  We describe a way of representing finite biquandles with n elements as 2n x
2n block matrices. Any finite biquandle defines an invariant of virtual knots
through counting homomorphisms. The counting invariants of non-quandle
biquandles can reveal information not present in the knot quandle, such as the
non-triviality of the virtual trefoil and various Kishino knots. We also
exhibit a virtual knot which is distinguished from its obverse and its reverse
by a finite biquandle counting invariant. We classify biquandles of order 2, 3
and 4 and provide a URL for our Maple programs for computing with finite
biquandles.
