  For a binary matrix X, the Boolean rank br(X) is the smallest integer k for
which X equals the Boolean sum of k rank-1 binary matrices, and the isolation
number i(X) is the maximum number of 1s no two of which are in a same row,
column and a 2x2 submatrix of all 1s. In this paper, we continue Lubiw's study
of firm matrices. X is said to be firm if i(X)=br(X) and this equality holds
for all its submatrices. We show that the stronger concept of superfirmness of
X is equivalent to having no odd holes in the rectangle cover graph of X, the
graph in which br(X) and i(X) translate to the clique cover and the
independence number, respectively. A binary matrix is minimally non-firm if it
is not firm but all of its proper submatrices are. We introduce two matrix
operations that lead to generalised binary matrices and use these operations to
derive four infinite classes of minimally non-firm matrices. We hope that our
work may pave the way towards a complete characterisation of firm matrices via
forbidden submatrices.
