# Homework Help: Diagonalizable Matrix

1. Mar 17, 2008

### rjw5002

1. The problem statement, all variables and given/known data
Let A =
[4 0 1;
2 3 2;
1 0 4]
Let n >= 1 be an integer. Compute the matrix A^n with entries depending on n.

2. Relevant equations

3. The attempt at a solution
First I need to show that A is diagonalizable, and find a matrix S such that D = (S^-1)(A)(S) is diagonal. I am having serious trouble finding S. First I found the eigenvalues of A to be 3,3, and 5. Next I found the eigenvectors to be for $$\lambda = 3$$: (1 0 -1) and $$\lambda = 5$$ : (1 2 1). There's a theorem in my book that says that a matrix is only diagonalizable if and only if there are n linearly independent eigenvectors. I only have 2. I'm not quite sure how to get the third.
Now, assuming I had three eigenvectors, I would combine them to form the matrix S, and if the $$det(S)\neq 0$$ then the eigenvectors would be linearly independent, and
i could find S^-1. Then A^n would just be (S^-1)(D^n)(S). Does this all sound correct? I guess I'm just a little confused about the eigenvalue/eigenvector situation....
Thanks for any help.

2. Mar 17, 2008

### Dick

There is a third linearly independent eigenvector, you are just overlooking it. Hint: try (0,1,0). Then do exactly what you propose.

3. Mar 17, 2008

### steelphantom

Wouldn't An = SDnS-1?

4. Mar 17, 2008

### Dick

Yep. Sorry, I overlooked that.