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Math Help - A question about nonsingular matrices

  1. #1
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    A question about nonsingular matrices

    Hello, I'm kind of stuck on this question where I have the following facts:
    A, B & A+B is nonsingular

    How do I prove that (A^-1+B^-1) is nonsingular as well ?

    Thank you.
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  2. #2
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    Re: A question about nonsingular matrices

    Hey Osp85.

    For this try and consider that if (A^(-1) + B^(-1)) is non-singular then (A^(-1) + B^(-1))^(-1) exists.

    We know that A^(-1) exists since A is non-singular. We also know that B^(-1) exists since B is non-singular.

    We know that (A+B)^(-1) exists since (A+B) is nonsingular.

    Let det(A) = a, det(B) = b, det(A+B) = c.

    det[A^(-1)] = 1/a, det[B^(-1)] = 1/b and det[A^(-1) + B^(-1)] = c

    From this paper

    http://www.math.umbc.edu/~gowda/papers/trGOW10-02.pdf

    We get 1/a + 1/b <= 1/c

    Proof by contradiction: assume that a = -b then 1/a + 1/b = 0 but 1/c can never be zero unless c = infinity which should never happen.

    So we rule out this case.

    The only other case is that 1/a + 1/b != 0 and thus 1/c is non-zero which implies non-singular matrix.

    Thus the statement is proven.

    I just realized that another solution could be to assume that 1/c was just non-zero.
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