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.

Pascal's triangle as a grid

If you’ve seen this before, please let us know where.

Advertisements

About girlsangle

We're a math club for girls.
This entry was posted in math and tagged , , , , . Bookmark the permalink.

3 Responses to An Amusing Determinant

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

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s