The common-submatrix Laplace expansion

TL;DR

This paper presents a classical Laplace expansion theorem variant where all submatrices share a common submatrix, showing that the restricted expansion equals the original determinant times the CSM's determinant.

Contribution

It introduces a new version of the Laplace expansion theorem with submatrices sharing a common submatrix, extending classical determinant expansion techniques.

Findings

The restricted Laplace expansion equals the product of the original determinant and the CSM's determinant.

The theorem is proven with sign considerations included.

The result generalizes classical Laplace expansion by incorporating a common submatrix constraint.

Abstract

We state and prove a classical version of the Laplace expansion theorem where all submatrices in the expansion are restricted to contain a specified common submatrix (CSM). The result states that (accounting for signs) this restricted expansion equals the determinant of the original matrix times the determinant of the CSM.