Is there a matrix $A \neq 0$ such that $A\in F^{2\times 2}$ and $A^2=0$?

Question

Any hints? i don't know how to disprove the statement I looked at the multiplication with parameters and looked at the different cases but there were not enough information

$F$ is a field

Answer 1

Take the field $F_2$,

then consider the matrix,

$$\left( \begin{matrix}1 & 1\\ 1 & 1\end{matrix} \right)$$

Answer 2

$\left( \begin{matrix}0 & 1\\ 0 & 0\end{matrix} \right)$ for $F=R$

Answer 3

HINT: Consider the matrix $$ B= \begin{pmatrix} 0 & 0\\ 1 & 0 \end{pmatrix} \in F^{2 \times 2} $$ for an arbitrary field $F$.

The property of a square matrix $A$ such that $A^k = 0$ for some integer $k$ is called "nilpotence".