  An inversion in a matrix of zeros and ones consists of two entries both of
which equal $1$, and one of which is located to the top-right of the other. It
is known that in the class $\mathcal{A}(R,S)$ of $(0,1)$--matrices with row sum
vector $R$ and column sum vector $S$, the number of inversions in a matrix is
monotonic with respect to the secondary Bruhat order. Hence any two matrices in
the same class $\mathcal{A}(R,S)$ having the same number of inversions, are
incomparable in the secondary Bruhat order. We use this fact to construct
antichains in the Bruhat order of $\mathcal{A}(n,2)$, the class of all $n\times
n$ binary matrices with common row and column sum~$2$.
  A product construction of antichains in the Bruhat order of
$\mathcal{A}(R,S)$ is given. This product construction is applied in finding
antichains in the Bruhat order of the class $\mathcal{A}(2k,k)$ of square
$(0,1)$--matrices of order $2k$ and common row and column sum~$k$.
