  As an application of linear algebra for enumerative combinatorics, we
introduce two new ideas, signed bigrassmannian polynomials and bigrassmannian
determinant. First, a signed bigrassmannian polynomial is a variant of the
statistic given by the number of bigrassmannian permutations below a
permutation in Bruhat order as Reading suggested (2002) and afterward the
author developed (2011). Second, bigrassmannian determinant is a $q$-analog of
the determinant with respect to our statistic. It plays a key role for a
determinantal expression of those polynomials. We further show that
bigrassmannian determinant satisfies weighted condensation as a generalization
of Dodgson, Jacobi-Desnanot and Robbins-Rumsey (1986).
