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Recall that the adjacency matrix of a network with $k$ nodes $N_1,$dots,$N_k$ is defined to be the $kk$ matrix A whose $(i,j)$ th entry equals $1$ if there is an edge $($i$.$e$.$ a direct link$)$

between the nodes $N_i$ and $N_j$ and $0$ otherwise. Remarkably, the $n-$fold product of A with itself has a concrete meaning in terms of the network. Namely, the $(i,j)$ th entry of

$A^n$ is the numbers of $n-$step paths joining the node $N_i$ to the node $N_j,$ where taking one step entails traversing one edge. Now consider the network shown below.

Use the adjacency matrix of the network to determine the number of three$-$step paths from the red node to the blue node.