Complete the dominance matrix, \begin{bmatrix} 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \\ ⬚ & ⬚ & ⬚ & 1 \\ ⬚ & ⬚ & ⬚ & ⬚ \end{bmatrix} .
The given dominance matrix represents the results of a darts competition between four people:
Which player lost the most games?
Who was the winner between Yuri and Edward?
Which player has the most wins?
The results of a tennis competition between four friends are listed below:
Create a dominance matrix for the tennis competition.
- Sally beat Bob
- Valentina beat Lain
- Lain beat Sally
- Bob beat Valentina
- Sally beat Valentina
- Lain beat Bob
The results of a handball competition between four friends are listed below:
Create a dominance matrix, M, for the handball competition.
Hence find the second stage dominance matrix, M^2.
Find M + M^2.
- Patricia beat Mohamad
- Joanne beat Ned
- Joanne beat Patricia
- Ned beat Mohamad
- Patricia beat Ned
- Mohamad beat Joanne
The results of an arm wrestling competition between four friends are listed below:
Create the dominance matrix, M, for the arm wrestling competition.
Hence find the second stage dominance matrix, M^2.
Find M + \dfrac{1}{2} M^2.
- Skye beat Aaron
- Sally beat James
- Skye beat James
- Sally beat Aaron
- Sally beat Skye
- James beat Aaron
Five soccer teams, A, B, C, D and E play a tournament where no draws are allowed (if necessary, games are decided by penalty shoot outs). The results are displayed below:
Construct a network diagram that represents the tournament results.
Create a dominance matrix, M, for the competition.
Hence find the second stage dominance matrix M^2.
Find T = M + M^2.
Hence rank the teams from highest to lowest dominance.
- E defeated B
- C defeated D
- A defeated C
- E defeated D
- B defeated D
- E defeated A
- E defeated C
- A defeated B
- A defeated D
- B defeated C
For each network diagram:
Create a dominance matrix, M, for the competition.
Hence find the second stage dominance matrix M^2.
Find T = M + M^2.
Hence determine the dominant node for the network.
A salesperson works in the towns A, B, C and D as shown in the network diagram:
Create an adjacency matrix, M, for this network.
How many one stage paths are there between C and A?
Determine the second stage adjacency matrix, M^2.
How many two stage paths are there between A and C?
The network diagram indicates a competition between five people. An arrow from A to B means that A defeated B.
Create a dominance matrix, M, for the competition.
Determine the second stage dominance matrix M^2.
Determine M + M^2.
Consider the following network diagram:
Create a dominance matrix, M, for this network.
Determine the second stage dominance matrix M^2 for the network.
Determine T = M + M^2.
Hence rank the nodes from highest to lowest dominance according to T.
The network diagram below represents a river system flowing from I to F:
Create a dominance matrix, M, for this network.
Determine the second stage dominance matrix M^2 for the system.
Determine the third stage dominance matrix M^3 for the system.
Find T = M + M^2 + M^3.
Use a matrix to determine the number of three stage paths there are from I to F.
Use a matrix to determine the number of two stage paths there are from I to F.
Use a matrix to determine the total number of paths from I to F.
Confirm your results to parts (e), (f) and (g) using the diagram of the river system.
