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Question: How can I get the general form for A^k? where A =([[r1*x,(1-r1)*x],[1-r2,r2]]) and k is an arbitrary interger

Question:How can I get the general form for A^k? where A =([[r1*x,(1-r1)*x],[1-r2,r2]]) and k is an arbitrary interger

chengzhi li 24 Maple
constants
January 20 2007
0
Dear Sir/Madam:

I try to find a close form for A^k,
where k is an arbitrary interger and A is 2 by 2 matrix as the following:
A = ([[r1*X, (1-r1)*X],[1-r2,r2]]);
r1 and r2 are symbolic constants, X is a variable.

First, I run the following commands in maple:

A:=matrix([[r1*X, (1-r1)*X],[1-r2,r2]]);
Z3:=A^3;
simplify(evalm(Z3),symbolic);

I hope to simplyfy the entries of Z3 in the order of X, i.e.,

z11 = a0*X^3 + a1*X^2 + a2*X +a3;
z12 = ....... (the same as the above)
z21 = .......
z22 = .......

But I do not know how to do it.

I try to find a law for representations of the entries of A^k by looking at the cases k=3,4,....

Any comments and hints and help are highly appreciated.

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