We consider the problem of testing bipartiteness in the adjacency
matrix model. The best known algorithm, due to Alon and Krivelevich,
distinguishes between bipartite graphs and graphs that are
$\epsilon$-far from bipartite using $O((1/\epsilon^2)polylog(1/epsilon)$
queries. We show that this is optimal for non-adaptive algorithms,
up to the polylogarithmic factor. We also show a lower bound of
$\Omega(1/\epsilon^{3/2})$ for adaptive algorithms.