An Amusing Determinant

A student and I were playing with determinants recently and we stumbled upon an amusing one. I have a very strong feeling that this must have been stumbled upon many times in the past so we’re making no claim to originality. All we’re claiming is that if you haven’t seen this before, you might be in for a little amusement.

Let $P = (P_{ij})$ be an $n$ by $n$ matrix where $P_{ij} = \binom{i + j - 2}{i-1}.$

Compute the determinant of $P$.

In other words, if you write out Pascal’s triangle as a grid instead of a triangle as shown below, then $P$ is the upper left $n$ by $n$ corner. If you’ve seen this before, please let us know where. We're a math club for girls.
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3 Responses to An Amusing Determinant

1. Marcus Neal says:

Nice!

2. Mitch Richling says:

This is the pascal matrix with k=1. It has a nice inverse when k is a reciprocal of an integer.

• girlsangle says:

Thank you for identifying it!
Googling “Pascal Matrix” brought me to the nice Wikipedia entry which references a nice Monthly article by Edelman and Strang, Pascal Matrices.