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Math Help - Inverse of upper triangular matrix with diagonal elements all 1

  1. #1
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    Inverse of upper triangular matrix with diagonal elements all 1

    Prove that the inverse of an upper triangular matrix whose diagonal entires are all 1 is itself an upper triangular matrix whose diagonal entires are all 1.

    I know that we can use Cramer's rule and just crunch out the result but I was hoping that someone could provide a short and simple reason.
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  2. #2
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    Using matrix multiplication, it is (fairly) easy to show that upper triangular matrices are closed under multiplication. If you haven't seen this before, let A,B be upper triangular and C=AB. Then letting
    A=[a_{ij}], B=[b_{ij}], C=[c_{ij}] , the rules of matrix multiplication show c_{ij}=\displaystyle\sum_{k=1}^n a_{ik}b_{kj}. But for k<i we have a_{ik}=0 since upper triangular. Similarly for k>j we have b_{kj}=0. But if i>j then the previous arguments imply a_{ij}=0.
    But these are just the strictly lower triangular entries.

    The same argument yields that c_{ii}=a_{ii}b_{ii}.
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