(Solved) YALinearAlgebraT: Applications

[Ricemarine]

***SOLVED***
Thanks TuxDave!:beer:

- There are 3 players playing a game
- In each game, there are always two winners and one loser
- The loser has to give up an amount equal to the winners prize money
- To make it easier, Player 1 loses the first, Player 2 second, Player 3 third, etc.
- After three games, each player has $24.

What was the original amount each player started with?

So what I'm having trouble with is finding the coefficients to solve for it. If a person won, would the coefficient be doubled since he won? Also, since the loser has to subtract their amount from the sum of the other two, how would I go about this?...

From what I gathered, I've tried for my matrix

[ -x2-x3 1 1 ]
[ 1 -x1-x3 1 ]
[ 1 1 -x1-x2 ]

But then writing the general solution would make this look weird because then all the variables cancel out, therefore 0 = 24... which is not true...

Can anyone spare a hand? Thanks.

 

[Ricemarine]

I've asked my professor, but she only told me to not think of it as a matrix equation... Hmm.. Any ideas?

 

[TuxDave]

Why not just work your way backwards?

After 3 games, everyone had $24 and player 3 just lost. That means player 1 and player 2 doubled their money (so they each got $12) so to figure out how much they had in game #2, take away the $12 they just won and give back $24 to player 3....

After 2 games, Player 1 had $12, Player 2 had $12 and Player 3 had $48.... repeat

 

[Ricemarine]

Originally posted by: TuxDave
Why not just work your way backwards?

After 3 games, everyone had $24 and player 3 just lost. That means player 1 and player 2 doubled their money (so they each got $12) so to figure out how much they had in game #2, take away the $12 they just won and give back $24 to player 3....

After 2 games, Player 1 had $12, Player 2 had $12 and Player 3 had $48.... repeat

Yeah I know that (not to come off as rude, thanks though), and the answer should be Player 1 = 39, Player 2 = 21, Player 3 = 12...
But the question is how do I put it as a system of equations. So if every player doubled and the loser lost the sum of the other two players, how do I insert that?

 

[TuxDave]

Actually i have a better way that's more "linear algebra" oriented.. one sec.

So using the notation:
A0 = Player1's Money after 0 Rounds (starting money)
B0 = Player2's Money after 0 Rounds (starting money)
...
A1 = Player1's Money after 1 Round...


[1 -1 -1] [A0] .... [A1]
[0 2 0] X [B0] = [B1]
[0 0 2] [C0] .... [C1]


And then you multiply it again by 2 more 3x3 matrix to get to 3 rounds

[2 0 0] [2 0 0] [1 -1 -1] [A0] .... [24]
[0 2 0] [-1 1 -1] [0 2 0] [B0] .... [24]
[-1 -1 1] [0 0 2] [0 0 2] [C0] = [24]


I hope my examples above is basic enough for you to understand what I'm doing Every game is another matrix multiplication where the last game is the last matrix that you multiply against. Writing matrix multiplications in text format sucks.

 

[Ricemarine]

There has got to be an easier way to do that... Hmm...
Trying to work backwards and simplify the equations before using your method TuxDave (which is good though, thanks). I'll try and see if I can come up with anything before the end of the day.

 

[TuxDave]

Originally posted by: Ricemarine
There has got to be an easier way to do that... Hmm...
Trying to work backwards and simplify the equations before using your method TuxDave (which is good though, thanks). I'll try and see if I can come up with anything before the end of the day.

I hope you're not reading my post wrong because what I wrote is pretty easy by itself unless you don't know how to find inverse matrices. Then I guess it could be a pain in the butt.

 

[Ricemarine]

Originally posted by: TuxDave

Originally posted by: Ricemarine
There has got to be an easier way to do that... Hmm...
Trying to work backwards and simplify the equations before using your method TuxDave (which is good though, thanks). I'll try and see if I can come up with anything before the end of the day.

I hope you're not reading my post wrong because what I wrote is pretty easy by itself unless you don't know how to find inverse matrices. Then I guess it could be a pain in the butt.

Yeah, we never actually went over how to do inverse matrices or when to use it. It's also not in the textbook, since they focus more on more scientific topics like balancing equations. So, we never really had to utilize separate matrices to solve just one problem.

Thanks.