Eigenvalue-less matrices from SL${}_3({\mathbb{F}}_7)$ to PSL${}_3({\mathbb{F}}_7)$

We analyse the set of matrices in SL${}_3({\mathbb{F}}_7)$ without eigenvalues explicitly, extracting nice bijections between the 18 equally sized conjugacy classes contained within. In doing so, we discover a set of $18$ commuting matrices for which every conjugacy class is represented and tells us how to decide when collections of commuting matrices are simultaneously conjugate. The main innovation is the proofs are accessible to undergraduates and do not rely on computer calculations nor very general theory. It is also rare that the details of special cases are written down.