A Formula for the Determinant

05/31/2022

∙

by   Nicholas Pippenger, et al.

∙

0

∙

share

We give a formula for the determinant of an n× 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^4log 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((log n)^2).