  We show that the product rank of the $3\times 3$ determinant is $5$, and the
product rank of the $3\times 3$ permanent is $4$. As a corollary, we obtain
that the tensor ranks of the $3 \times 3$ determinant and permanent are $5$ and
$4$, respectively. We show moreover that the border product rank of the $n
\times n$ permanent is larger than $n$ for any $n\geq 3$.
