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Math Help - prove inverse of upper triangular matrix

  1. #1
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    prove inverse of upper triangular matrix

    Can you help me with the following question please

    Prove that the inverse of a non singular N X N upper triangular matrix is an upper triangular matrix
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  2. #2
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    We can assume that the matrix A is upper triangular and invertible, since

    A^{-1}=\frac{1}{det(A)}\cdot adj(A)

    We can prove that A^{-1} is upper triangular by showing that

    the adjoint is upper triangular or that the matrix of cofactors is lower

    triangular. Do this by showing every cofactor C_{ij} with

    i<j(above the diagonal) is 0.

    Since C_{ij}=(-1)^{i+j}M_{ij} it suffices to show that

    each minor M_{ij} with i<j is 0.

    Start by letting B_{ij} be the matrix we get when the ith row

    and jth column of A are gotten rid of (deleted).

    Can you go further?.

    There is a handy theorem which we won't prove, but use, which says:

    "If A is an nXn triangular matrix then det(A) is the product of the entries on the main diagonal of the matrix. What I mean is det(A)=a_{11}a_{22}.....a_{nn}
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