1.
These answers follow directly from the definitions of matrix algebra.
a.
b.
c.
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:
Then the associated agumented matrix is as follows:
To reduce this to echelon form, we multiply the first row by a constant such that, when we add it to the second row, it will eliminate the
. This implies that
, which in turn implies that
. Multiplying the first row by this constant, we get,
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+ |
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= |
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= |
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+ |
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= |
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4.
5.
Here is the matrix raised to the 10th power:
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 be the tax collected from the Grendals,
, from the Hymlocks,
, from the Wyrotts, and
, from the shails.
We are given four facts which can be expressed as equations:
We can arrange the system as follows:
G | + | H | + | W | + | S | = |
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G | + |
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= |
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H | + | S | = |
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W | + | S | = |
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2. Add 2/3 times the second row to the third row.
3. Add 3/2 times the third row to the fourth row.
The associated system is shown below:
This system can be solved via backsubstitution (not shown) to yield the following solution:
The Shails pay no taxes but instead receive 2 million credits.