  We introduce a new notion of the determinant, called symmetrized determinant,
for a square matrix with the entries in an associative algebra $\AA$. The
monomial expansion of the symmetrized determinant is obtained from the standard
expansion of the commutative determinant by averaging the products of entries
of the matrix in all possible orders. We show that for any fixed
finite-dimensional associative algebra $\AA$, the symmetrized determinant of an
$n\times n$ matrix with the entries in $\AA$ can be computed in polynomial in
$n$ time (the degree of the polynomial is linear in the dimension of $\AA$).
Then, for every associative algebra $\AA$ endowed with a scalar product and
unbiased probability measure, we construct a randomized polynomial time
algorithm to estimate the permanent of non-negative matrices. We conjecture
that if $\AA=\Mat(d, {\Bbb R})$ is the algebra of $d\times d$ real matrices
endowed with the standard scalar product and Gaussian measure, the algorithm
approximates the permanent of a non-negative $n \times n$ matrix within
$O(\gamma_d^n)$ factor, where $\lim_{d \longrightarrow +\infty} \gamma_d=1$.
Finally, we provide some informal arguments why the conjecture might be true.
