Permanent of a $3 \times 3$ and $4 \times 4$ matrix from determinants Let $A$ be a $3 \times 3$ or a $4 \times 4$ matrix with entries $a_{ij}$. Can someone provide me a matrix $B$ so that $\operatorname{per}(A) = \det(B)$? What is the smallest explicit $B$ that is known such that $\operatorname{per}(A) = \det(B)$? Any references on this with explicit examples?

Some restrictions could be the following cases:

Case $(1)$ Only linear functionals are allowed as entries of $B$.

Case $(2)$ Non-linear functionals are allowed provided each term has atmost $O(log(n))$ degree (degree is sum of degree of variables) where $n$ is the size of the matrix involved. In our case, upto degree $2$.
[EDIT]


For consistency, I switched the notations from $c(n)$ to $dc(n)$.
It was asked by v s in the comments whether my answer generalize to higher dimensions. It does and gives an upper bound over any field:
$$dc(n)\le 2^n-1.$$
See my draft on this: An Upper Bound for the Permanent versus Determinant Problem.


[/EDIT] 

[A side comment: I think you could edit your previous question instead of creating a new one.]

I have the following answer for you: 

$$\operatorname{per}\begin{pmatrix} a&amp;b&amp;c\\d&amp;e&amp;f\\g&amp;h&amp;i\end{pmatrix}=
    \det\begin{pmatrix}
    0 &amp; a &amp; d &amp; g &amp; 0 &amp; 0 &amp; 0 \\
    0 &amp; 1 &amp; 0 &amp; 0 &amp; i &amp; f &amp; 0 \\
    0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 &amp; c &amp; i \\
    0 &amp; 0 &amp; 0 &amp; 1 &amp; c &amp; 0 &amp; f \\
    e &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 &amp; 0 \\
    h &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1 &amp; 0 \\
    b &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 0 &amp; 1
    \end{pmatrix}$$

Note that looking for such references about explicit examples, I could'nt find any and thus the example I give you is an example I built.

This question you are asking is commonly called the "Permanent vs Determinant problem". Suppose we are given an $(n\times n)$ matrix $A$, and we want the smallest matrix $B$ such that $\operatorname{per} A=\det B$. Let us denote by $dc(n)$ the dimensions of the smallest such $B$. Here are historical results:


[Szegö 1913] $dc(n)\ge n+1$ 
[von zur Gathen 1986] $dc(n)\ge n\sqrt 2-6\sqrt n$
[Cai 1990] $dc(n)\ge n\sqrt 2$
[Mignon &amp; Ressayre 2004] $dc(n)\ge n^2/2$ in characteristic $0$
[Cai, Chen &amp; Li 2008] $dc(n)\ge n^2/2$ in characteristic $\neq 2$.


This shows that $5\le dc(3)\le 7$ (the upper bound is the matrix given above). 

As I am lazy, I just give you one reference where you can find the other ones. It is the most recent paper I cited, by Cai, Chen and Li: A quadratic lower bound for the permanent and determinant problem over any characteristic $\neq 2$.

If you read French, you can also have a look to my slides on this subject: Permanent versus Déterminant.