Given a matrix , let denote its determinant. Then
where is the submatrix of formed by the intersection of the subset of columns and of rows. Bareiss (1968) writes the identity as
where
for .
When , this identity gives the Chió pivotal condensation method.
More things to try:
Weisstein, Eric W. "Sylvester's Determinant Identity." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SylvestersDeterminantIdentity.html