I have an initial value problem (henceforth IVP) as follows: $$\frac{d \Phi(t)}{dt}= A(t)\Phi(t)$$ subject to the initial condition $\Phi(t_0)=I$, where $\Phi(t), A(t), I$ are square matrices of same order and $t_0$ is the initial time. $I$ is the identity matrix.
Runge-Kutta method says one require the initial condition to be a vector. In my case it is a identity matrix $I$.
So $(a)$ is it 4th order Runge-Kutta method applicable here $?$ $(b)$ Is there any other way (Numerical method/Scheme)to solving the given IVP $?$
$\bullet$ Thanks in advance!
Split your $\Phi$ and $I$ into columns, so \begin{align} \Phi &= \left [ \phi_1, \phi_2, \ldots, \phi_n\right ] \\ I &= \left [e_1, e_2, \ldots, e_n \right ] \end{align} where $\phi_i, e_i \in \mathbb R^n,\ \forall i$.
So, instead one matrix system you'll get $n$ linear systems $$ \frac {d \phi_i}{dt} = A(t) \phi_i(t) \\ \phi_i(t_0) = e_i $$ for which you can apply regular RK4.
You also can do it simultaneously, since matrix multiplication is row-to-column based, i.e. $$ A(t) \Phi (t) = \left [ A \phi_1(t), A \phi_2(t), \ldots, A \phi_n(t)\right ] $$