Year 12 General Mathematics 2024 Modelling with Matrices Topic Test Notes

Topic Test: Modelling with Matrices

1. Table Tennis Competition

• Teams Played: Five teams play one match against every other team.
• Directed Graph Representation: An arrow from one team to another indicates the winning team.

a. Dominance Matrix

• Definition: A matrix representing the outcome of matches among teams.
• Example Matrix:
  D = \begin{bmatrix} 0 & 1 & 0 & 0 & 0 \
  0 & 0 & 1 & 0 & 0 \
  0 & 0 & 0 & 1 & 0 \
  0 & 0 & 0 & 0 & 1 \
  1 & 0 & 0 & 0 & 0 \end{bmatrix}
• Calculate Dominance: Fill in the matrix based on match outcomes.

b. Wins Calculation and Team Ranking

• Calculate Wins: Count the number of wins for each team based on the dominance matrix.
• Rank Teams: Order teams from highest wins to lowest wins.

c. Matrix Interpretation

• Interpretation of $D^2$: Find and interpret the square of the dominance matrix to analyze paths of victories.
• Example: If $D^2$ indicates multiple victories over different teams, this shows dominance in competition.

d. Supremacy Matrix Calculation

• Matrix Construction: Construct $S = D + \frac{1}{2}D^2$.
• Purpose: Use it to enhance the ranking of teams by considering more factors.

e. Full Team Ranking Using Supremacy Matrix

• Rank Teams: Based on supremacy matrix values to rank teams.

f. Limitation of Supremacy Matrices

• Limitation: Supremacy matrices may not account for match nuances; strong teams might have few encounters leading to misrepresentations in ranks.

2. Connectivity in the City of Robville

• City Layout: Five suburbs labeled A to E; include a lake region.

a. Connectivity Matrix

• Representation: Cell values represent land borders between suburbs.
• Interpret 2's in Matrix: Two means two land borders or connections exist between those suburbs.

b. Row and Column Zeros

• Explanation: Zero values in the matrix indicate no borders connecting the lake to any suburbs.

c. Directed Network Diagram

• Drawing Missing Edges: Complete the diagram by adding edges between relevant suburbs.
• Labeling Missing Node: Add missing nodes to the diagram.
• Directed Arrows: Indicate directionality of connections (from one suburb to another).

d. Calculation and Interpretation of $C + C^2$

• Calculate Connectivity Results: Understand the context in suburb connectivity through matrix summation.

e. Bus Service Impact

• Two-step Routes Analysis: How many routes operate from C to B?
• Paths Definition: Evaluate whether these routes qualify as distinct paths based on definitions.

f. Routes from A to D in 3 Steps

• Step Analysis: Use the connectivity matrix to find all possible routes in three steps.

g. Limitation of Connectivity Matrix

• Limitation: May not capture dynamic interactions, such as changes in borders or new transport routes.

3. Customer Travel Analysis for a Company

• Initial Travel Matrix T:
  T = \begin{bmatrix} 0.65 & 0.15 & 0.05 & 0.15 \
  0.25 & 0.60 & 0.10 & 0.05 \
  0.25 & 0.20 & 0.25 & 0.30 \
  0.50 & 0.15 & 0.20 & 0.15 \end{bmatrix}

a. Calculation of $S_0 T$

• Customer Matrix Calculation: Multiply initial travel choices by transition probabilities.

b. Interpretation of Results

• Contextual Understanding: Explain results regarding customer travel preferences for 2019 based on calculations.

c. Customer Choices for 2020 and 2030

• Yearly Analysis: Use matrix to forecast sea travel customers in each outlined year.

d. Assumptions Limiting Reliability

• Reliability Limitation: Assumes steady behavior without changes in external influences like trends or marketing.

e. Long-term Customer Prediction

• Matrix Method Calculation: Use to determine consistent travel choices by extended customer analysis.

f. New Transition Plans

• Updated Matrix After Sale:
  T_{new} = \begin{bmatrix} 0.65 & 0.15 & 0.05 & 0.15 \
  0.25 & 0.60 & 0.10 & 0.05 \
  0.5 & 0.15 & 0.20 & 0.15 \end{bmatrix}
• Evaluate Changes in Long-Term Customer Expectations: Discuss resulting impact on total customer flows for transport modes based on sales effects.