I am preparing to take an exam on Linear Algebra.

I found this question on previous exams.

Here is what I think is needed to be done.

1)First find the eigenvalues, I get:

2)There are exactly three eigenvalues, thus there are exactly three linearly independant eigenvectors. Since the eigenvectors are a subspace of the nullspace of which is of dimension three it tells us that there is basis for the eigenspace.

3)To find the basis for the eigenspace I need to substitute the eigenvalues and solve the homogenous system . Thus I need to,

Solve it by Gaussian elimination for each eigenvalue .

4)Then the basis I get for the eigenspace in #3 will be the diagnolizable matrix for by making the vectors as coloum vectors.

Is this approach correct?

I have a second question, how is "Eigenvalue" pronounced.

(At least it is not as bad, Lebesque. I never would have guess it, unless I heard someone pronouce it).