:Hi does anyone know whether it is generally true for A an m*n :matrix, and B an n*m matrix that det(I-AB)=det(I-BA)? And if it is, :what approach one might take to prove it? I have been able to show :that AB and BA have the same set of eigenvalues and that for each :eigenvalue the dimension of the corresponding eigenspaces are also :the same for AB and BA. But wasn't able to go further. Thanks. Consider the identity (fixed spacing required) [I-AB A] * [I 0] = [I 0] * [I A ] [0 I] [B I] [B I] [0 I-BA]