node j.