Let A be an adjacency matrix of a graph G. The ij entry in the matrix A^k , gives

The ij entry in the matrix A^k gives the number of distinct paths of length k from vertex i to vertex j in the graph G.

Here's the step by step explanation:

1. An adjacency matrix A of a graph G is a square matrix where the entry A[i][j] is one if there is an edge between the vertices i and j, otherwise it's zero.

2. When you multiply the adjacency matrix by itself (k times), you get the matrix A^k.

3. The entry in the i-th row and j-th column of A^k (denoted as A^k[i][j]) gives the number of distinct paths of length k from vertex i to vertex j in the graph G.

4. This is because each multiplication corresponds to taking one step along an edge in the graph. So, after k multiplications (or steps), you end up with the number of ways to get from vertex i to vertex j in exactly k steps.