Pascal matrix

The Pascal matrix is an infinite matrix containing the binomial coefficients as its elements. There are 3 ways this can be achieved - either as an upper-triangular matrix, a lower-triangular matrix, or as a symmetric matrix. The 5×5 truncations of these are shown below.

Upper triangular: $U_{5}={\begin{pmatrix}1&1&1&1&1\\0&1&2&3&4\\0&0&1&3&6\\0&0&0&1&4\\0&0&0&0&1\end{pmatrix}};$ lower triangular: $L_{5}={\begin{pmatrix}1&0&0&0&0\\1&1&0&0&0\\1&2&1&0&0\\1&3&3&1&0\\1&4&6&4&1\end{pmatrix}};$ symmetric: $S_{5}={\begin{pmatrix}1&1&1&1&1\\1&2&3&4&5\\1&3&6&10&15\\1&4&10&20&35\\1&5&15&35&70\end{pmatrix}}$

These matrices have the pleasing relationship S_n = L_nU_n. 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 L_n and U_n.

See also

• Pascal's triangle

References

• G. S. Call and D. J. Velleman, Pascal's matrices, American Mathematical Monthly 100 (April 1993) pp 372-376.