Determinants of Block Matrices with Noncommuting Blocks

TL;DR

This paper investigates conditions under which two different determinant evaluation procedures yield the same result for block matrices with noncommuting blocks, extending known results and identifying optimal commutativity conditions.

Contribution

It establishes new minimal commutativity conditions ensuring the equality of determinants for noncommuting block matrices and proves their optimality.

Findings

Identifies specific pairwise commutativity conditions for blocks

Proves the equality of determinants under these conditions

Demonstrates the optimality of the conditions

Abstract

Let $M$ be an $mn\times mn$ matrix over a commutative ring $R$. Divide $M$ into $m \times m$ blocks. Assume that the blocks commute pairwise. Consider the following two procedures: (1) Evaluate the $n \times n$ determinant formula at these blocks to obtain an $m \times m$ matrix, and take the determinant again to obtain an element of $R$; (2) Take the $mn \times mn$ determinant of $M$. It is known that the two procedures give the same element of $R$. We prove that if only certain pairs of blocks of $M$ commute, then the two procedures still give the same element of $R$, for a suitable definition of noncommutative determinants. We also derive from our result further collections of commutativity conditions that imply this equality of determinants, and we prove that our original condition is optimal under a particular constraint.