Using matrices to represent data: Networks | Matrices | Precalculus | Khan Academy

We're told this network diagram represents the different train routes between three cities. Each node is a city, and each directed arrow represents a direct bus route from city to city. So, for example, this arrow right over here, I guess, would represent a direct bus route that starts in city three and ends in city one.

While this arrow that has an arrow on both sides shows a route that both starts in city three and ends in city one, and a route that starts in city one and ends in city three. So it says complete the matrix so it represents the number of direct routes between the cities, where rows are starting points and columns are endpoints.

So this is the matrix right over here. I encourage you, if you feel so inspired, and I encourage you to feel so inspired, pause this video and see if you can fill out this matrix right over here. You have nine entries in this matrix for each of these combinations between the starting city and ending city. All right, now let's do it together.

So, what would go here? This would be the number of direct routes that start at city one and end at city one. So if we start at city one, are there any things that then end at city one? Well, no, it doesn't look like there's anything that starts at city one and ends at city one, so I will put a zero there.

What about this one right over here? Well, this is to start at city one and end at city two. So let's see, this starts at city one and ends at city two, so that's one. We get 2, and then we get 3, and then we get 4 because you can start at city one here and then end at city two. So we get 4.

Now, how many start at city one and end at city three? Pause this video and think about that actually. All right, we're going to start at city one and end at city three. Let me get another color out here. So I could start here and go on this route, and because this arrow ends at city three, so that's one.

This middle one does not start at city one and end at city three; it goes the other way around, so I'm not going to count that. This one right over here, I can go either way. So I could start at city one and end at city three because we have that arrow there, and those look like the only two that start at city one and end at city three.

So that looks like, go back to the original color, two routes right over there. Now, what about starting at city two and ending at city one? Well, if we start at city two and end at city one, these three over here, all of these start at one and end at two. They don't go the other way, but this one on top with the double arrows, you can go either way. So you could start at city two and end at city one, so there's one route here.

Let's see, start at city two, end at city two. Well, I don't see anything that looks like that for city two, so this is going to be a zero. And then, starts at city two, ends at city three. So starts at city two, ends at city three. This arrow doesn't count because this starts at three and ends at two, not the other way around, so we get a zero there as well.

And then let's go to city three. How many start at three and end at one? So start at three and end at one. So this two-way arrow, you could do that. You could start at three and end at one, so that's one. Then this one right over here starts at three and ends at one because we can see the arrow points to one right over there.

And then it looks like—actually, this one, I have so much that I've written here that I actually can't see too well the original. Let me erase it actually so I can make sure I'm seeing things properly. Yep, that one too looks like—so this one I can do, and then this one I can start at city three and end at city one as well, so it looks like we have three paths there.

Now start at city three, end at city two. That one's a little bit more straightforward; that's that path there, so that is one. And then starts at city three, ends at city three. Well, we have this one right over here; that's the only one, so I would put one.

So there you have it; we have filled in this matrix. So which city has the most incoming routes? Pause the video and think about that.

So the city with the most incoming routes, we can look at the cities that are the endpoints. And so city one has a total of zero plus one plus three, has four incoming routes. City two has a total of four plus one, five incoming routes, and city three has a total of two plus zero plus one, has three incoming routes.

So it looks like this would be city two. Sorry, yep, city two with five incoming routes. Which city has the most outgoing routes? Well, then we would just look the other way, actually. Pause the video and think about that.

Well, looks like city one has six outgoing routes. City two only has one outgoing route—I'm just adding up along the row—and city three has, looks like, five outgoing routes. So city one with zero plus four plus two, there's a total of six routes that start at city one and go out of the city. So that is city one with six routes.