# 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

which seems to be related with how `NDSolveValue` is interpreting `coeff`

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?