Solving an equation matrix with NDSolve

Imagine I want to solve the following matrix system of equations
$$
begin{pmatrix}
dot{x}_{11} & dot{x}_{12} & dot{x}_{13}\
dot{x}_{21} & dot{x}_{22} & dot{x}_{23}\
dot{x}_{31} & dot{x}_{32} & dot{x}_{33}
end{pmatrix}=
begin{pmatrix}
a_{11}x_{11} & a_{12}x_{12} & a_{13}x_{13}\
a_{21}x_{21} & a_{22}x_{22} & a_{23}x_{23}\
a_{31}x_{31} & a_{32}x_{32} & a_{33}x_{33}
end{pmatrix}
$$

where $x_{ij}equiv x_{ij}(t)$ and ${a_{ij}}$ are some real coefficients. When ${a_{ij}}=1$, we can solve this with NDSolveValue as follows

ini = RandomReal(1, {3, 3});
sol = NDSolveValue({
   x'(t) == x(t),
   x(0) == ini
   }, x, {t, 0, 1})

for some random initial conditions ini (also a matrix). Now, how can I include custom coefficients? Specifically, what if the coefficient matrix is such that the diagonal is zero and all the other entries are 1? That is,

$$
begin{pmatrix}
dot{x}_{11} & dot{x}_{12} & dot{x}_{13}\
dot{x}_{21} & dot{x}_{22} & dot{x}_{23}\
dot{x}_{31} & dot{x}_{32} & dot{x}_{33}
end{pmatrix}=
begin{pmatrix}
0 & x_{12} & x_{13}\
x_{21} & 0 & x_{23}\
x_{31} & x_{32} & 0
end{pmatrix}
$$

I tried setting

coeff = ConstantArray(1, {3, 3}) - IdentityMatrix(3)

followed simply by

ini = RandomReal(1, {3, 3});
sol = NDSolveValue({
   x'(t) == coeff * x(t),
   x(0) == ini
   }, x, {t, 0, 1})

However this doesn’t seem to work and I get the error message

enter image description here

which seems to be related with how NDSolveValue is interpreting coeff

enter image description here

I’ve noticed that, although x(t) within the NDSolve environment can be tretaed as a list or a list of lists, it fails its interpretation in some cases. Any ideas?