**Question.** Let $\mathcal{M}_3$ be the set of $3\times 3$ matrices with non-negative entries and unit determinant. What is the number of $M\in \mathcal{M}_3$ with fixed sum of entries? What is the answer if we allow only matrices with strictly positive entries?

The rate of growth is also interesting.

**Origin.** For the similar question in the case of $2\times 2$ matrices we get $\phi(n)$ and $\phi(n) - 2$ respectively, see this recent post: https://mathoverflow.net/q/441469/498423.

**Computation of the first elements.**
If we denote the answer sequence for non-negative case by $a_n$ then the first elements are

| $n$  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
| ---  | -- | -- | -- | -- | -- | -- | -- | -- | -- | -- | -- | -- | -- | -- |
| $a_n$| 0 | 0 | 3 | 18 | 54 | 126 | 261 | 404 | 667 | 955 | 1417 | 1709 | 2603 | 2979 |

$a = [0, 0, 0, 3, 18, 54, 126, 261, 404, 667, 955, 1417, 1709, 2603, 2979, \ldots]$.

And if $b_n$ is the answer for the strictly positive case then $b_n = 0$ if $n \le 10$ and further we have
| $n$  | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |
| ---  |  -- | -- | -- | -- | -- | -- | -- | -- | -- | -- | -- | -- | -- | -- |
| $b_n$|  9 | 18 | 49 | 69 | 169 | 239 | 388 | 423 | 848 | 902 | 1231 | 1559 | 2283 | 2002 |

$b = [0,\ldots ,0, 9, 18, 49, 69, 169, 239, 388, 423, 848, 902, 1231, 1559, 2283, 2002, \ldots]$.

Unfortunately, both sequences do not seem to be in OEIS.

**Thoughts on the growth.** Evidently, $a_n\ge b_n$, and $a_n\le Cn^9$ because the number of partitions of $n$ into $9$ terms is $O(n^9)$. If we consider only upper-triangular matrices with ones on the diagonal, we will get the bound $a_n\ge Cn^3$.

Thus, the answer is of polynomial order. Note that this is different to the $2\times 2$ case where the number-theoretic properties played the main role.