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We show that the product rank of the $3\times 3$ determinant is $5$, and theproduct rank of the $3\times 3$ permanent is $4$. As a corollary, we obtainthat 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 $n\geq 3$.

Product Ranks of the $3\times 3$ Determinant and Permanent