Matrix is either null or similar to the elementary matrix $E_{2,1}$
Solution 1:
If $A^2=0$, then $det(A^2) = [det(A)]^2 = 0$, implying $det(A) = 0$. Since $A$ is a $2 \times 2$ matrix, what can you conclude about $A$?
Solution 2:
If $A$ is non-zero,there is $X$ such that $AX$ is non-zero.Now $\{X,AX\}$ is linearly independent,forms a basis of $C^2$.Then the matrix representation of $A$ with respect to the basis is the given matrix. [QED]