  We reveal an intriguing connection between the set of 27 (disregarding the
identity) invertible symmetric 3 x 3 matrices over GF(2) and the points of the
generalized quadrangle GQ(2,4). The 15 matrices with eigenvalue one correspond
to a copy of the subquadrangle GQ(2,2), whereas the 12 matrices without
eigenvalues have their geometric counterpart in the associated double-six. The
fine details of this correspondence, including the precise algebraic
meaning/analogue of collinearity, are furnished by employing the representation
of GQ(2,4) as a quadric in PG(5,2) of projective index one. An interesting
physical application of our findings is also mentioned.
