Q: If A and B are both nxn matrices and AB-I is invertable then prove that BA-I is also invertable.
if A is invertible iff |A|<>0
The Attempt at a Solution
I've been thinking about this for over an hour I've only managed to prove it if either A or B are invertable. because if let's say A is invertable then:
|AB-I|<>0 => |AB-I||A|<>0 => |ABA-A|<>0 => |A||BA-I|<>0 => |BA-I|<>0 and so it's invertable. if B is invertable then you do pretty much the same thing on starting on the left side.
But what if they're both singular?