A Formula for the Determinant

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 $O\big((\log n)^2\big)$.