Three types of  matrices can be obtained by writing Pascal's triangle as a lower triangular matrix and truncating appropriately: a symmetric matrix  with , a lower triangular matrix  with , and an upper triangular matrix  with , where , 1, ..., . For example, for , these would be given by

			(1)
			(2)
			(3)

The Pascal -matrix or order  is implemented in the Wolfram Language as LinearAlgebra`PascalMatrix[n].

These matrices have some amazing properties. In particular, their determinants are all equal to 1

(4)

and

(5)

(Edelman and Strang).

Edelman and Strang give four proofs of the identity (5), the most straightforward of which is

			(6)
			(7)
			(8)
			(9)

where Einstein summation has been used.

REFERENCES:

Abbott, P. "Tricks of the Trade: Pascal Matrices." Mathematica J. 9, 691-694, 2005.

Edelman, A. and Strang, G. "Pascal Matrices." http://web.mit.edu/18.06/www/pascal-work.pdf.

Strang, G. Introduction to Linear Algebra, 3rd ed. Wellesley-Cambridge Press, 2003.