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Diagonalizing and expressing as A^k

  1. Nov 3, 2011 #1
    1. The problem statement, all variables and given/known data
    Let [tex] A=\left(\begin{array}{cc}11 & 6\\-12 & -6\end{array}\right) [/tex] and let k be a positive integer. By diagonalizing A, express Ak as an explicit 2x2 matrix in terms of k. Your answer should take the form [tex] A=\left(\begin{array}{cc}f_1 (k) & f_2 (k)\\f_3 (k) & f_4 (k)\end{array}\right) [/tex]


    2. Relevant equations
    P-1AP=D and Ak=PDkP-1.


    3. The attempt at a solution
    I don't have a problem diagonalizing, I have a problem with expressing it in terms of functions (at least I think). I diagonalized normally and got [tex] P=\left(\begin{array}{cc}-3 & -2\\4 & 3\end{array}\right) [/tex] and [tex] D=\left(\begin{array}{cc}3 & 0\\0 & 2\end{array}\right) [/tex] After using the second formula above, I get [tex] A^k=\left(\begin{array}{cc}9*3^k-8*2^k & 6*3^k-6*2^k\\-12*3^k+12*2^k & -8*3^k+9*2^k\end{array}\right) [/tex] I am just not sure if this is "good enough" or even right? Is there some way I can combine this? Just want to double check my answer basically :smile:
     
  2. jcsd
  3. Nov 3, 2011 #2

    micromass

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    This looks good!! :smile:
     
  4. Nov 3, 2011 #3
    Thank you :smile:
     
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