Countexample: Sum and Product of two non-zero nilpotent operators is nilpotent?

Question

I am asked to come up with a proof or a counterexample of the following two statements:

1. The sum of two non-zero nilpotent matrices is nilpotent.

2. The product of two non-zero nilpotent matrices is nilpotent.

I think both of these claims are false, and a counterexample exists, but I can't seem to come up with one

Answer 1

Both claims are indeed false. you can look at: $$A_1 =\begin{pmatrix} 0 & 1\\ 0 & 0\\ \end{pmatrix}\space\space\space A_2=\begin{pmatrix} 0 & 0\\ 1 & 0\\ \end{pmatrix}$$ $$(A_1)^2 = (A_2)^2 = \begin{pmatrix} 0 & 0\\ 0 & 0\\ \end{pmatrix}$$ and thus, both matrices are nilpotent, but $$A_1+A_2 = \begin{pmatrix} 0 & 1\\ 1 & 0\\ \end{pmatrix}$$ $$(A_1+A_2)^2 = \begin{pmatrix} 1 & 0\\ 0 & 1\\ \end{pmatrix}$$ and thus, for any n $\ge2$ $(A_1+A_2)^n=I$, Which is not nilpotent.
$$A_1A_2 = \begin{pmatrix} 1 & 0\\ 0 & 0\\ \end{pmatrix}$$ and for any n $\ge2$ $(A_1A_2)^n=\begin{pmatrix} 1 & 0\\ 0 & 0\\ \end{pmatrix}$ , Which again, is not nilpotent.