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.

  • A000011210
  • B200001011
  • C000101011
  • D000201011
  • E100001011

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.

  • A112100200
  • B011100100
  • C002011211
  • D111100100
  • E012100200

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.

  • A1111
  • B1001
  • C1111
  • D0110
  • E1111

Q5:

In a soccer league, there are 3 teams. Every team played 4 games.

The first team won 1 game, lost 3 games, and drew no games.

The second team won 2 games, lost 1 game, and drew 1 game.

The third team won 2 games, lost 1 game, and drew 1 game.

Write down the matrix that represents the results of the league. Every team is represented by a different row. The first column contains the number of wins, the second contains the number of losses, and the third contains the number of draws.

  • A301112112
  • B310121121
  • C130211211
  • D103211211
  • E122311011

Q6:

The table below shows the prices, in pounds, of 3 types of sandwiches in 3 different sizes in a fast-food restaurant. Arrange this data in a matrix such that the prices for each type of sandwich are arranged in the rows, with the small in the first column, medium in the second column, and large in the third column. The rows of the matrix should also be arranged in ascending order.

SmallMediumLarge
Burger81218
Hot Dog91520
Fried Chicken71016
  • A789101215161820
  • B710168121891520
  • C812187101691520
  • D812189152071016
  • E897121510182016

Q7:

Rock, paper, scissors is a 2-player game where players make gestures with their hands in the shapes of either a rock, a paper, or a pair of scissors. It is played as follows:

  1. Rock beats scissors.
  2. Scissors beat paper.
  3. Paper beats rock.
  4. If the two players make the same gesture, it is a tie.

Write down the payoff matrix that represents the outcome of the game such that every choice made by player 1 is represented by a different row and every choice made by player 2 is represented by a different column. The first row/column represents rock, the second row/column represents paper, and the third row/column represents scissors. 1 represents a win for player 1, 0 represents a tie, and -1 represents a loss for player 1.

  • A101110011
  • B011101110
  • C001100010
  • D011101110
  • E101110011

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