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# Pascal matrix

Definition The *Pascal matrix* $P$ of order $n$ is the real
square $n\times n$ matrix whose entries are [1]

$P_{{ij}}={i+j-2\choose j-1}.$ |

For $n=5$,

$P=\begin{pmatrix}1&1&1&1&1\\ 1&2&3&4&5\\ 1&3&6&10&15\\ 1&4&10&20&35\\ 1&5&15&35&70\end{pmatrix},$ |

so we see that the Pascal matrix contains the Pascal triangle on its antidiagonals.

Pascal matrices are ill-conditioned. However, the inverse of the $n\times n$ Pascal matrix is known explicitly and given in [1]. The characteristic polynomial of a Pascal triangle is a reciprocal polynomial [1].

# References

- 1
N.J. Higham,
*Accuracy and Stability of Numerical Algorithms*, 2nd ed., SIAM, 2002.

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## Mathematics Subject Classification

65F35*no label found*15A12

*no label found*15A09

*no label found*15A57

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