linear algebra – Is this a correct solution? Jordan form of a marix

I am given a matrix $ A $ defined as follows: the element of line i and column j is $ alpha $ if i = j, 1 if j = i + 2 and 0 otherwise. I need to find Jordan Form. This is my solution, I feel that it is too simple, so I want to check if I understood correctly: by calculating the characteristic polynomial, we obtain that $ alpha $ is the only eigenvalue, so the elements in the form of a diagnosis are all $ alpha $, so the form jorden instead i, j is $ alpha $ if i = j, $ 1 $ if i = j + 1 and if not 0. Is the solution completely correct?