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Combinatorial Matrix problem - REALLY NEAT!

  1. Jan 23, 2008 #1
    Let [tex]A\Large[/tex] be the [tex]n \times n[/tex] matrix [tex]\left[a_i_j][/tex] given by

    [tex]a_i_j = \left(\stackrel{m_j + i - 1}{j - 1}\right)[/tex]

    where [tex]i,j = 1, 2, ..., n[/tex] and [tex]n[/tex] and [tex]m_j[/tex] is a natural number. Find the determinant of [tex]A\Large[/tex].

    So, I've been looking at this problem for the past couple days and it is a really interesting problem. I discovered that the determinant is actually 1, however I'm wondering how to show this result in terms of a proof. If anyone has any suggestions, they would be greatly welcomed! ;)

    I personally was thinking that when you transpose the matrices, something interesting happens with the nested matrices in the upper left corner moving outward, but I'm not entirely sure this is a right approach to be taking. Also, finding determinants by cofactors is really only helpful in smaller matrices, so I was wondering if it might also need to be expanded using the Big Formula, which in this case would consist of n! different terms.
    Last edited: Jan 23, 2008
  2. jcsd
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