I got this problem:
Is it true or not that for every matrix $A\in M_2( \mathbb C)$ there is a matrix $X\in M_2( \mathbb C)$ such that $X^2=A$? prove the answer.
I know that we can pass to Jordan form, and I proved that this claim is true for all the Jordan forms except $$ \begin{pmatrix} 0 & 1 \\ 0 & 0 \\ \end{pmatrix} $$ any suggestions what to do with this matrix? and also, there is a general version of this problem? thanks for helpers!
