Matrix Year 11 Draft for Reference(Not Plagiarism)
Mat
MPS 1
Part A: Determining fair rank between teams(no ties)
Let win=3, draw=1 and lose=0. The reason for this weighting method is because it is natural for the loser not to get a point. Also, there must be a visible gap between winners and draw players. This can be proved by contradiction proof. Blue | B | Crimson | C | Green | G | Orange | O | Red | R | Yellow | Y |
Assume that win=2 draw=1 and lose=0. If the supremacy matrix is calculated, tie always occurs, which proves that if there is a small gap between win and draws, that supremacy matrix isn’t valid.
This is the teams and their initial letters, arranged by alphabetical order.
Then, the diagraph below is converted into the following matrix. M=
In order to find the supremacy matrix with no ties, three cases with different coefficients are examined. Let coefficient of M is Mc, and coefficient of M^2 is M^2c; Mc>M^2c (S=M+0.5M^2), Mc=M^2c (S=M+M^2), and Mc<M^2c (S=M+2M^2). The result is followed by calculation.
S=M+0.5M^2
=
The rank order is Oranges, Blues, Reds, Yellows, Crimsons and Greens. As it can be seen, there is no tie amongst the teams. This means, the supremacy S=M+0.5M^2 is valid. To identify that this matrix is valid, other cases are examined.
S=M+M^2
= The rank order is Oranges, Blues, Reds, Yellows, Crimsons=Greens. As it can be seen in the vector, there is tie between Crimson and Green. This shows that the Supremacy Matrix S=M+M^2 is invalid.
S=M+2M^2
=
The rank order is Oranges, Blues, Reds, Yellows, Crimsons and Greens. As it can be seen, there is no tie amongst the teams. However, this matrix is less reliable than S=M+0.5M^2 because in most of matrices including this matrix, the first order matrix is more reliable than second order matrix because M^2 has lesser reliability because M is more related to the original data. M is the matrix derived from the actual data, whilst M^2 is the data generated from M through
MPS 1
Part A: Determining fair rank between teams(no ties)
Let win=3, draw=1 and lose=0. The reason for this weighting method is because it is natural for the loser not to get a point. Also, there must be a visible gap between winners and draw players. This can be proved by contradiction proof. Blue | B | Crimson | C | Green | G | Orange | O | Red | R | Yellow | Y |
Assume that win=2 draw=1 and lose=0. If the supremacy matrix is calculated, tie always occurs, which proves that if there is a small gap between win and draws, that supremacy matrix isn’t valid.
This is the teams and their initial letters, arranged by alphabetical order.
Then, the diagraph below is converted into the following matrix. M=
In order to find the supremacy matrix with no ties, three cases with different coefficients are examined. Let coefficient of M is Mc, and coefficient of M^2 is M^2c; Mc>M^2c (S=M+0.5M^2), Mc=M^2c (S=M+M^2), and Mc<M^2c (S=M+2M^2). The result is followed by calculation.
S=M+0.5M^2
=
The rank order is Oranges, Blues, Reds, Yellows, Crimsons and Greens. As it can be seen, there is no tie amongst the teams. This means, the supremacy S=M+0.5M^2 is valid. To identify that this matrix is valid, other cases are examined.
S=M+M^2
= The rank order is Oranges, Blues, Reds, Yellows, Crimsons=Greens. As it can be seen in the vector, there is tie between Crimson and Green. This shows that the Supremacy Matrix S=M+M^2 is invalid.
S=M+2M^2
=
The rank order is Oranges, Blues, Reds, Yellows, Crimsons and Greens. As it can be seen, there is no tie amongst the teams. However, this matrix is less reliable than S=M+0.5M^2 because in most of matrices including this matrix, the first order matrix is more reliable than second order matrix because M^2 has lesser reliability because M is more related to the original data. M is the matrix derived from the actual data, whilst M^2 is the data generated from M through