1. The problem statement, all variables and given/known data Show that two similar matrices A and B share the same determinants, WITHOUT using determinants 2. The attempt at a solution A previous part of this problem not listed was to show they have the same rank, which I was able to do without determinants. The problem is I can't think of how to show they have the same eigenvalues without going to the characteristic polynomial (derived from the determinant of |A-lamba*I|. My other idea was to think of both A and B as the same linear map with respect to a different basis. After that I draw a blank.