Let [itex]A\Large[/itex] be the [itex]n \times n[/itex] matrix [itex] (a_{ij}) [/itex] given by

[tex]a_{ij} = \binom{m_j + i - 1}{j - 1} [/tex]

where [itex]i,j = 1, 2, ..., n[/itex] and [itex]n[/itex] and [itex]m_j[/itex] are natural numbers. Find the determinant of [itex]A\Large[/itex].

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.

[tex]a_{ij} = \binom{m_j + i - 1}{j - 1} [/tex]

where [itex]i,j = 1, 2, ..., n[/itex] and [itex]n[/itex] and [itex]m_j[/itex] are natural numbers. Find the determinant of [itex]A\Large[/itex].

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.

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