Sylvester's determinant theorem

The theorem states that if A, B are matrices of size p × n and n × p respectively, then

$det(I\_p\; +\; AB)\; =\; det(I\_n\; +\; BA),$

where I_a is the identity matrix of order a.

This theorem is useful in developing a Bayes estimator for multivariate Gaussian distributions.

Sylvester (1857) stated this theorem without proof.