Find P_300 Given P_299: Initial State Vector Homework

[alexcc17]

Homework Statement

Suppose that the initial vector P0 is not given and instead we are given that at time t=299 the state vector is P_299=[.1 .1 .8].
Than find P_300, the state vector at time t=300

Homework Equations

I also know from the problem it is referencing that the transition matrix is:

[.3 .4 .3]
T= [.5 0 .5]
[.5 .5 0]

The Attempt at a Solution

I thought I would just set up the equation as such:

P_299=P0 * T^299

and just take the inverse of T^299 and multiply it by P_299 to get the P0, then use that to find P_300, but my graphing calculator won't even calculate the 299th power of the transition matrix, so there must be another way to do this.

Any help would be great!

 

[haruspex]

I don't see any need to worry about P0. T describes how the state changes from one timestamp to the next, right? And you know the state at time 299. And you want the state at time 300? Either I'm missing some subtlety or you are missing the obvious.

 

[alexcc17]

Would it just be p_299*T to get p_300?

 

[HallsofIvy]

Yes, that's what an exponent means isn't it? T³⁰⁰v= T(T²⁹⁹v)

 

[alexcc17]

Wasn't sure if that applied to matrices. Thanks