# A linear algebra problem.

1. Nov 25, 2008

### bluewhistled

1. The problem statement, all variables and given/known data
A is a 2x2 matrix with eigenvectors 1,-1 and 1,1 with respective eigenvalues 2 and 3. x is 2,0. Find (A^4)x

2. Relevant equations
I know (A^k)v=(lamda^k)v
But I just don't know how to solve this to find A and then multiple it by x

3. The attempt at a solution
See above

Thanks a lot for anyone's help or input!

2. Nov 25, 2008

### Vid

The eigenspace is two dimensional since there are two distince eigenvalues. Thus, A is diagonalizable. Then....

3. Nov 25, 2008

### HallsofIvy

Staff Emeritus
More simply, <2, 0>= <1, -1>+ <1, 1>. Apply A to that. You don't need to determine A itself.

4. Nov 25, 2008

### Dick

You don't need to find A. (2,0)=(1,-1)+(1,1). How would you find A^4((1,-1))?

5. Nov 25, 2008

### bluewhistled

Ah I got it, (A^k)x=(lamda1^4)v1+(lamda2^4)v2

Thanks a lot for pointing out that those two vectors added up to x, I overlooked that. Would this be possible otherwise?

6. Nov 25, 2008

### Dick

If the only information you have about A is it's eigenvectors, and you can't express the vector as a linear combination of eigenvectors, then, no, you don't have enough information about A.

7. Nov 25, 2008

### bluewhistled

Thank you very much you guys are awesome.