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!=*x*^{0}=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 *F ^{r}*, 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.