
COMMENTS

An upper bound is the total number of {0,1}matrices, 2^(n^2).
Comment from Michael Kleber, Jan 05 2006: A lower bound is 2^(n^2n), A053763. For given the principal n1 X n1 submatrix A, the 2n2 further entries (excluding the bottom right corner) can be filled in arbitrarily and then there is always at least one choice for the last entry which makes the matrix invertible.
Comment from N. J. A. Sloane, Jan 06 2006: Let the matrix be [A b; c d], where A is n1 X n1, b is n1 X 1, c is 1 X n1, d is 0 or 1. The matrix is singular iff d = c A^(1) b, which for given A, b, c has at most one solution d.
Suppose A = identity, as in A125586. Then if d=0 there are 3^(n1) choices for b and c, while if d=1 there are (n1)*3^(n2) choices for b and c. This proves the formula in A125586.
