Is this still a vector space?

Computer Science Level pending

Let us define the product of two matrices \(a \times b = c \) as follows : (in terms of the element at the \(i\)th row \(j\)th column).

\[c_{ij} = \min( a_{ik} + b_{kj} ) \ \ \ 1 \leq k \leq n \]

Now suppose we are given the adjacency matrix \(A\) of some directedgraph \(G\). Using our definition of multiplication above, what would \((A^5)_{34} \) represent? (Suppose there exists a path between node \(3\) and \(4\))

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