Question 14
Consider the set $M=\left\{\left.\left(\begin{array}{ll}a & b \\ c & d\end{array}\right) \in \mathcal{M}_2(\mathbb{C}) \right\vert\, a b=c d\right\}$.
a) Give an example of a matrix $A \in M$ such that $A^{2017} \in M$ and $A^{2019} \in M$, but $A^{2018} \notin M$.
b) Show that if $A \in M$ and there exists an integer $k \geq 1$ such that $A^{k} \in M, A^{k+1} \in M$ and $A^{k+2} \in M$, then $A^{n} \in M$ for every integer $n \geq 1$.
a) Give an example of a matrix $A \in M$ such that $A^{2017} \in M$ and $A^{2019} \in M$, but $A^{2018} \notin M$.
b) Show that if $A \in M$ and there exists an integer $k \geq 1$ such that $A^{k} \in M, A^{k+1} \in M$ and $A^{k+2} \in M$, then $A^{n} \in M$ for every integer $n \geq 1$.