<s>[INST] Given $A^2$, find all $A$ that satisfy Find the set of all matrices $A\in M_{n\times n}(\mathbb R)$ satisfying $A^2=\begin{pmatrix} 0 &1& 0 \\ 0& 0& 0 \\ 0& 0& 0 \end{pmatrix}$
I have been given this problem. Do I just need to take $A=\begin{pmatrix} a &b& c \\ d& e& f \\ g& h& i \end{pmatrix}$ and solve for the variables? or is there a better way
 [/INST] Let $$A=\begin{pmatrix} a &b& c \\ d& e& f \\ g& h& i \end{pmatrix}$$ as OP suggests. Now clearly, 
$$A^4=0$$ and thus the minimal polynomial of $A$ divides $x^4$. However the minimal polynomial has degree $\leq 3$ and therefore the minimal polynomial divides $x^3$. Thus we have 
$$A^3=0$$ and so
$$\begin{pmatrix} a &b& c \\ d& e& f \\ g& h& i \end{pmatrix}
\begin{pmatrix} 0 &1& 0 \\ 0& 0& 0 \\ 0& 0& 0 \end{pmatrix}=0$$
And $$\begin{pmatrix} 0 &1& 0 \\ 0& 0& 0\\0& 0& 0 \end{pmatrix}
\begin{pmatrix} a &b& c \\ d& e& f \\ g& h& i \end{pmatrix}
=0$$. Which imply that 
$$A=\begin{pmatrix} 0 &b& c \\ 0& 0& 0 \\ 0& h& i \end{pmatrix}$$
 The original equation now becomes much easier to solve we get
$$A=\begin{pmatrix} 0 &b& c \\ 0& 0& 0 \\ 0& c^{-1}& 0 \end{pmatrix}
$$
</s>