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## Homework Statement

Let A and B be nxn matrices over reals. Show that I - BA is invertible if I - AB is invertible. Deduce that AB and BA have the same eigen values

## Homework Equations

det(AB) = det(A).det(B)

## The Attempt at a Solution

given: (I-AB) is invertible

-> det(I-AB) is not equal to 0

i.e. (-1)^n times det(AB-I) is not equal to 0

i.e. (-1)^n times det(AB)-det(I) is not equal to 0

i.e. (-1)^n times det(A).det(B)-det(I) is not equal to 0

i.e. (-1)^n times det(B).det(A)-det(I) is not equal to 0

i.e. (-1)^n times det(BA)-det(I) is not equal to 0

i.e. (-1)^n times det(BA-I) is not equal to 0

-> det(I-BA) is not equal to 0

and hence (I - BA) is invertible, if (I - AB) is invertible

as regards to the second part i have not got it yet...