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    I'm not sure how I would go about doing this one:

    For 0<a<1 and 0<b<1 consider the matrix:

    P=|\begin{array}{cc}1-a&a\\b&1-b  \end{array}|

    Show that for any n>=0,

    P^n=\frac{1}{a+b}|\begin{array}{  cc}b&a\\b&a\\\end{array}|+\frac{  (1-a-b)^n}{a+b}|\begin{array}a&-a\\-b&b\end{array}|


    Any help would be much appreciated
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    (Original post by Goldenratio)
    I'm not sure how I would go about doing this one:

    For 0<a<1 and 0<b<1 consider the matrix:



    Show that for any n>=0,




    Any help would be much appreciated
    Could you not show that it's true by induction?? I'm just glancing at it, but show it's true for the case n=1, then assume it's true for the case of n, and show that it's true for the case n+1, and then it should hold true for all n. That's a bit of a guess, but I think it would work!!
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    erm does it make any difference if it is a stochastic matrix?
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    I've just done it by induction (I think!!), and it's quite messy, but took only 20 minutes. I haven't done Markov Chains yet, so I can't be sure that it's the easiest method (in fact, it most probably isn't!!). Hope it all goes well!!
 
 
 

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