I want to know how $Z(\mathrm{PGL}_n( \mathbb{K} )) = \left\{ \overline{I_n} \right\} $ and there is a proof that I don't understand. The proof is the following \begin{align*} & \overline{A} \in Z(\mathrm{PGL}_n( \mathbb{K} )) \\ \Leftrightarrow & \forall \overline{B} \in \mathrm{PGL}_n( \mathbb{K} ) \mbox{, } \overline{AB} = \overline{BA}\\ \Leftrightarrow & \exists \lambda_B \neq 0 \mbox{, } AB = \lambda_B BA \\ \Leftrightarrow & A \in Z( \mathrm{GL}_n(\mathbb{K}) ) = \mathbb{K}^* I_n = \overline{I_n}\\ \Leftrightarrow & \overline{A} = \overline{I_n} \\ \end{align*}
But I don't understand the second fact : $\forall \overline{B} \in \mathrm{PGL}_n( \mathbb{K} ) \mbox{, } \overline{AB} = \overline{BA} \Leftrightarrow \exists \lambda_B \neq 0 \mbox{, } AB = \lambda_B BA \\$. I tried to compute $\overline{AB} = \overline{BA}$ in many ways but I don't get the result. Is there somebody to give me an indication ? Thank you