Given two $m \times n$ matrices $A$ and $B$ over the complex numbers, prove that $\det(I_m + A B^T ) = det(I_n + B^T A)$.

Question

Let $A$ and $B$ be two $m\times n$-matrices over the complex numbers. I'd like to prove that $$\det\big( I_m + AB^T\big) = \det\big( I_n + B^T A\big)\,.$$

Answer 1

The trick is to create a block matrix with the above mentioned matrices in a particular order, and then to observe the determinant.

This link shows how to prove the answer: https://en.wikipedia.org/wiki/Sylvester%27s_determinant_theorem

You have to fill in the blanks a bit, but should be okay.