Here is the question I am trying to answer:
Find the Jordan form of $$A = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 0 & 0\\ 0 & 0 & -1 \end{pmatrix}$$
I know that the first step is to find the eigenvalues, which I did and I found them to be $\lambda = 0,1,-1.$ I know that by a theorem we know that $A - \lambda I $ is a nilpotent matrix, but when I tried to compute the index of nilpotency incase of $\lambda = 0$ i.e., of the given matrix $A$ I kept on multiplying now until the 5th time but it is still not giving me zero, I am wondering if someone can clarify this to me, is my matrix $A$ contains a typo? if not, what is its index of nilpotency?