SEEMOUS 2010 Problem 3

Denote by $\mathcal M_2(\mathbb R)$ the set of all $2\times2$ matrices with real entries. Prove that:

a) for every $A\in\mathcal M_2(\mathbb R)$ there exist $B,C\in\mathcal M_2(\mathbb R)$ such that $A=B^2+C^2$ ; b) there do not exist $B,C\in\mathcal M_2(\mathbb R)$ such that $\begin{pmatrix}0&11&0\end{pmatrix}=B^2+C^2$ and $BC=CB$ .