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Let $M$ be a square matrix over a fixed ground field, partitioned in block form as
$$
M = \left({ \begin{array}{cc} P & Q \\ R & S \end{array} }\right) \ ,
$$
where $P$ is a square non-singular submatrix.

The complement of $P$ is
$$
M/P = S - R P^{-1} Q \ .
$$

The [[Schur determinant lemma]] may be expressed in the form
$$
\det(M) = \det(P) \det(M/P) \ .
$$

====References====
* Hogben, Leslie. ''Handbook of linear algebra'' (2nd enlarged ed.) Discrete Mathematics and its Applications, Chapman & Hall/CRC (2014) ISBN 978-1-4665-0728-9 {{ZBL|1284.15001}}
* Zhang, Fuzhen (ed.) ''The Schur complement and its applications'', Numerical Methods and Algorithms '''4''' Springer (2005) ISBN 0-387-24271-6 {{ZBL|1075.15002}}