Pascal matrix

In mathematics, particularly matrix theory and combinatorics, the Pascal matrix is an infinite matrix containing the binomial coefficients as its elements. There are three ways to achieve this: as either an upper-triangular matrix, a lower-triangular matrix, or a symmetric matrix. The 5×5 truncations of these are shown below. Upper triangular: U 5 = ( 1 1 1 1 1 0 1 2 3 4 0 0 1 3 6 0 0 0 1 4 0 0 0 0 1 ) ; {displaystyle U_{5}={egin{pmatrix}1&1&1&1&1\0&1&2&3&4\0&0&1&3&6\0&0&0&1&4\0&0&0&0&1end{pmatrix}};,,,} Lower triangular: L 5 = ( 1 0 0 0 0 1 1 0 0 0 1 2 1 0 0 1 3 3 1 0 1 4 6 4 1 ) ; {displaystyle L_{5}={egin{pmatrix}1&0&0&0&0\1&1&0&0&0\1&2&1&0&0\1&3&3&1&0\1&4&6&4&1end{pmatrix}};,,,} Symmetric: S 5 = ( 1 1 1 1 1 1 2 3 4 5 1 3 6 10 15 1 4 10 20 35 1 5 15 35 70 ) . {displaystyle S_{5}={egin{pmatrix}1&1&1&1&1\1&2&3&4&5\1&3&6&10&15\1&4&10&20&35\1&5&15&35&70end{pmatrix}}.} These matrices have the pleasing relationship Sn = LnUn. From this it is easily seen that all three matrices have determinant 1, as the determinant of a triangular matrix is simply the product of its diagonal elements, which are all 1 for both Ln and Un. In other words, matrices Sn, Ln, and Un are unimodular, with Ln and Un having trace n. The elements of the symmetric Pascal matrix are the binomial coefficients, i.e.