Diagonalizing and expressing as A^k

  • #1

Homework Statement


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]


Homework Equations


P-1AP=D and Ak=PDkP-1.


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:
 

Answers and Replies

  • #2
22,089
3,296
This looks good!! :smile:
 
  • #3
Thank you :smile:
 

Related Threads on Diagonalizing and expressing as A^k

  • Last Post
Replies
3
Views
1K
Replies
0
Views
1K
Replies
4
Views
2K
Replies
6
Views
2K
Replies
2
Views
2K
  • Last Post
Replies
8
Views
1K
Replies
5
Views
5K
  • Last Post
Replies
9
Views
7K
  • Last Post
Replies
10
Views
853
Top