# Is this still a vector space?

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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