Define a step as the time it takes a person to e$-$mail everyone on his or her list. $($Thus$,$ in one step, gossip gets from Ann to both Carla and Ehaz.$)$ If Bert hears a rumor, how many steps will it take for everyone else to hear the rumor? What matrix calculation reveals this?

If Ann hears a rumor, how many steps will it take for everyone else to hear the rumor? What matrix calculation reveals this?

In general, if A is the adjacency matrix of a digraph, how can we tell if vertex i is connected to vertex j by a path $($of some length$)?$