Is a column-wise and row-wise Gaussian matrix, jointly Gaussian?

Question

Let $Ainmathbb{R}^{ntimes m}$ be a random matrix that each of its rows or columns is a Gaussian vector with iid components. More formally we have $A_{i,cdot}simmathcal{N}(0,I_m),A_{cdot,j}simmathcal{N}(0,I_n)$ for all $iin[n],jin[m]$, and furthermore, $mathrm{law}(A)$ has support almost everywhere. Is $mathrm{law}(A)$ a multivariate Gaussian?
Example ($m=n=2$): let $A=left[begin{matrix}a& b\ c & dend{matrix}right]$. Assuming that $(a,b),(c,d),(a,c),(b,d)simmathrm{N}(0,I_2)$, and $mathrm{support}(mathop{law}(A))$ is non-zero almost everywhere. Is $mathop{law}(a,b,c,d)$ a multivariate Gaussian?
Possible generalization: if each row and column follow a multivariate p-Stable distribution, namely Cauchy for $p=1$, would the matrix follow a joint p-Stable distribution?

kimchi lover · Answer

No.  Answer to a previous version of the problem:
Let $X,Ysim N(0,1)$ iid, let $S=pm1$ be independent of $X,Y$ and let $$A=pmatrix{X&Y\Y&S X}.$$  Note that $SXsim N(0,1)$, and  $(X,Y)sim N(0,I_2)$,
$(Y,SX)sim N(0,I_2)$ but $(X,SX)nsim N(0,I_2)$.
With the current, changed version of the problem:  There are rvs $X,Z$ which are marginally $N(0,1)$ but not jointly Gaussian, whose joint density function is positive everywhere.
(For example, let $f(x,z)=3phi(x,z)/2$ in the  1st and 3d quadrants and $f(x,z)=phi(x,z)/2$ in the 2nd and 4th quadrants, where $phi$ is the density of $N(0,I_2)$.
Now construct $$A=pmatrix{X&Y_1\Y_2&Z},$$ where $Y_1,Y_2$ are independent $N(0,1)$ rv.s

Is a column-wise and row-wise Gaussian matrix, jointly Gaussian?

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