Four cities A, B, C, D have these one-way direct flights: A → B, A → C, B → A, B → C, B → D, C → D…

Four cities $A, B, C, D$ have these one-way direct flights: $A \to B$, $A \to C$, $B \to A$, $B \to C$, $B \to D$, $C \to D$ and $D \to A$. Write the matrix $N$ with a $1$ where there is a flight from the row city to the column city. Use $N^3$ to find the number of routes from $B$ to $C$ with exactly three flights, and the number of three-flight routes that start at $A$.