Combinatorial Matrix problem - REALLY NEAT!

1. Jan 23, 2008

rbzima

Let $$A\Large$$ be the $$n \times n$$ matrix $$\left[a_i_j]$$ given by

$$a_i_j = \left(\stackrel{m_j + i - 1}{j - 1}\right)$$

where $$i,j = 1, 2, ..., n$$ and $$n$$ and $$m_j$$ is a natural number. 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.

Last edited: Jan 23, 2008