Game Mathematics

Week 8 Official Solutions

1.

These answers follow directly from the definitions of matrix algebra.

a.

[Graphics:Images/solutions_gr_1.gif]

b.

[Graphics:Images/solutions_gr_2.gif]

c.

[Graphics:Images/solutions_gr_3.gif]

2.

Matrices can be used to quickly solve the kinds of differential equations that arise in physics simulations.

3.

Let the lines be defined by the following equations:

[Graphics:Images/solutions_gr_4.gif]

Then the associated agumented matrix is as follows:

[Graphics:Images/solutions_gr_5.gif]

To reduce this to echelon form, we multiply the first row by a constant [Graphics:Images/solutions_gr_6.gif] such that, when we add it to the second row, it will eliminate the [Graphics:Images/solutions_gr_7.gif]. This implies that [Graphics:Images/solutions_gr_8.gif], which in turn implies that [Graphics:Images/solutions_gr_9.gif]. Multiplying the first row by this constant, we get,

[Graphics:Images/solutions_gr_10.gif] [Graphics:Images/solutions_gr_11.gif] [Graphics:Images/solutions_gr_12.gif]


Adding this to the second row, we get the matrix shown below:

[Graphics:Images/solutions_gr_13.gif]

The associated system of linear equations is as follows:

[Graphics:Images/solutions_gr_14.gif] + [Graphics:Images/solutions_gr_15.gif] = [Graphics:Images/solutions_gr_16.gif]
[Graphics:Images/solutions_gr_17.gif] = [Graphics:Images/solutions_gr_18.gif]


From the bottom equation, we see that [Graphics:Images/solutions_gr_19.gif]. Substituting this into the first equation, we find,

[Graphics:Images/solutions_gr_20.gif] + [Graphics:Images/solutions_gr_21.gif] = [Graphics:Images/solutions_gr_22.gif]


Solving for x, we get,

[Graphics:Images/solutions_gr_23.gif]

So the intersection point is [Graphics:Images/solutions_gr_24.gif].

This works for all lines, even if they are perfectly horizontal or vertical, since the general form of the line equation can represent all lines.

4.

                              [Graphics:Images/solutions_gr_25.gif]
[Graphics:Images/solutions_gr_26.gif]

5.

Here is the matrix raised to the 10th power:

[Graphics:Images/solutions_gr_27.gif]

The interpretation of the matrix is as follows: after 10 transitions, the character is most likely to be in a disinterested state, and second most likely to be in the peaceful, talkative, or jovial states (roughly speaking -- there is some variance between the probabilities but not much). The character is least likely to be in the aggressive state (which itself is slightly more unlikely than the unhappy state).

6.

Let [Graphics:Images/solutions_gr_28.gif] be the tax collected from the Grendals, [Graphics:Images/solutions_gr_29.gif], from the Hymlocks, [Graphics:Images/solutions_gr_30.gif], from the Wyrotts, and [Graphics:Images/solutions_gr_31.gif], from the shails.

We are given four facts which can be expressed as equations:

[Graphics:Images/solutions_gr_32.gif]

We can arrange the system as follows:

G + H + W + S = [Graphics:Images/solutions_gr_33.gif]
G + [Graphics:Images/solutions_gr_34.gif] = [Graphics:Images/solutions_gr_35.gif]
H + S = [Graphics:Images/solutions_gr_36.gif]
W + S = [Graphics:Images/solutions_gr_37.gif]


The associated augmentex matrix is shown below:

[Graphics:Images/solutions_gr_38.gif]

Here are the steps involved in reducing the matrix to echelon form:

1. Add -1 times the first row to the second row.

[Graphics:Images/solutions_gr_39.gif]

2. Add 2/3 times the second row to the third row.

[Graphics:Images/solutions_gr_40.gif]

3. Add 3/2 times the third row to the fourth row.

[Graphics:Images/solutions_gr_41.gif]

The associated system is shown below:

[Graphics:Images/solutions_gr_42.gif]
[Graphics:Images/solutions_gr_43.gif]
[Graphics:Images/solutions_gr_44.gif]
[Graphics:Images/solutions_gr_45.gif]

This system can be solved via backsubstitution (not shown) to yield the following solution:

[Graphics:Images/solutions_gr_46.gif]

The Shails pay no taxes but instead receive 2 million credits.


Converted by Mathematica      December 6, 2001