Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system $$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$ We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. We set for all $t \in \mathbb { R } , W ( t ) = \operatorname { det } ( E ( t ) )$. Show that for all real $t , W ^ { \prime } ( t ) = \operatorname { tr } ( A ( t ) ) W ( t )$.
Let $A$ be a continuous function, periodic of period $T > 0$. We are interested in the homogeneous differential system
$$\forall t \in \mathbb { R } , \quad X ^ { \prime } ( t ) = A ( t ) X ( t ) \tag{IV.1}$$
We fix $t _ { 0 } \in \mathbb { R }$ and set $E ( t ) = [ U ( t ) , V ( t ) ]$ where $U, V$ are the solutions of (IV.1) with $U(t_0) = \binom{1}{0}$, $V(t_0) = \binom{0}{1}$. We set for all $t \in \mathbb { R } , W ( t ) = \operatorname { det } ( E ( t ) )$. Show that for all real $t , W ^ { \prime } ( t ) = \operatorname { tr } ( A ( t ) ) W ( t )$.