The entries if B are called ``cofactors'' of A. These are (up to a plus-or-minus sign) special cases of ``minors'' of a matrix. A minor of a (not necessarily square) matrix A is the determinant of a square matrix obtained by omitting some rows and/or columns of A. [A determinant of order 0 is deemed to equal 1, as in 0!=x0=1.] We have seen already that a square matrix of order n has rank <n if and only if its determinant vanishes. This generalizes as follows: the rank of any matrix A is the largest integer r such that some order-r minor of A does not vanish.
Proof: It is clear that the rank is at least r, because we have r linearly independent columns. We'll show that, conversely, if there are r linearly independent columns then at least one of the minors formed with these columns is nonzero. This is because the submatrix of A formed with these columns has rank r; therefore, so does its transpose. So, its rows span Fr, and include a basis. Using these rows gives the desired minor.
Corollary: the rank of the cofactor matrix of A is n,1,0 according as the rank of A is n, n-1, or less than n-1.