Word problem with matrix (3 dwarves)

[Enemy of my enemy]

Hello guys,
we have word problem on university and unfortunately I don't know how to deal with it

We have 3 dwarves: Gimli, Dain and Thorin. They are sitting around circular table. And they have kilograms of Mithril. Gimli has 80 kg, Dain has 40 kg and Thorin only 20 kg. They want to divide mithril, because they have lot of time and they don't want be bored. They made up, they will do following operation 7 times: Each of them will divide their mithril on 2 (same) halves. First half they will give to dwarf on left and second half to dwarf on right. How many of kilograms of mithril will have each of them on end of this operation?

I tried to do it logically, but our objective is use matrix (like part of algorithm). And in this case I am lost.

 

[Romsek]

clearly the state should be a 3 vector of the amounts of mithril each dwarf currently has.

Given that coming up with a state transition matrix should be pretty simple. For example the first row will be

\(\displaystyle T_{1j} = \left(0,~\dfrac 1 2,~\dfrac 1 2\right)\)

Finish up the transition matrix. Take it to the 7th power. Multiply it by the initial state. Boom done.
Alternatively you can multiply the transition matrix by the initial state to obtain a new state.
Then multiply that new state again by the transition matrix, rinse and repeat 6 more times.

Interestingly enough this method does indeed spread the mithril around so as to equalize the amounts among the dwarves.

 

[Enemy of my enemy]

Thank you sir, it really helps!