Is there a faster method to solve this?

Me and my friend are new to linear algebra (we posted earlier, thanks for the help) and we're a little confused on another question here:

http://i.imgur.com/qKEtL.jpg

How do we solve this with that big scary exponent hanging around in the corner there? Doing it manually would take ages, is there some specific method to solving with large exponents? My friend said she googled it and said something about a "eigen value" or something - we haven't learned that yet.

Can anyone provide insight into solving this? Sorry if we're bugging you all, we've managed to get most of this stuff down except this one puzzled us.

Thanks a lot!

Re: Is there a faster method to solve this?

Quote:

Originally Posted by

**Damuffinator** Me and my friend are new to linear algebra (we posted earlier, thanks for the help) and we're a little confused on another question here:

http://i.imgur.com/qKEtL.jpg
How do we solve this with that big scary exponent hanging around in the corner there? Doing it manually would take ages, is there some specific method to solving with large exponents? My friend said she googled it and said something about a "eigen value" or something - we haven't learned that yet.

Can anyone provide insight into solving this? Sorry if we're bugging you all, we've managed to get most of this stuff down except this one puzzled us.

Thanks a lot!

Well here is a hint:

Calculate A^1 and A^2.

Now calculate A^3 and A^4.

What's the pattern? Make a conjecture. Prove your conjecture. Now state the answer to the posted question.

Re: Is there a faster method to solve this?

Aah yes! We got it! Thank you for your help :D Maybe we won't fail first year after all!

Thanks again!