Worksheet: Applications on Representing Data Using Matrices

In this worksheet, we will practice using matrices to model some applications like the payoff matrix of a game and the incidence and adjacency matrices of a graph.

Q1:

Write down the adjacency matrix of the network shown.

  • A 0 0 0 0 1 1 2 1 0
  • B 2 0 0 0 0 1 0 1 1
  • C 0 0 0 1 0 1 0 1 1
  • D 0 0 0 2 0 1 0 1 1
  • E 1 0 0 0 0 1 0 1 1

Q2:

When a connection in a network does not have an arrow, it is said to be ‘undirected’. An undirected connection between nodes 𝑎 and 𝑏 is equivalent to a directed connection from 𝑎 to 𝑏 together with a directed connection from 𝑏 to 𝑎. Determine the adjacency matrix of the network shown.

  • A 1 1 2 1 0 0 2 0 0
  • B 0 1 1 1 0 0 1 0 0
  • C 0 0 2 0 1 1 2 1 1
  • D 1 1 1 1 0 0 1 0 0
  • E 0 1 2 1 0 0 2 0 0

Q3:

Players 𝑅 and 𝐶 play a game in which they both call “heads” or “tails” at the same time. They use the payoff matrix 2211, where the first row and column represent “heads” and the second row and column represent “tails.” Describe the payoff rules for this game.

  • AIf both players call “heads,” player 𝐶 gets 1 point from player 𝑅; if both players call “tails,” player 𝐶 gets 2 points from player 𝑅; and if both players call different sides, player 𝑅 gets 2 points from player 𝐶 if he calls “heads” or 1 point if he calls “tails.”
  • BIf both players call “heads,” player 𝐶 gets 2 points from player 𝑅; if both players call “tails,” player 𝐶 gets 1 point from player 𝑅; and if both players call different sides, player 𝑅 gets 2 points from player 𝐶 if he calls “heads” or 1 point if he calls “tails.”
  • CIf both players call “heads,” player 𝐶 gets 2 points from player 𝑅; if both players call “tails,” player 𝐶 gets 1 point from player 𝑅; and if both players call different sides, player 𝑅 gets 1 point from player 𝐶 if he calls “heads” or 2 points if he calls “tails.”
  • DIf both players call “heads,” player 𝐶 gets 1 point from player 𝑅; if both players call “tails,” player 𝐶 gets 2 points from player 𝑅; and if both players call different sides, player 𝑅 gets 1 point from player 𝐶 if he calls “heads” or 2 points if he calls “tails.”
  • EIf both players call “heads,” player 𝐶 gets 2 points from player 𝑅; if both players call “tails,” player 𝐶 gets 2 points from player 𝑅; and if both players call different sides, player 𝑅 gets 1 point from player 𝐶 if he calls “heads” or 1 point if he calls “tails.”

Q4:

Players 𝑅 and 𝐶 play a game in which they both call “heads” or “tails” at the same time. If they both call the same thing, player 𝐶 wins one point from player 𝑅. If they call different things, player 𝑅 wins one point from player 𝐶. Write the payoff matrix for this game.

  • A 1 1 1 1
  • B 1 0 0 1
  • C 1 1 1 1
  • D 0 1 1 0
  • E 1 1 1 1

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