# Rank of Submatrices of the Pascal Matrix

## TL;DR

This paper derives a formula for the rank of submatrices of the Pascal matrix, extending previous invertibility conditions, and applies these results to polynomial approximation problems.

## Contribution

It introduces a general formula for the rank of Pascal matrix submatrices and provides bases for their row and column spaces, expanding understanding of their structure.

## Key findings

- Derived a formula for the rank of Pascal matrix submatrices
- Provided bases for row and column spaces of these submatrices
- Applied results to polynomial approximation problems

## Abstract

In a previous paper, we derived necessary and sufficient conditions for the invertibility of square submatrices of the Pascal upper triangular matrix. To do so, we established a connection with the two-point Birkhoff interpolation problem. In this paper, we extend this result by deriving a formula for the rank of submatrices of the Pascal matrix. Our formula works for both square and non-square submatrices. We also provide bases for the row and column spaces of these submatrices. Further, we apply our result to one-point lacunary polynomial approximation.