In matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix, an upper-triangular matrix, or a symmetric matrix. For example, the 5 × 5 matrices are:

$${\displaystyle L_{5}={\begin{pmatrix}1&0&0&0&0\\1&1&0&0&0\\1&2&1&0&0\\1&3&3&1&0\\1&4&6&4&1\end{pmatrix}}\,\,\,}$$$${\displaystyle U_{5}={\begin{pmatrix}1&1&1&1&1\\0&1&2&3&4\\0&0&1&3&6\\0&0&0&1&4\\0&0&0&0&1\end{pmatrix}}\,\,\,}$$$${\displaystyle S_{5}={\begin{pmatrix}1&1&1&1&1\\1&2&3&4&5\\1&3&6&10&15\\1&4&10&20&35\\1&5&15&35&70\end{pmatrix}}=L_{5}\times U_{5}}$$ There are other ways in which Pascal's triangle can be put into matrix form, but these are not easily extended to infinity.

The non-zero elements of a Pascal matrix are given by the binomial coefficients:

$${\displaystyle L_{ij}={i \choose j}={\frac {i!}{j!(i-j)!}},j\leq i}$$ $${\displaystyle U_{ij}={j \choose i}={\frac {j!}{i!(j-i)!}},i\leq j}$$ $${\displaystyle S_{ij}={i+j \choose i}={i+j \choose j}={\frac {(i+j)!}{i!j!}}}$$

such that the indices i, j start at 0, and ! denotes the factorial.

The 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. In other words, matrices S_n, L_n, and U_n are unimodular, with L_n and U_n having trace n.

The trace of S_n is given by

with the first few terms given by the sequence 1, 3, 9, 29, 99, 351, 1275, ... (sequence A006134 in the OEIS).