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Properties of Inverse Matrices

  1. May 11, 2008 #1
    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

    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?
     
    Last edited: May 11, 2008
  2. jcsd
  3. May 11, 2008 #2
    The ones that are false all have trivial counterexamples. What have you tried so far?
     
  4. May 11, 2008 #3
    Hi, I'm a little rusty on Linear Algebra, so forgive my ignorance on some of the approaches. Can you elaborate by what you mean by "trivial"?

    I tried random matrices (often with easy numbers, such as the identity matrix) for A, B, C, D, E, and F, but only found counterexamples for A, B, and F (but knowing for a fact that C, D, and E are not all correct).

    By one of the simple properties of inverse matrices, I am almost certain that C is correct.
     
    Last edited: May 11, 2008
  5. May 12, 2008 #4
    C is true, since otherwise A^(-1) is not invertible in the first place, with [itex]A^{-5}=A^{-1}......A^{-1}[/itex] hence if the determinant is zero then one of them has also a zero determinant.

    D holds only for reals and from the positive eigenvalues of A^(2) .[itex]trace(A^2)>0, \det(A^2)>0[/itex]
    [tex]
    (A+A^{-1})= (I+A^{2})A^{-1} = (A(I+A^{2})^{-1})^{-1}
    [/tex]

    E follows from the distribution property,

    I think I have to sleep so what is the the counterexample for F?
     
    Last edited: May 12, 2008
  6. May 12, 2008 #5

    HallsofIvy

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    Any two matrices that do not commute will give an counterexample to F.
     
  7. May 12, 2008 #6
    Haha, Of course! That was lame, sorry!
     
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