Given deterministic functions of one variable $f(t)$ and $g(t)$, the stochastic differential equation $$ dX= f(t) X dt + g(t) X dW $$ has the following solution: $$ X(t) = \exp \left \{ \int_0^t f(s) - \frac{1}{2}g(s)^2ds + \int_0^t g(s) dW \right \}, \ \ \ t > 0. $$ I would like to know if this solution can be generalized to the following stochastic differential equation: $$ dX = F(t) X dt + G X dW, $$ where $X \in \mathbb{C}^n$ and $F(t), G \in \mathbb{C}^{n \times n}$ and importantly $dW$ is still 1-dimensional.
Would the solution be simply $$ \tag{1} X(t) = \exp \left \{ \int_0^t F(s) - \frac{1}{2}G^2ds + \int_0^t G dW \right \} \ ? $$ If not, would it be possible to somehow generalize a $1$-d solution to $n$-d solution given that $dW$ is still $1$-d?
Edit:
As pointed out in the comments, the LHS of (1) is a vector, and the RHS of (1) is a matrix. A naive fix to this issue could be treating the RHS of (1) as an "evolution" operator applied to some initial condition vector $X_0 \in \mathbb{C}^n$ like so:
$$
\tag{2} X(t) = \exp \left \{ \int_0^t F(s) - \frac{1}{2}G^2ds + \int_0^t G dW \right \} X_0.
$$