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Math Help - Find Matrix A^n...

  1. #1
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    Post Find Matrix A^n...

    Hello,

    I have a very small doubt on this problem..

    A = (1 4 2 3) in M(2x2) R, find an expression for A^n, where n is an arbitrary positive integer.


    I know its a pretty simple concept but I am getting hard time figuring it out!

    when I multiply A*A I get = (9 16 8 17)

    and then next will be A^2 * A = (41 84 42 83)

    and then next will be A^3*A = (209 416 208 417)

    and then next will be A^4*A = (1041 2084 1042 2083)

    I am sure there is a pattern but I cant seem to understand and simplify this problem!


    thanks a lot!
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  2. #2
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    Google: the cayley hamilton theorem
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  3. #3
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    There won't be any simple pattern. You need to do it a different way.

    That matrix has eigenvalues -1 and 5. That means that if we define "P" to be the matrix having the corresponding eigenvectors as columns, we have
    \begin{bmatrix}1 & 4 \\ 2 & 3\end{bmatrix}= P^{-1}\begin{bmatrix}-1 & 0 \\ 0 & 3\end{bmatrix}P

    Writing that as A= P^{-1}DP, A^2= (P^{-1}DP)(P^{-1}DP)= P^{-1}D^2P, A^3= A(A^2)= (P^{-1}DP)(P^{-1}D^2P)= P^{-1}D^3P, etc. And, because D is diagonal, D^n is easy.
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