<s>[INST] Quick Question on Zero Matrix Find all $2\times2$ matrices
$$B =\begin{pmatrix} \alpha & \beta\\ \gamma & \delta\\ \end{pmatrix}$$
such that $B^2$ is the zero matrix.
Any help would be greatly appreciated. Thanks
 [/INST] $B^2=0$ means $\operatorname{im}(B)\subseteq\ker(B)$. In particular, $B$ needs to be of rank at most one: If $\dim\operatorname{im}(B)=2$, then $\dim\ker(B)=0$ and the above inclusion is impossible. Furthermore, $B^2=0$ is equivalent to $(SBS^{-1})^2=0$, so we can change bases as we please.  Let us assume $B\ne0$ and $B^2=0$, then the kernel and image of $B$ are both of dimension one. We change bases so that $\ker(B)$ is spanned by 
$$e_1=\begin{pmatrix}1\\0\end{pmatrix}.$$
Then,
$$\exists a,b\colon\quad B=\begin{pmatrix} 0 & a \\ 0 & b  \end{pmatrix}.$$
Since $\operatorname{im}(B)\subseteq\ker(B)$, we have $b=0$. As $B\ne0$, we have $a\ne 0$. By a change of bases, we can achieve that $a=1$, because
$$
\begin{pmatrix} a^{-1} & 0 \\ 0 & 1  \end{pmatrix} 
\begin{pmatrix} 0 & a \\ 0 & 0  \end{pmatrix}
\begin{pmatrix} a & 0 \\ 0 & 1  \end{pmatrix}
=
\begin{pmatrix} 0 & 1 \\ 0 & 0  \end{pmatrix}.
$$
Hence, aside from the zero matrix, every matrix $B$ with $B^2=0$ is of the form
$$S \cdot \begin{pmatrix} 0 & 1 \\ 0 &0\end{pmatrix}\cdot S^{-1}$$
for an invertible matrix $S$.
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