  Denote by $A_n$ the set of square $(0,1)$ matrices of order $n$. The set
$A_n$, $n\le8$, is partitioned into row/column permutation equivalence classes
enabling derivation of various facts by simple counting. For example, the
number of regular $(0,1)$ matrices of order 8 is 10160459763342013440. Let
$D_n$, $S_n$ denote the set of absolute determinant values and Smith normal
forms of matrices from $A_n$. Denote by $a_n$ the smallest integer not in
$D_n$. The sets $\mathcal{D}_9$ and $\mathcal{S}_9$ are obtained; especially,
$a_9=103$. The lower bounds for $a_n$, $10\le n\le 19$, (exceeding the known
lower bound $a_n\ge 2f_{n-1}$, where $f_n$ is $n$th Fibonacci number) are
obtained. Row/permutation equivalence classes of $A_n$ correspond to bipartite
graphs with $n$ black and $n$ white vertices, and so the other applications of
the classification are possible.
