Show that a nonzero $2 \times 2$ matrix $A$ such that $A^2 = 0$ is similar to $\begin{pmatrix}0&1\\0&0\end{pmatrix}$

Question

Let $A$ be a $2 \times 2$ non-zero matrix such that $${A}^{2}=0.$$ How do I find an invertible matrix P such that $${P}^{-1}AP=\begin{bmatrix}0&1\\0&0\end{bmatrix} ?$$

Anyone? Please provide a clear explanation, I really appreciate it!

Thank you.

Answer 1

Hint Since $A \neq 0$ we can pick an element ${\bf e}_2$ such that $A {\bf e}_2 \neq 0$.

Additional hint Since the only eigenvalue of $A$ is $0$, $A {\bf e}_2$ cannot be a multiple of ${\bf e}_2$.