Odd Degenerate

Determinant of an Odd Anti-symmetric Matrix is 0

To flip the signs of a row of a matrix flips the sign of its determinant. Flipping the signs of all the elements flips the sign of the determinant if there are an odd number of rows. The determinant of a matrix is the same as the determinant of its transpose. The transpose of an antisymmetric matrix is the same as the matrix with all of its signs flipped. The determinant of an odd antisymmetric matrix is minus the determinant of its transpose. That determinant is thus 0!

For a degenerate differential form Ω, there is a non-zero vector x such that for all y, Ω(x, y) = 0. For the tensor ω corresponding to Ω this amounts to a linear combination of the rows of the tensor which yields the vector 0. The determinant of the tensor is thus 0.

A non-degenerate anti-symmetric tensor has thus an anti-symmetric inverse. A non-degenerate anti-symmetric bilinear form has thus an anti-symmetric bilinear dual inverse.