Finding the nth Power of a 2x2 Matrix | A^n = PD^nP^-1 Formula Not Required

[tracedinair]

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 ?

 

[whybother]

Is it specifically $${1\, 1 \choose 0\, 0}$$ or a more general problem?

Have you tried calculating any powers of that particular matrix?

$${1\, 1 \choose 0\, 0}{1\, 1 \choose 0\, 0} = {1\, 1 \choose 0\, 0}$$

So...

 

[tracedinair]

Let's say more general.

Is it just calculating successive powers and finding some pattern to base a formula off of?

 

[whybother]

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).

 

[tracedinair]

This is Calculus 4. Pattern finding is probably the most obvious solution. Thanks for the help.