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Math Help - Matrices (A+B)^-1 = A^-1+B^-1 is not true for all square matrices A,B

  1. #1
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    Matrices (A+B)^-1 = A^-1+B^-1 is not true for all square matrices A,B

    Hi,

    I have this question and I am not sure how to tackle it:

    Construct an infinite number of examples that demonstrate (A+B)^-1 = A^-1 + B^-1 is not true for all square matrices A and B of the same size.

    So I am aware from the Linear Algebra unit we have been doing that if A or B are not invertible then their inverse does not exist but I am unsure how to use this to construct an infinite number of examples. Any help would be greatly appreciated! Thanks in advance
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  2. #2
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    Re: Matrices (A+B)^-1 = A^-1+B^-1 is not true for all square matrices A,B

    Hey Cotty.

    One example is setup two matrices: the first has a full row of zeros in the first row and the second has a full row of zeroes in the second row. Together though, the sum of the matrices gives a matrix that is invertible, however each matrix on its own is not.
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  3. #3
    Junior Member Nehushtan's Avatar
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    Re: Matrices

    I think the question (from the way itís written) assumes that all of A, B and A+B are invertible.

    Hint: Let A and B be multiples of the identity matrix.
    Last edited by Nehushtan; March 10th 2013 at 07:06 AM.
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    Re: Matrices

    You're amazing I got it, thank you so much!!!!! Thanks Chiro as well
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