MAT 280
Adjacency Matrices and Power of Matrices (Movie: Good Will Hunting)
1. Find the adjacency matrix A. (This describes the number of ways to travel from one endpoint to another
endpoint using a one-step walk and the matrix records our results.)
1 2 3 4
1 _
2  _
3 _

4 _
_
_
_
_
_
_
_
_
_
_ 
_

_
1 2 3 4
1 0
2 1
3 0

4 1
**Notice the adjacency matrix A is symmetric!
1 0 1
0 2 1 
2 0 0

1 0 0
0
1
A
0

1
1 0 1
0 2 1 
2 0 0

1 0 0
2. First find the matrix giving the number of two-step walks.
From 1 to 1:
From 1 to 2:
From 1 to 3:
From 1 to 4:
1 2 3 4
1 2
2  _
3 _

4 _
1
2
_
_
_
_
_
_
1
_ 
_

_
2
1
2
2
Now find A by hand. Notice that A = 
2

1
**Notice A2 is also symmetric!
1 2 1
6 0 1 
0 4 2

1 2 2
2: Continued
Now find the matrix giving the number of three-step walks.
From 1 to 1:
From 1 to 2:
1 2 3 4
1 2
2  _
3 _

4 _
7
_
_
_
_
_
_
_
_
_ 
_

_
 2 7 2 3
7 2 12 7 

Now find A3 by hand. Notice that A3 = 
 2 12 0 2


 3 7 2 2
**Notice A3 is also symmetric!