If $X(t)$ is a $2\times 2$ fundamental matrix for $X'=AX$ and $W(t)=\det(X(t))$, where $W(t)$ denotes the Wronskian, show that $$W(t)=W(t_0)\exp\left(\int_{t_0}^{t} \text{tr}(\underline{A}(s)) \ ds\right).$$
This is a specific case of my previous post Explanation on the Proof of $W(t)=W(t_0)\exp\left(\int_{t_0}^{t} \text{tr}(\underline{A}(s)) \ ds\right)$. Although the steps should be similar, I'm unsure of how to begin.
For instance, in the general case we approximated $X(t)$ using the Taylor expansion about $t_0$, $$X(t)=X(t_0)+X'(t_0)(t-t_0)+\mathcal{O}\big((t-t_0)^2\big).$$ Now, if $X(t)$ is now a $2\times 2$ matrix, can this be approximated using the following Taylor expansion about $t_0$, $$X(t)=X(t_0)+X'(t_0)(t-t_0)+\frac{X''(t_0)}{2}(t-t_0)^2?$$ I'm a bit unsure of how the proof for the $2\times 2$ case differs from the $n\times n$ case.