Sylvester's_determinant_theorem
Weinstein–Aronszajn identity
For two suitable matrices, A and B, I+AB and I+BA have the same determinate
In mathematics, the Weinstein–Aronszajn identity states that if and are matrices of size m × n and n × m respectively (either or both of which may be infinite) then, provided (and hence, also ) is of trace class,
where is the k × k identity matrix.
It is closely related to the matrix determinant lemma and its generalization. It is the determinant analogue of the Woodbury matrix identity for matrix inverses.