Question: Let G = (V, E) be a finite simple graph with V' = {1,2,...,n}. The adjacency matrix for G is the n. X n matrix

Let G = (V, E) be a finite simple graph with V' = {1,2,...,n}. The adjacency matrix for G is the n. X n matrix A such that e 1, if there is an edge between i and j 7710, if there is no edge between i and j where A;; denotes the entry in row i and column j. Prove that for all positive integers m and all vertices p, q V, there is a walk from p to q of length m if and only if (Am]m # 0 (ie. the (p, q) -entry of A is non-zero). You may use the fact that for two n x n matrices A, B, the (%, j)-entry is given by then sum n (AB);; = ZAakBkj =1 Consider using induction

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