Question

Can I have this as soon as possibe please? I WILL GIVE A RATE!! (This is for linear Algebra)

9. Rugby i s a very popular sport in New Zealand. Rivalries among teams can go back decades. A student want each other. After the first month of the season, Taranaki had beaten Manavw Manawatu had beaten Waikato (28-20), an (24-14). U ranked highest after the first month of the season? ted to rank three teams, Manawatu, Taranaki, and Waikato, in their matches against d Manawatu had beaten Taranaki in the re-match se the Massey Method to rank the teams. Show your work. Which team would be (12 points)

Answer 1

Given, below is the diagram showing the point differences for each game played between the team. The outgoing node from team A to team B denotes that that team A beaten team B with the point differences mentioned on the node.

From the diagram, we will construct the below matrix, where row 1, 2, and 3 are for teams Taranaki, Manawatu and Waikato respectively. Similarly, Column 1, 2, and 3 are for teams Taranaki, Manawatu and Waikato respectively. Each diagonal entry denotes how many games that team have played. A negative sign in cell (r, c) denotes the number of matches Team r have played with team c.

$\begin{bmatrix} & Taranaki & Manawatu &Waikato \\ Taranaki & 2 & -2 &0 \\ Manawatu & -2 &3 &-1 \\ Waikato & 0 &-1 & 1 \end{bmatrix}$

We will create a matrix with overall point differences for each team from the diagram.

For Taranaki, overall point differences = 15 - 10 = 5

For Manawatu, overall point differences = -15 + 10 + 8 = 3

For Waikato, overall point differences = -8

$\begin{bmatrix} 5\\ 3\\ -8 \end{bmatrix}$

Let r1, r2 and r3 are the rankings of the teams Taranaki, Manawatu and Waikato respectively. Then,

$\begin{bmatrix} 2 & -2 &0 \\ -2 &3 &-1 \\ 0 &-1 & 1 \end{bmatrix}\begin{bmatrix} r_1\\ r_2\\ r_3 \end{bmatrix} = \begin{bmatrix} 5\\ 3\\ -8 \end{bmatrix}$

Since, the left hand matrix is singular, we will change the last row of the matrix to a row of all ones and the corresponding entry in the point differential vector to a zero to get a unique solution.

$\begin{bmatrix} 2 & -2 &0 \\ -2 &3 &-1 \\ 1&1&1 \end{bmatrix}\begin{bmatrix} r_1\\ r_2\\ r_3 \end{bmatrix} = \begin{bmatrix} 5\\ 3\\ 0 \end{bmatrix}$

$\begin{bmatrix} r_1\\ r_2\\ r_3 \end{bmatrix} =\begin{bmatrix} 2 & -2 &0 \\ -2 &3 &-1 \\ 1&1&1 \end{bmatrix}^{-1} \begin{bmatrix} 5\\ 3\\ 0 \end{bmatrix}$

$\begin{bmatrix} r_1\\ r_2\\ r_3 \end{bmatrix} =\begin{bmatrix} 2/3 & 1/3 &1/3 \\ 1/6 &1/3 &1/3 \\ -5/6&-2/3&1/3 \end{bmatrix} \begin{bmatrix} 5\\ 3\\ 0 \end{bmatrix}$

$\begin{bmatrix} r_1\\ r_2\\ r_3 \end{bmatrix} = \begin{bmatrix} 4.33\\ 1.83\\ -6.17 \end{bmatrix}$

Based on rankings order, the teams (from best to worst) would be ranked Taranaki, Manawatu, Waikato