Using transitive closure to determine acyclic property on directed graph
It is possible to use Warshall's algorithm for that also. If there is is a negative entry on the diagonal, then the graph contains a negative cycle. If both the shortest path from i to j and the shortest path from j to i is not infinite, then the graph contains a cycle. If there are no i and j with this property, then the graph is acyclic. (This takes only $O(|V|^2)$ steps to check, while Warshall's algorithm itself takes $O(|V|^3)$ steps.)

However, it's faster to use depth first search and check that there are no backward edges in the depth first search tree. Also note that there exists an algorithm for computing the transitive closure (including the transitive reduction) in $O(|V|\cdot|E|)$ steps. (I think it uses a topological order as a starting point.)