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3.30 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 connect- ing some pairs of towns. For in- stance, there is no way to go from Winooski to Grand Isle without go- ing through Colchester. (To sim- plify the graph many other roads and towns have been omitted. From top to bottom of this map is about forty miles.) Grand Isle Swanton Colchester I Winooski Burlington (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, taking 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?

3.30 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 connect- ing some pairs of towns. For in- stance, there is no way to go from Winooski to Grand Isle without go- ing through Colchester. (To sim- plify the graph many other roads and towns have been omitted. From top to bottom of this map is about forty miles.) Grand Isle Swanton Colchester I Winooski Burlington (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, taking 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?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please do part A,B,C and please show step by step and explain

3.30 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 connect- ing some pairs of towns. For in- stance, there is no way to go from Winooski to Grand Isle without go- ing through Colchester. (To sim- plify the graph many other roads and towns have been omitted. From top to bottom of this map is about forty miles.) Grand Isle Swanton Colchester I Winooski Burlington (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, taking 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:3.30 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 connect- ing some pairs of towns. For in- stance, there is no way to go from Winooski to Grand Isle without go- ing through Colchester. (To sim- plify the graph many other roads and towns have been omitted. From top to bottom of this map is about forty miles.) Grand Isle Swanton Colchester I Winooski Burlington (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, taking 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?
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