Question 77
a) Let $A$ be a matrix from $M_2(\mathbb{C})$, $A \neq aI_2$, for any $a \in \mathbb{C}$. Prove that the matrix $X$ from $M_2(\mathbb{C})$ commutes with $A$, that is, $AX = XA$, if and only if there exist two complex numbers $\alpha$ and $\alpha'$, such that $X = \alpha A + \alpha' I_2$.
b) Let $A$, $B$ and $C$ be matrices from $M_2(\mathbb{C})$, such that $AB \neq BA$, $AC = CA$ and $BC = CB$. Prove that $C$ commutes with all matrices from $M_2(\mathbb{C})$.
b) Let $A$, $B$ and $C$ be matrices from $M_2(\mathbb{C})$, such that $AB \neq BA$, $AC = CA$ and $BC = CB$. Prove that $C$ commutes with all matrices from $M_2(\mathbb{C})$.