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Properties of Inverse Matrices

  1. May 11, 2008 #1
    1. The problem statement, all variables and given/known data

    Determine which of the formulas hold for all invertible nxn matrices A and B

    A. AB=BA
    B. (A+A^–1)^8=A^8+A^–8
    C. A^5 is invertible
    D. A+A^–1 is invertible
    E. (In+A)(In+A^–1)=2In+A+A^–1 (where In is the identity matrix)
    F. (A+B)^2=A^2+B^2+2AB

    2. Relevant equations

    Certain properties of inverse matrices can be used. For example, if A is invertible, then A^k is invertible for all k greater or equal to 1 (this proves C to be correct).

    3. The attempt at a solution

    I was able to find counterexamples to prove A and B and F incorrect. However, the webwork program (designed for practicing basic Linear Algebra) I am using states that C, D, and E are not all correct ... what am I missing?
  2. jcsd
  3. May 12, 2008 #2
    Option D is not always true. Try to find a matrix A where [tex]A^{-1} = -A[/tex].
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