  In this note, we study linear determinantal representations of smooth plane
cubics over finite fields. We give an explicit formula of linear determinantal
representations corresponding to rational points. Using Schoof's formula, we
count the number of projective equivalence classes of smooth plane cubics over
a finite field admitting prescribed number of equivalence classes of linear
determinantal representations. As an application, we determine isomorphism
classes of smooth plane cubics over a finite field with 0, 1 or 2 equivalence
classes of linear determinantal representations.
