I wish I can understand you

What do you mean by i-j-k walk?

This is difficult to explain and messy.2) Show that the number of walks of length 2 from vertex i to vertex j in G is the ijth entry of the matrix A^2.

But the general theorem is that the number of walks of length k between two verticies is the entry in the adjancey matrix A^k.

Try to think of it like this:

The a_ij entry in the adjancey matrix is the dot product between the i-th row and j-colomum. Thus, since it consists of only 1's and 0's is the sum of these numbers (after the dot product). The sum is the number of the entries which are common to both (otherwise is zero) this is your walk from these two verticies.