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Powers of a 2x2 matrix

  • #1

Homework Statement



Find the nth power of the matrix A,

|1 1|
|0 0|

Homework Equations



The Attempt at a Solution



My guess is the A^n = PD^nP^-1 formula. But my prof says not use to eigenvalues and eigenvectors. Is it possible to solve this without using the formula A^n = PD^nP^-1 ???
 

Answers and Replies

  • #2
166
0
Is it specifically [tex] {1\, 1 \choose 0\, 0} [/tex] or a more general problem?

Have you tried calculating any powers of that particular matrix?

[tex] {1\, 1 \choose 0\, 0}{1\, 1 \choose 0\, 0} = {1\, 1 \choose 0\, 0} [/tex]

So...
 
  • #3
Let's say more general.

Is it just calculating successive powers and finding some pattern to base a formula off of?
 
  • #4
166
0
Let's say more general.

Is it just calculating successive powers and finding some pattern to base a formula off of?
It depends what level a course the problem is being asked in, really. For something like this, the pattern is incredibly obvious. In general, there may not be a pattern though... But, if your teacher/professor is looking for a simple solution without much technical mathematics, which it sounds like, doing some sample calculations and finding a pattern is a solid plan (especially for this particular matrix).
 
  • #5
This is Calculus 4. Pattern finding is probably the most obvious solution. Thanks for the help.
 

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