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# Pascal Matrix

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.

Pascal's Triangle

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## 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.

Pascal Matrix

## Cite this as:

Weisstein, Eric W. "Pascal Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PascalMatrix.html