# Matrix Algebra

1. Dec 18, 2008

### jkeatin

In a certain country, there are three political parties: The conservative Butter-side-up party
(the Uppers), the radical Butter-side-down party (the Downers), and the progressive Pitapocket
party (the Pocketeers). Recent polls show that each year, 70% of Uppers remain
Uppers, 80% of Downers remain Downers, 40% of Pocketeers remain Pocketeers. Everyone
else switches, and they divide equally among the other two parties.
(a) Write the transition matrix representing the party-switching process.
(b) In the long run, how will the people be distributed among the political parties?

2. Dec 18, 2008

### Vid

What have you tried?

3. Dec 18, 2008

### jkeatin

.7 .15 .15
.8 .2 .2
.4 .3 .3

is that say matrix A?

4. Dec 18, 2008

### HallsofIvy

Staff Emeritus
No, that's not correct. Assuming (A, B, C) means that one year A% uppers, B% downers, C% pitas, then you must have
$$\left[\begin{array}{ccc}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a{_33}\end{array}\right]\left[\begin{array}{c}1.0 \\ 0.0 \\ 0.0\end{array}\right]= \left[\begin{array}{c} 0.7 \\ 0.15 \\ 0.15\end{array}\right]$$

$$\left[\begin{array}{ccc}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]\left[\begin{array}{c}0.0 \\ 1.0 \\ 0.0\end{array}\right]= \left[\begin{array}{c} 0.1 \\ 0.8 \\ 0.1\end{array}\right]$$
and
$$\left[\begin{array}{ccc}a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\end{array}\right]\left[\begin{array}{c}0.0 \\ 0.0 \\ 1.0\end{array}\right]= \left[\begin{array}{c} 0.3 \\ 0.3 \\ 0.4\end{array}\right]$$

You can determine the "a"s from that.

5. Dec 18, 2008

### jkeatin

.7 .1 .3
.15 .8 .3
.15 .1 .4

then do i just reduce the matrix equation and find the values?

6. Dec 18, 2008

### jkeatin

or do i find the eigenvalues and eigenvectors?

7. Dec 19, 2008

### Staff: Mentor

Have you tried what Halls suggested?

8. Dec 19, 2008

### HallsofIvy

Staff Emeritus
C'mon now, you are not coming to this having no idea what you are doing are you? WHY would you "reduce the matrix equation" (what matrix equation) or "find the eigenvalues and eigenvectors"? What would either of those tell you?

Intrepret "in the long run as meaning "after many years". After n years, You will have An X where X is the initial distribution. How can you most easily find An for large n?

He did- in post #5.

9. Dec 19, 2008

### jkeatin

I took .7 , .8 , and .4 and subtracted 1 from each, then i got x1=6t x2=3t and x3=t
so 60% uppers, 30% downers 10% pocketeers, is that close or am i just way off

10. Dec 20, 2008

### HallsofIvy

Staff Emeritus
What reasoning did you use to arrive at that answer?