Question 6
Define the set $M = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathcal{M}_2(\mathbb{C}) \mid ab = cd \right\}$.
a) Give examples of $A \in M$ having the property $A^{2017} \in M$ and $A^{2019} \in M$, but $A^{2018} \notin M$.
b) Prove that if for an $A \in M$ there is $k \ge 1$ such that $A^k \in M$, $A^{k+1} \in M$ and $A^{k+2} \in M$, then $A^n \in M$, for all integers $n \ge 1$.
a) Give examples of $A \in M$ having the property $A^{2017} \in M$ and $A^{2019} \in M$, but $A^{2018} \notin M$.
b) Prove that if for an $A \in M$ there is $k \ge 1$ such that $A^k \in M$, $A^{k+1} \in M$ and $A^{k+2} \in M$, then $A^n \in M$, for all integers $n \ge 1$.