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
, let 
denote its determinant. Then
is the submatrix of 
formed by the intersection of the subset 
of columns and 
of rows. Bareiss (1968) writes the identity as
![|A|[a_(kk)^((k-1))]^(n-k-1)=|a_(k+1,k+1)^((k)) ... a_(k+1,n)^((k)); | ... |; a_(n,k+1)^((k)) ... a_(n,n)^((k))|,](/images/equations/SylvestersDeterminantIdentity/NumberedEquation2.svg)
.
,
this identity gives the Chió pivotal
condensation method.
