Reachability matrix in time $O(|V| \cdot |E|)$
This algorithm doesn't work. Consider a graph with four vertices 1,2,3,4 and edges $1\to2\to3\to4$. Your algorithm outputs $T(1,4)=0$ even though 4 is reachable from 1.

The running time analysis of the algorithm can be improved to $O(VE)$. The first loop takes time $O(V^2)$. The number of pairs $(v,u)$ in the second loop is the number of edges $E$, and for each pair $(v,u)$, the maximal number of $w$s is $V$. So the second loop runs in time $O(VE)$.