A Formula for the Determinant

TL;DR

This paper presents an efficient formula for computing the determinant of an n×n matrix over a commutative ring, using only addition, subtraction, and multiplication, with bounded operation count and staged computation.

Contribution

It introduces a determinant formula that avoids divisions and branching, enabling efficient computation with a fixed operation count and staged evaluation.

Findings

Operation count bounded by O(n^4 log n)

Number of computation stages bounded by O((log n)^2)

Applicable to matrices over any commutative ring with unity

Abstract

We give a formula for the determinant of an $n\times n$ matrix with entries from a commutative ring with unit. The formula can be evaluated by a "straight-line program" performing only additions, subtractions and multiplications of ring elements; in particular it requires no divisions or conditional branching (as are required, for example, by Gaussian elimination). The number of operations performed is bounded by a fixed power of $n$, specifically $O(n^4\log n)$. Furthermore, the operations can be partitioned into "stages" in such a way that the operands of the operations in a given stage are either matrix entries or the results of operations in earlier stages, and the number of stages is bounded by a fixed power of the logarithm of $n$, specifically $O\big((\log n)^2\big)$.