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

  1. Jul 31, 2010 #1
    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. jcsd
  3. Jul 31, 2010 #2
    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
     
  4. Jul 31, 2010 #3

    Mark44

    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.
     
  5. Jul 31, 2010 #4
    Nevertheless, we can save this reasoning provided you are allowed to use det(AB)=det(A)det(B).
     
  6. Jul 31, 2010 #5
    I am allowed
     
  7. Jul 31, 2010 #6
    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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