# Combinatorial Matrix problem

#### rbzima

Let $A\Large$ be the $n \times n$ matrix $(a_{ij})$ given by
$$a_{ij} = \binom{m_j + i - 1}{j - 1}$$
where $i,j = 1, 2, ..., n$ and $n$ and $m_j$ are natural numbers. Find the determinant of $A\Large$.

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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#### fresh_42

Mentor
2018 Award
I would first check whether it is still true for all $m_j =1$. Then an induction over $n$ should be the way to prove it. Also formulas for the various diagonals might help a lot.

"Combinatorial Matrix problem"

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