We say that a matrix over a finite field Fq is positive definite if it is symmetric and each of its leading principal minors is a nonzero square in Fq. In previous work of the authors [J. Algebra, 2025], the entrywise positivity preservers on Mn(Fq)
Nearby in the stack
were classified for every
n≥2
, with one remaining case:
n=2
,
q≡1(mod4)
, and
q
not a square. We settle this case by proving that every positivity preserver on
M2(Fq)
is injective on the set
Fq+
of nonzero squares whenever
q≡1(mod4)
. The proof combines an idempotent reduction of positivity preservers with a well-known property of quadratic characters. This yields the complete classification of entrywise positivity preservers over every finite field and in every fixed dimension.