TITLE: A matrix equation, are there solutions?
QUESTION [5 upvotes]: I try to prove that: there are no matrices $A, B \in M_{3 \times 3}(\mathbb{R})$ such that:
$ \begin{pmatrix} 0&1&0\\ 0&0&1\\ 0&0&0 \end{pmatrix} = A^2 + B^2 $
How can I do that? I have no idea.

REPLY [10 votes]: The claim is false. As a counterexample consider $$A = \left(\begin{matrix}0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 1 & 0\end{matrix}\right), B = \left(\begin{matrix}0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 0 & 0\end{matrix}\right).$$