The need to take linear combinations of rows and columns in tables of numbers arises often in practice. For instance, this is a map of part of Vermont and New York.
In part because of Lake Champlain, there are no roads directly connecting some pairs of towns. For instance, there is no way to go from Winooski to Grand Isle without going through Colchester. (To simplify the graph many other roads and towns have been omitted. From top to bottom of this map is about forty miles.)

(a) The adjacency matrix of a map is the square matrix whose i, j entry is the number of roads from city i to city j (all (i, i) entries are 0). Produce the adjacency matrix of this map, with the cities in alphabetical order.
(b) A matrix is symmetric if it equals its transpose. Show that an adjacency matrix is symmetric. (These are all two-way streets. Vermont doesn't have many one-way streets.)
(c) What is the significance of the square of the incidence matrix? The cube?

Transcribed Image Text:

Swanton Grand Isle Colchester Winooski Burlington