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

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