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