# If AB^2-A Invertible Prove that BA-A Invertible

1. Jul 31, 2010

### ThankYou

1. The problem statement, all variables and given/known data
As in the tile
If AB^2-A is a Invertible matrix Prove that BA-A is also a Invertible matrix

2. Relevant equations
liner algebra 1 , only the start...

3. The attempt at a solution
Made many things noting work

Thank you

2. Jul 31, 2010

### ThankYou

I've solve it..
Thank you..
AB^2-A = A(B^2-I)
Means that A and (B^2-I) are Invertible
(B-I)(B+I) are Invertible and so
A(B-I) Is Invertible

3. Jul 31, 2010

### Staff: Mentor

You're supposed to show that BA - A is invertible.

A(B - I) = AB - A, which is not necessarily equal to BA - A.

4. Jul 31, 2010

### losiu99

Nevertheless, we can save this reasoning provided you are allowed to use det(AB)=det(A)det(B).

5. Jul 31, 2010

### ThankYou

I am allowed

6. Jul 31, 2010

### losiu99

Then we can just say det(AB2-A)=det(A)det(B-I)det(B+I) is nonzero, so det(BA-A)=det(B-I)det(A) is nonzero.

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