Higher Power of a square Matrix

Guidenable

Homework Statement

Given the matrix A=
-1/5 7/5
-3/5 -4/5

find A43.

The Attempt at a Solution

It's obvious that I can't go and actually compute A43 so there must be a more elegant way of doing this. The only notes I have on the subject is Ak=P-1DkP, where D is a diagonal matrix. However, I have no clue what P is supposed to to, nor why this would work in the first place.

You're on the right track. If you've studied eigenvalues and eigenvectors, you should be able to calculate D and P.

Guidenable
I've never heard of eigenvalues or eigenvectors. I'm in a college level Linear Algebra I class, so I don't know if I should or not.

Mentor
You need to have some understanding of eigenvalues and eigenvectors to be able to diagonalize a matrix. In your formula, the columns of matrix P are the eigenvectors of matrix A, and P-1 is the inverse of P. Matrix D is a diagonal matrix whose entries are the eigenvalues of A.

If you're expected to work a problem like this, there must be similar problems in your textbook, and some presentation of these ideas must have been given in class.

Guidenable
Ah ok, it's quite possible that the prof mentioned it but I missed it. Thanks for the link, I'll put it to good use.